Physics-Lasers and Relativity and Superconductor and Quantum

7/31/2019 Physics-Lasers and Relativity and Superconductor and Quantum

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Chapter-1

LASERThe word LASER is an acronym for "Light Amplification by Stimulated Emission of Radiation". As the name suggests, in lasers, the light is amplified with the help of aprocess called "Stimulated emission" Thus, laser is based on the principle of stimulatedemission.

In other words, it is a device to produce a monochromatic beam, highly intense andcoherent beam of light. The theoretical basis for the development of laser was providedgreat scientist Albert Einstein in 1917, when he predicted the possibility of stimulatedemission of radiations.

1. STIMULATED ABSORPTION

When a photon of light having energy E2 – E1 = hυ is incident on an atom in the groundstate, the atom in the ground state E1 may absorb the photon and jump to higher energystate E2. This process is called stimulated absorption or induced absorption. This is calledso because the incident photon has stimulated the atom to absorb the energy.

Figure 1. Stimulated absorption

2. SPONTANEOUS EMISSION

If the atom in the excited state automatically decays to the ground state by emitting a

photon of energy (E2 – E1 = hυ), then this process is called spontaneous emission.Generally an atom/electron in excited state can stay for 10-9 – 10-8 seconds.



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Figure 2. Spontaneous Emission

Characteristics of spontaneous emission:

a) The emitted photon of energy E2 – E1 = hυ can move in any random direction.

b) There will not be any phase relationship between the photons emitted from various

atoms.Hence, the radiations coming out due to spontaneous emissions are incoherent.

3. STIMULATED EMISSION

If the atom is in the excited state E2 and a photon of energy exactly equal to E2 – E1 = hυ is incident on it, then the incident photon interacts with the atom in the excited state andthen it stimulates or induces the atom to come down to the ground state E1. A freshphoton is emitted in this process. Therefore, when an atom ejects a photon due to its

interaction with a photon incident on it, the process is called stimulated emission (orinduced emission).

Figure 3. Stimulated emission

Characteristics of stimulated emission:a) For each incident photon, there are two outgoing photons moving in the same

direction.b) As the emitted photon has exactly the same energy, phase and direction as the

incident photon, we will achieve an amplified and unidirectional coherent beam.



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Note: The laser is based on the principle of stimulated emission.

4. POPULATION INVERSION IN LASERS

Normally the population density of atoms/electrons is more in the ground state than the

excited state. But if the process of stimulated emission dominates over the process of spontaneous emission, then it may be possible that N2 > N1.

Where N1 is the number of atoms in the ground state andN2 is the number of atoms in the excited state.

The process of achieving greater population density of atoms in the higher energy state ascompared to lower energy state is called population inversion. The atoms from lowerenergy states are raised to excited states by external energy.

Figure 4. Population inversion

5. METASTABLE STATE

It is also the excited state but having life time of 10-5 to 10-3 seconds. Populationinversion occurs only in between the metastable state and lower state. The two states inbetween the population inversion occurs and laser is achieved, the upper one is known asupper laser level (ULL) and lower is known as lower laser level (LLL).

Note: Two levels lasers cannot be constructed. Can you explain why? Atleast three levelsare required to produce laser.

6. TYPES OF LASERS

The lasers can be classified by various ways, as explained below:a) State of laser medium: According to the state of laser medium, we have solid

state lasers like ruby laser, gas lasers like He-Ne laser and liquid laser like dyelaser.

b) Mechanism of pumping: According to it, we have optical pumping based laserslike ruby laser, electric discharge based lasers like He-Ne laser. Pumping can alsobe done through chemical reaction.



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c) Nature of output: According to nature of output, we have pulsed lasers like rubylaser and continuous wave lasers like He-Ne laser.

d) Spectral region (wavelength) of output: According to it, we have ultra violet,visible and infrared lasers.

Note: Do you know what is pumping or how atoms/electrons are excited?

7. COMPONENTS OF A LASER

Three main components of laser device are:a) Active medium:b) Pumping sourcec) Optical resonator system

7.1 Active medium of laser

When energy is given to laser medium (solid, liquid or gas), then only a small fraction of laser medium shows lasing action. This part of laser medium is called active medium oractive centre. Thus, due to this reason, the laser medium is also called the heart of alaser. For example, in case of ruby laser, Al2O3 is doped with Cr2O3. The laser is due todoped chromium ions. Thus Cr3+ ions are active centres. In He-Ne laser, laser is produceddue to Ne atoms, therefore, Ne atoms are active centres.

7.2 Pumping Source

As we have discussed in the previously, that principle of laser is stimulated emission andfor it to take place, population inversion has to be achieved and maintained. For this,there must be a source of external energy, which can continuously supply energy toenergy to laser medium, so that population inversion can be achieved. Such a source of external energy is called pump or pumping source and the process of supplying externalenergy to laser medium so as to achieve the population inversion is called pumping.

Types of pumping source

Depending upon the type of laser, the most commonly used pumping methods are listed

below:

a) Optical pumping: In this, the population inversion is achieved by means of lightenergy delivered from appropriate pumping source such as gaseous discharge orflash tubes. For example, in ruby laser, xenon flash tube is used.

b) Electric discharge: pumping: this type of pumping accomplished by means of intense electrical discharge in the medium and is particularly suited to gas medialike He-Ne laser and CO2 laser. The electric discharge coverts the gas into a



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plasma where active centers collide inelastically with free electrons andpopulation inversion is achieved.

c) Chemical pumping: It raises active centers into the higher levels by means of suitable exothermal chemical reactions in the active medium.

d) Heat pumping: In this type of pumping, the active material is first brought to a

high temperature then rapidly cooled down.

7.3 Optical Resonator System

An optical resonator is a system or set up, which is used to obtain amplification of stimulated photons by oscillating them back and forth between system of two mirrors.Thus, it consists of two plane or concave mirrors. One of the mirrors is partiallyreflecting (having reflectivity less than 100%) and other is totally reflecting (havingreflectivity 100%). Laser output is received from partially reflecting mirror. The space

between the two mirrors is called cavity.

Figure 5. working of optical resonator system

8. TYPES OF LASERS

The lasers can be classified by various ways, as explained below:



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a) State of laser medium: According to state of laser medium, we have solid state laserslike ruby laser, gas lasers like helium-neon laser.

b) Mechanism of pumping: According to pumping mechanism, we have flash light oroptical pumping based lasers like ruby laser, electric discharge based laser like He-Ne

laser.c) Nature of output: According to nature of output, we have pulsed lasers like ruby laserand continuous wave lasers like he-Ne laser.

d) Wavelength of output: According to wavelength of lasers, we have ultra violet, visibleand infrared lasers.

9. PROPERTIES OF LASER

A laser beam has following important characteristics:

• Divergence or directionality: laser is highly directional beam.• monochromaticity: laser is monochromatic that is it has single wavelength• brightness• coherence:

9.1 Laser Coherence

When a laser beam is emitted from a source, there may be phase difference between thebeams at different points that is it may be possible that there may be some time gapbetween the two atoms to come from upper energy level to lower energy level. The term

coherence refers to the degree of co-relation between the phases.

In other words, two or more light waves are said to be coherent if they bear a constantphase relation among themselves.

Coherence can be classified into two ways:

a) Temporal coherence: consider a light wave traveling along +X axis. Consider twodifferent points A and B along the same wave train that is along +X axis.

Figure 6. Temporal coherence

If Φ(A) is phase of point A at any time and Φ(B) is phase of point B at any time, then



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phase difference between these point is given by

Φ = Φ(A) - Φ(B)

If Φ is independent of time then points A and B are said to exhibit temporal coherence or

longitudinal coherenceIn other words, a beam of laser is said to exhibit temporal coherence if the phasedifference of the waves crossing the two points lying on a plane parallel to the directionof the propagation of beam is independent of time.

b) Spatial coherence: consider a light wave traveling along +X axis. Draw a lineperpendicular to the direction of the beam. Consider two different points C and D on thisline.

If Φ(C) is phase of point C at any time and Φ(D) is phase of point D at any time, then

phase difference between these point is given by

Φ = Φ(C) - Φ(D)

Figure 7. Spatial Coherence

If Φ is independent of time then points C and D are said to exhibit spatial coherence ortransverse coherence or lateral coherence.



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In other words, a beam of laser is said to exhibit spatial coherence if the phase differenceof the waves crossing the two points lying on a plane perpendicular to the direction of thepropagation of beam is independent of time.

10. THREE LEVEL LASER SYSTEM

In the three level laser system, the atoms are pumped from the ground state E1 to higherstate E3 with the help of pumping source. The E3 is called pumping state. The life timeof atoms is least in the energy level E3. It means E3 is unstable stable state and hereatoms stay for 10-9 (ten power -9) to 10-8 seconds.

Atoms make transition from level E3 to E2. Energy level E2 is the metastable state(having life time 10-5 to 10-3 seconds). Thus, population of atoms become more in theenergy state E2 as compared to E1.

Figure 8 Three level laser system

The stimulated emission occurs between E2 and E1 producing laser. E2 is known asupper laser level (ULL) and E1 is known as lower laser level (E1).

The example of three level laser is ruby laser.

10.1 Drawback of three level laser system

In the three level laser system, the terminal level is ground level and hence more than half of the atoms are to be transferred to level E2. This requires more pumping power. If thedifference of atoms in two levels is small, the power required is also small. But thepresence of large number of atoms in level E2 give rise to large number of spontaneous



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radiationless transitions. This energy is usually carried by lattice photons due to whichthe efficiency of three level laser system is less.

11. FOUR LEVEL LASER SYSTEM

In the four level laser system, the atoms from ground state E1 are raised to excited stateE4 with the help of pumping. From the E4, the atoms decay to energy state E3 byspontaneous emission. The transition rate of atoms from E4 to E3 is much faster ascompared to transition rate of atoms from E3 to E2. This is due to the reason that E4 is anexcited state with life time of the atoms of the order of 10-8 seconds.

Figure 9. Four level laser system

E3 is the metastable state. Thus the number of atoms in E3 exceeds the number of atomsin E2. The population inversion is achieved between E3 and E2. The laser action takesplace between E3 and E2 by stimulated emission. The atoms from energy state E2 getdeexcited to E1. The atoms from E1 are again pumped to E2.

Example: Helium-Neon laser is four level laser system.

11.1 Advantage of four level laser:

The rate of relaxation of the atoms from E2 to E1 should be faster than the rate of arrivalof atoms from E4 or E2.

This is required for better efficiency of laser system. Thus, four level laser is better thanthree level laser.

12. RUBY LASER



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The first laser to be operated successfully was ruby laser. First demonstration of laseraction using ruby crystal was given by T.H. Maiman in 1960. It is a solid state laser.

12.1 Construction

Figure 10a. Construction of Ruby laser

Ruby is a crystal of aluminium oxide (Al2O3) in which some of the aluminium ions(Al3+) are replaced by chromium ions (Cr3+). This is done by doping small amounts of

chromium oxide (Cr2O3) in the melt of purified Al2O3.

These chromium ions give the crystal a pink or red color depending upon theconcentration of chromium ions. Laser rods are prepared from a single crystal of pinkruby which contains 0.05% (by weight) chromium. Al2O3 does not participate in the laseraction. It only acts as the host.

The ruby crystal is in the form of cylinder. Length of ruby crystal is usually 2 cm to 30cm and diameter 0.5 cm to 2 cm. As very high temperature is produced during theoperation of the laser, the rod is surrounded by liquid nitrogen to cool the apparatus.

Active medium or active center: Chromium ions act as active centers in ruby crystal. Soit is the chromium ions that produce the laser.

Pumping source: A helical flash lamp filled with xenon is used as a pumping source.The ruby crystal is placed inside a xenon flash lamp. Thus, optical pumping is used toachieve population inversion in ruby laser.



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Optical resonator system: The ends of ruby crystal are polished, grounded and madeflat. The one of the ends is completely silvered while the other one is partially silvered toget the output. Thus the two polished ends act as optical resonator system.

12.2 Working

Ruby is a three level laser system. Suppose there are three levels E1, E2 and (E3 & E4).E1 is the ground level, E2 is the metastable level, E3 and E4 are the bands. E3 & E4 areconsidered as only one level because they are very closed to each other

Pumping: The ruby crystal is placed inside a xenon flash lamp and the flash lamp isconnected to a capacitor which discharges a few thousand joules of energy in a fewmilliseconds. A part of this energy is absorbed by chromium ions in the ground state.Thus optical pumping raises the chromium ions to energy levels inside the bands E3 andE4. This process is called stimulated absorption. The transition to bands E3 and E4 arecaused by absorption of radiations corresponding to wavelengths approximately 6600

angstroms and 4000 angstroms respectively. The levels inside the bands E3 and E4 arealso known as pumping levels.

Figure 10b: Energy level diagram of Ruby laser

Achievement of population inversion: Cr3+ ions in the excited state loose a part of theirenergy during interaction with crystal lattice and decay to the metastable state E2. Thus,the transition from excited states to metastable state is non-radiative transition or in otherwords there is no emission of photons. As E2 is a metastable state, so chromium ions will

stay there for longer time. Hence, the number of chromium ions goes on increasing in E2state, while due to pumping , the number in the ground state E1 goes on decreasing. As aresult, the number of chromium ions become more in excited state(metastable state) ascompared to ground state E1. Hence, the population inversion is achieved between statesE2 and E1.

Achievement of laser: Few of the chromium ions will come back from E2 to E1 by theprocess of spontaneous emission by emitting photons. The wavelength of a photon is



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6943 Å. This photon travels through the ruby rod and if it is moving in a directionparallel to the axis of the crystal, then it is reflected to and fro by the silvered ends of theruby rod until it stimulates the other excited ions and cause it to emit a fresh photon inphase with the stimulating photon. Thus, the reflections will result in stimulated emissionand it will result in the amplification of the stimulated emitting photons. This stimulated

emission is the laser transition.The two stimulated emitted photons will knock out more photons by stimulating thechromium ions and their total number will be four and so on. This process is repeatedagain and again, thus photons multiply. When the photon beam become sufficientlyintense, then a very powerful and narrow beam of red light of wavelength 6943 Åemerges through the partially silvered end of the ruby crystal.

In the energy level diagram, E2 is the upper laser level and E1 is the lower laser levelbecause laser beam is achieved in between these levels. Thus, the ruby laser fits into thedefinition of three level laser system.

Output: The output wavelength of ruby laser is 6943 Å and output power is 10 raise topower 4 to 10 raise to power 6 watts and it is in the form of pulses.

12.3 Spiking in Ruby laser:

As we have discussed in working of ruby laser that the terminus of laser action is theground state E1 in ruby laser. Therefore it is difficult to maintain the populationinversion. This will result in the depletion of upper laser level E2 population (due tostimulated emission) more rapidly than it can be restored by the flash light that is opticalpumping source. The laser emission is made up of spikes of high intensity emissions.

This phenomenon is called spiking of the laser.After the depletion of E2 state, the laser action ceases for a few microseconds. Since theflash lamp is still active, it again pumps the ground state chromium ions to upper leveland again laser action begins. A series of such pulses is produced until the intensity of theflash light has fallen to such a level that it can no linger rebuild the necessary populationinversion. So the output laser will be in the form of pulse in ruby laser or in other words,it will not be continuous.

12.4 Drawbacks of ruby laser

1. As the terminus of laser action is the ground state, it is difficult to maintain thepopulation inversion. This fact results in ruby laser’s low efficiency.2. The ruby laser requires high power pumping source.3. The laser output is not continuous but occurs in the form of pulses of microsecondduration.4. The defects due to crystalline imperfection are also present in ruby laser.

12.5 Uses of ruby laser



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1. Ruby laser has very high output power of the order of 104 – 106 watts. It haswavelength of 6943 Angstroms.2. Ruby lasers are used for holography, industrial cutting and welding.

13. Nd: YAG LASER

Nd: YAG is a solid state laser four level laser. Nd stands for neodymium and YAG forYttrium aluminium garnet (Y3Al5O12). It is developed by J.E. Geusic, H.M. Marcos andL.G. Van Vitert in 1964. The rod of Y3Al5O12 is doped 1% with triply ionizedneodymium. Nd3+ ions will replace the Y3+ ions in the crystal. Maximum length of therod is about 10 cm and diameter is 6-9 cm.

13.1 Construction:

Active medium: Nd3+

ions act as active medium or active centers. YAG is just the host.

Pumping source: The pumping of Nd3+ ions to upper levels is done by krypton arc lamp.Xenon lamp can also be used as pumping source. Thus, the optical pumping is used toachieve population inversion.

Optical resonator system: The ends of the Nd:YAG rod are polished and silvered so asto act as the optical resonator system.

Nd:YAG is a four level laser system. The pumping of neodynium (Nd3+) ions to upperstate (E4) is done using krypton arc lamp. Thus optical pumping is used in this laser. The

wavelength of light of wavelength 7200 Å to 8000 Å excites the ground state (E1) Nd3+ions to E4 states. From E4 states, they make a non-radiative transition and come to E3state. E3 is the metastable state so population inversion is achieved between the levels E3and E2.

Note: Try to make construction diagram yourself. It is same as ruby laser.

13.2 Working

After this, the process of stimulated emission will occur. (I have already explained theprocess of stimulated emission in detail in my previous articles).Thus, the laser emission

will occur in between the levels E3 and E2 with the process of stimulated emission. SoE3 is the upper laser level and E2 is the lower laser level. Then Nd3+ ions come back tothe ground state E1. Laser emission will have wavelength of 10600 Å so occur in theinfrared range of spectrum.

Note: Diagram is same as ruby laser.

13.3 Nd YAG laser has following applications or uses or advantages:



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1. They produce continuous laser at room temperature

2. They can be used as portable systems since the rods are small.3. they have surgical applications.4. they are used material processing such as drilling, spot welding and marking.

5.

They are used as pumping tunable visible light lasers.6. They have applications in military such as including range finders and targetdesignators.

7. Research applications such as Raman spectroscopy, remote sensing, massspectrometry.

14. HELIUM- NEON LASER

The helium-neon laser was the first gas laser to be operated successfully. It wasfabricated by Ali Javan and his co-workers in Bell Telephone Laboratories in the USA in1961.

Helium neon laser used a mixture of 10:1 for its active medium. It consists of a long andnarrow discharge tube of diameter of about 1 cm and about 80 cm long. The mixture is ata pressure of about 1 mm of Hg, the partial pressure of helium gas being 5 to 10 timesthat of neon.

14.1 Construction

Figure 11. Construction He-Ne laser



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Therefore, the electric discharge through the gas mixture continuously populates the Neexcited levels E4 and E6. This helps to create a state of population inversion between thelevels E4 (or E6) and lower energy levels E5 and E3. Therefore the purpose of He atomsis to help in achieving a population inversion in the Ne atoms.

Achievement of laser: The following three transitions will occur:E6 to E5 with laser wavelength of 3.39 µm or 33900 Angstroms.E6 to E3 with laser wavelength of 6328 Angstroms.E4 to E3 with laser wavelength of 1.15 µm or 11500 Angstroms.

The wavelengths of 3.39 µm and 1.15 µm corresponds to infrared region and wavelength6328 Angstroms corresponds to red light wavelength (visible region).Mirrors of the optical resonators are so designed to show low reflectivity for wavelengths3.39 µm and 1.15 µm. Thus photons of these wavelengths will be eliminated. Therefore,the photons of wavelengths 6328 Angstroms will move back and forth in optical

resonator system and thus laser of wavelength 6328 Angstroms emerges through thepartially reflected mirror.The excited Ne atoms drop down from levels E3 to E2 through spontaneous emission andthis process will emit a photon of wavelength 0.6 µm. As the level E2 is also Metastable,there is a probability of excitation of Ne atoms from E2 to E3 leading to quenching of thepopulation inversion. To eliminate quenching, the narrow discharge tube is used becauseNe atoms de-excited to level e1 from E2 through collisions with the walls of the tube.

Output: The helium neon laser output is continuous and it lies between 1mW and 50mW for input of about 5-10 W.

Disadvantage: As internal mirrors are used in He-Ne laser to act as optical resonators,but these mirrors are usually eroded by the gas discharge and have to be replaced.

This problem is eliminated by placing two windows at the ends of the discharge tube atthe Brewster angle (Figure 13). For such windows, the light which is polarised in theplane of figure passes through without suffering any reflection loss while light withperpendicular polarisation suffers reflection. Due to use of these windows, the outputlaser beam is polarised.



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Figure 13

Applications: He-Ne lasers are used in research, spectroscopy, holography,communications and weaponry.

15. WHY HELIUM-NEON LASER IS BETTER THAN RUBY LASER

OR WHY FOUR LEVEL LASER IS BETTER THAN THREE LEVEL

LASER

Helium neon laser is better than ruby laser or most of the four level lasers are better than

three level lasers due to the following reasons:

1. The laser output is continuous in the case of helium-neon laser. But it is in theform of pulse in the ruby laser.

2. Ruby laser requires high power pumping source, whereas Helium-neon laserrequires low power pumping source like electric discharge.

3. Efficiency of helium-neon laser is more than ruby laser.4. The defects due to crystalline imperfections are also present in the ruby laser. But

it is not so in the helium-neon laser.

16. CARBONDIOXIDE LASER

Carbondioxide is a four level molecular laser. The CO2 molecules consist of two oxygenatoms and a carbon atom between them. These molecules undergo three different types of vibrational oscillations. These three vibrational configurations are called vibrationalmodes.The three vibrational modes are:



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i) Symmetric stretching mode: In this mode, the oxygen atoms oscillate along the axis of the molecule simultaneously departing or approaching the carbon atom which isstationary.ii) Asymmetric stretching mode: In this mode, all the three atoms oscillate but oxygenatoms move in one direction and carbon atom moves in the opposite direction.

iii) Bending mode: In bending mode, atoms move perpendicular to the molecular axis.As a result of vibrational modes, CO2 molecule is characterized not only byelectronic levels but also by vibrational and rotational levels. Each electronic level is splitinto various vibrational sublevels and each vibrational level is further splitted intorotational sublevels. The energy difference between various electronic levels correspondsto visible and ultraviolet region. The energy difference between various vibrational levelscorresponds to the infrared region while the energy difference between various rotationallevels corresponds to far infrared region of the spectrum.

16.1 Construction

Carbon dioxide laser consists of a discharge tube having a diameter of 2.5cm and a lengthof about 5m. The discharge tube is filled with a mixture of carbon dioxide, nitrogen andhelium gases in the ratio of 1:2:3 with water vapors. Pressures maintained are about P(for He)= 7 Torr, P (for N2)= 1.2 Torr and P (for CO2 = 0.33 Torr).

Active medium and active centers= The active medium is the CO2, N2 and He in theratio of 1:2:3. The active centers are the carbon dioxide molecules because laser will beachieved due to these molecules.

Pumping source= Electric discharge method is used for pumping and achievingpopulation inversion. In this method, electrons will collide with CO2 molecules and

pump them to excited states.The purpose of N2 is to help in excitation of CO2 molecules by colliding with CO2molecules and transferring the energy to them. So N2 molecules increases the pumpingefficiency.

Optical resonator system= All the gas mixtures is enclosed between a set of mirrorswhich forms the optical resonator system. One of mirrors is completely reflecting and theother is partially reflecting.

Note: Try to make the construction diagram. It is same as He-Ne laser, just replace He-Ne with CO2, N2 and He in the ratio of 1:2:3.

16.2 Working

In the earlier articles, I have explained the modes and construction of carbon dioxidelaser. Now I will explain the working of the carbon dioxide laser.



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Pumping of nitrogen molecules: As electric discharge is used as pumping source andwhen electric discharge is passed through the mixture of CO2, N2 and He, electrons areaccelerated down the tube. These accelerated electrons collide with the N2 molecules andexcite them to higher vibrational energy levels. Let us say the N2 excited from level F1 toF2.

The level F2 happens to be Metastable and thus the N2 molecules excited to F2 spend asufficient amount of time before getting de-excited.

Figure- Energy level diagram of carbon dioxide laser

Achievement of population inversion of CO2 molecules: The excited level (Metastablelevel) of CO2 molecules corresponds approximately to the same energy as the energy of

the excited level F2 of nitrogen. Thus when N2 molecules in level F2 collide with theCO2 in the ground state E1, an energy exchange takes place and this result in theexcitation of CO2 molecules to level E5 and de-excitation of N2 to the ground level F1.

Thus population inversion is achieved between vibrational levels E5 and E4 or E5 andE3. Thus E5 is the upper laser level. E3 and E4 are lower laser levels.

Achievement of laser: The following transitions will occur:

E5 to E4 with laser wavelength of 10.6 µm.


Thus, these transitions produce lasers of wavelength 10.6 µm and 9.6 µm which lie in thefar infra-red region.

The CO2 molecules in the states E4 and E3 deexcite to state E2 through inelastic collisionwith unexcited CO2 molecules. This process is very fast so there will be accumulation of



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CO2 in this level and they can break the population inversion in upper levels becausethere is probability of excitation of molecules from E2 to E3 and/ or E4.

To stop the accumulation of CO2 molecules in E2 special additives like He and watervapors are added into the gas mixture. CO2 molecules return to the ground state E1

through collisions with the He to which it transfers the excitation energy.Other function of He is to conduct the heat away to the walls keeping CO2 cold, this isbecause helium has high thermal conductivity.

16.3 Why helium is doped in carbon dioxide laser

The CO2 molecules in the states E4 and E3 deexcite to state E2 through inelastic collisionwith unexcited CO2 molecules. This process is very fast so there will be accumulation of CO2 in this level and they can break the population inversion in upper levels becausethere is probability of excitation of molecules from E2 to E3 and/ or E4.

To stop the accumulation of CO2 molecules in E2 special additives like He and watervapors are added into the gas mixture. CO2 molecules return to the ground state E1through collisions with the He to which it transfers the excitation energy.

Other function of He is to conduct the heat away to the walls keeping CO2 cold, this isbecause helium has high thermal conductivity.

16.4 Wavelength and output of carbon dioxide laser:



Thus, these transitions produce lasers of wavelength 10.6 µm and 9.6 µm which lie in thefar infra-red region.

Output of 10KW is achieved with transition of 10.6 µm and it is in continuous wavemode. Efficiency of CO2 laser is approximately 30%. The CO2 laser is more efficientthan other gas lasers because in CO2 laser, the levels taking part in laser transitions arethe vibrational rotational levels of the lowest electronic level and as these levels are veryclose to the ground level, thus a large part of the input energy is converted into output

laser energy. Thus CO2 laser is more efficient than other gas laser.16.5 Applications of CO2 laser: CO2 laser have wide applications in industry forwelding, cutting and for hole drilling. High energy CO2 lasers are used to destroy cancertissues.



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17. SEMICONDUCTOR LASER

Before discussing the construction and working of semiconductor laser, let us discuss the

review of semiconductors as it will be necessary for you to write the basics of semiconductors in case of exams:

A semiconductor is a material whose conductivity lies between those of conductor andinsulator. Semiconductors are of two types:

a) Intrinsic semiconductors or pure semiconductors

b) Extrinsic semiconductors or doped semiconductors

Extrinsic semiconductors are further classified into two types depending upon the type of

majority carriers:

i) n-type semiconductors where electrons are majority carriers.

ii) p- type semiconductors where holes are majority carriers.

When a p-type semiconductor and a n- type semiconductor is joined by specialtechniques, there will be flow of electrons from n side to p side and flow of holes from pside to n side. After some time, an electric field will be created which will oppose thisflow and flow stops. Thus, there will be formation of depletion region. This region iscalled so because it is depleted from charge carriers.

17.1 Construction of a semiconductor laser



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Figure 14

Example of semiconductor laser: One of the examples of semiconductor lasers isgallium arsenide (GaAs). It is heavily doped semiconductor. Its n-region is formed byheavily doping with tellurium in a concentration of 3 x 1018 to 5 x 1018 atoms/cm3 while

its p-region is formed by doping with zinc in concentration around 10

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atoms/cm

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.Active medium: The active medium in GaAs is GaAS. But it is also commonly said thatdepletion region is the active medium in semiconductor laser. The thickness of thedepletion layer is usually very small (0.1 µm).Pumping Source: Forward biasing is used as pumping source. The p-n junction is madeforward biased that is p side is connected to positive terminal of the battery and n side tonegative. Under the influence of forward biased electric field, conduction electrons willbe injected from n side into junction area, while holes will enter will enter the junctionfrom the p side. Thus, there will again be recombination of holes and electrons indepletion region and thus depletion region becomes thinner.Optical resonator system: The two faces of semiconductor which are perpendicular to

junction plane make a resonant cavity. The top and bottom faces of diode, which areparallel to junction plane are metallised so as to make external connections. The front andback faces are roughned to suppress the oscillations in unwanted direction.

17.2 Working of semiconductor laser

Achievement of population inversion: When p-n junction diode is forward biased, thenthere will be injection of electrons into the conduction band along n-side and productionof more holes in valence band along p-side of the junction. Thus, there will be morenumber of electrons in conduction band comparable to valence band, so populationinversion is achieved.Therefore, when the electrons and holes are injected into the junction region fromopposite sides with forward biasing, then population inversion is achieved between levelsnear the bottom of the conduction band and empty levels near the top of the valenceband.Achievement of laser: When electrons recombine with the holes in junction region, thenthere will be release of energy in the form of photons. This release of energy in the formof photons happen only in special types of semiconductors like GaliumArsenide (GaAs).Otherwise in semiconductors like silicon and germanium, whenever holes and electronsrecombine, energy is released in the form of heat, thus Si and Ge can not be used for theproduction of laser.



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Figure 15. No-biasing

Figure 16. Forward biasing

The spontaneously emitted photon during recombination in the junction region of GaAs

will trigger laser action near the junction diode. The photons emitted have a wavelengthfrom 8200 Å to 9000 Å in the infrared region.

Output of semiconductor laser: The output powers of about 10mW are achieved incontinuous wave operation and in pulsed opration the peak power runs to 100W. It hasoutput wavelength from 8200 to 9000 angstroms. So the wavelength of semiconductorlaser lies in the infrared region.



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Advantages/Applications: Semiconductor lasers are compact and have efficiency of about 50-60%. They can be used as sources for light wave communication systems. Theyexpected to find applications in optical radar equipment and space communications.

Disadvantages. It is difficult to control the mode pattern and mode structure of the

semiconductor laser action due to small size of laser region that is junction region.As in semiconductor, the laser emission occurs between two bands of energies instead of two well defined energy levels (like in He-Ne laser), thus the laser emission is not asmonochromatic as that from a gas laser.

18. DYE LASER

Dye lasers use liquid organic dyes. These organic dyes are dissolved in solvents likewater, ethyl alcohol, methanol. The most widely used dye is rhodamine-6G, also knownas Xanthene dye. These lasers are discovered by Sorokin and his colleagues. Dye lasers

operate without the intervening metastable state.

18.1 Construction

Figure 17

The dye laser consisted of a 1cm long quartz glass tube filled with solutions of organicdyes such as rhodamine- 6G.Active Medium. Rhodamine-6G dissolved in a suitable liquid like water, ethyl alcohol ormethanol is used as the active medium. The dye solution used in the dye lasers typicallyhas a concentration in the range of 10-2 to 10-4 M. Rhodamine-6G emits in yellow-redregion.

Pumping source: Energy to excite the dye is supplied by a strong light source that maybe a flash lamp or another laser like N2 laser or argon-ion laser. Thus, optical pumping isused to excite the dye and to achieve population inversion.

Optical resonator System. The most useful feature of dye lasers is their tunability. Thetunability means that the lasing wavelength for a dye may be varied over a wide range.Due to this reason, dye lasers are also called tunable lasers. Tuning over 500 angstromhas been obtained.



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One of the teachniques to obtain tuning is to replace one of the mirrors of the resonantcavity with a diffraction grating. Thus a dye cell is usually placed inside a cavityconsisting of a partially reflective mirror onthe front and a diffraction grating on the rear. A source light is focused onto the dye toexcite it and stimulate laser action.

By rotating the diffraction grating , wavelength of laser output can be altered. Thustuning is obtained. Therefore, this combination of partially reflective mirror anddiffraction grating will act as optical resonator system. For radiation to be reflected backalong the laser cavity axis, the angle θ that the normal to the diffraction grating makeswith the cavity must satisfy the condition.

2dsinθ =nλ (n = 1, 2, 3, ...)

Where d is grating spacing

λ is wavelength of radiation.By rotating the grating, angle θ will be changed and thus the output wavelength will bechanged that is tuning of output wavelength will be achieved.

18.2 Working of dye laser

The molecules have singlet as well as triplet states. Each electronic state comprises of several vibrational states and each vibrational state comprises of several rotational levels.

Due to absorption of light from pumping source, dye molecules get excited from the

ground state E1 to upper vibrational rotational levels of excited state E2 which is upperlaser level. Most of the dye molecules decay to the lowest vibrational level L of E2 in atime of about 10-11 seconds. This process is due to thermal redistribution in level E2, thus,it is a non-radiative process. Population inversion is achieved at level L.

From level L of E2, the dye molecules decay to any higher lying vibrational sublevel of E1. When this process occurs, then the radiation is emitted. This is termed asfluorescence. The life time of level L for dye molecules is about 10-9 seconds. As mostof the molecules decay from level E2 by fluorescence, thus laser action occurs at thefluorescence wavelength. Thus laser output is achieved in between states E2 and E1.

Molecules from the state E2 can also make a non-radiative transition to the triplet levelT1. This transition is known as intersystem crossing. This process of intersystemcrossing can limit the laser action because it will lead to reduction of the population of E2which is upper laser level and thus there will be accumulation of molecules in state T1.As the transition T1 to T2 is allowed, and the wavelength corresponding to absorptionspectrum of T1 to T2 usually overlaps the emission spectrum of E2 to E1. Thus,intersystem crossing will lead to reduction of number of molecules in upper laser level



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E2 and it will reduce the laser gain or laser output. Sometimes, it may even prevent laseroscillation.

Thus for good laser action, the number of molecules in state E2 should reach thethreshold level before a significant number of molecules have dropped to level T1.

Therefore, it requires very intense and rapid pumping to maintain population inversion.Addition of oxygen to solution can also reduce the life time T1. Thus oxygen acts as atriplet quenching additive.

I have already discussed the construction and working of dye laser. Let us discuss theoutput and applications of dye lasers.

Output: The dye laser provides 3nsec pulses in the spectral range of 360 nm to 950 nm.The typical peak powers are on the order of about 10kW to 20kW. Dye lasers can beoperated in both pulsed and continue wave (CW) modes. If a flash lamp is used to pump

the dye laser, the output will be pulsed one whereas if the laser is pumped by acontinuous wave laser like argon-ion laser, the dye laser will also be continuous.

CW dye lasers produce emission with linewidth in the range of 20 to 40 GHz.

Applications: Organic dye lasers are used in spectroscopy, holography and in biomedicalapplications. A recent application of dye laser is its use in isotope separation.

19. APPLICATIONS OF LASERS

Lasers have applications in almost every field like medicine, industry, communication

and science and technology. These applications are due to the directional, coherent andmonochromatic properties of lasers.a) Holography: Holography is a technique to record the complete picture of an object,that is it will produce the three dimensional picture. The process of holography will bediscussed in detail later on.b) Measurement of long distance: The beam spreading in the laser light is very small,laser can travel along distances, without appreciable spreading. The time taken by laserpulse to travel from laser source to a given target and back is measured. As the velocityof light is known, the distance of the target can be calculated using the relation 2d = c x twhere d is the distance of the target and c is the velocity of light.c) Applications in scientific research: Due to the coherent nature of laser light, many new

optical phenomenon have been observed using laser. Using laser light we investigate thebasic laws of interaction of atoms and molecules with electromagnetic waves.d) Application in communication: In the fibre communication system, laser beam isused. The rate at which information is transmitted is proportional to the band width of theinformation carrier signal.The bandwidth is proportional to the frequency of the carrier. Since the frequency rangeof laser signal is quite high compared to the microwaves, large bandwidth can beobtained using optical region as compared to the microwave region.



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e) Applications in Industry: Due to the high intensity of laser beam, laser can be used inwelding, cutting and in producing very high temperatures. The other advantage of laser isthat the beam can be focused onto a fine spot. The small spot size implies that highenergy densities are possible. Lasers are also found suitable for machining and drillingholes.

f) Lasers in Isotope separation: The light emerging from a laser is extremelymonochromatic. When laser light falls on a mixture of two isotopes, the laser lightexcites the atoms of only one of the isotopes thus separating it from the other isotope.In addition to the high monochromaticity, the high intensity of the laser is alsoresponsible for its application for isotope separation because with low intensity beamsthe separation rate would be too low for practical use.g) Applications in Medicine:i) Laser Surgery: The focused laser beam is capable of bloodless surgery, since the beamnot only cuts but also welds blood vessels being cut. Laser surgery is painless becauseoperations are very fast and there is not enough time for the patient to respond to theincision and sense pain.

ii) Lasers in opthamology: Lasers are used for several years to treat the detachment of retina. The beam is focused on a certain point of the retina after it has passed throughthe lens of the eye and vitreous chamber without being absorbed in them. The greenbeam of laser is strongly absorbed by the red blood cells of the retina and the consequentthermal effects leads to re-attachment of the retina. The operation is carried out by a 0.01sec pulse and being very short, is virtually painless. Other illnesses treated by the focusedlaser beam are cataract, tumors and glaucoma.iii) Laser Therapy: He-Ne laser has produced curing effect on trophic ulcers, poorlyhealing wounds, and bone fractures. Laser can also be guided through optical fiber intoblood vessels to remove the clothings, in case of heart patients, through heating. It hasalso found application in treating the decaying teeth. Laser can replace dental drills.

20. HOLOGRAPHY

The word holography originates from the Greek words ``holos'' (complete) and``graphos'' (writing). Thus, it is the technique to record the complete picture of an object.The technique was proposed by Gabor in 1947.An ordinary photograph records the two dimensional image of the picture because itrecords only the amplitude or intensity distribution. But in holography technique, both,the intensity as well as phase of the light wave is recorded.In holography, the light waves reflected from an object is recorded. These light waves

consist of intensity and phase and the record is called a hologram. The hologram hasno resemblance to the original object but it contains all the information about the objectin a optical code.The formation of hologram is done by a process called recording process. The formationof three- dimensional image from hologram is done with a process called reconstructionprocess.Thus holography consists of two processes :I Recording of hologram



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22. Q-SWITCHING

Q-switching is a technique used to produce a high output pulse. It is accomplished byusing a device to prevent the reflection of photons back and forth in the active medium.This produces a higher population inversion in the metastable state. Then suddenly the

optical cavity is opened to permit a large fraction of stored energy to be emitted in theform of very intense pulse of laser radiation. Q-switched lasers produce pulses of 10 to250 nanoseconds.Q-switching is also known as Q-spoiling.As the quality factor Q of a laser cavity shows the ability of the cavity to store energy,thus, high Q means that high energy can be stored in the cavity and a low Q means thatthe cavity will rapidly dissipate its energy. As the technique of Q-switching involvesswitching the optical cavity quality factor Q from a low to a high value, therefore,it is known as Q-switching.

Techniques of the Q-Switching :

There are four different techniques of Q-switching known as:a) Mechanical Shutters

b) The rotating reflector method

c) Electro-optical shutters

d) Passive shutter

Let us discuss them one by one:

a) Mechanical Shutters. In this case, a shutter is introduced in front of one of the mirrors in laser cavity. If theactive medium is continuously pumped keeping the shutter closed, then the population

inversion in the cavity goes on increasing and reaches a high value. Now, if the shutter isopened suddenly, the population inversion would correspond to a value much above thethreshold and the energy stored in the cavity will be released in the form of a short pulseof light with a high value of intensity. If the shutter is opened in a time much shorter thanthe time required for the building of laser oscillation, the output would consist of a giantpulse of light. If the shutter opening is slow, the output would be a series of pulseshaving smaller peak power.

Last time I have discussed the basics of Q-switching and one of its techniques known asmechanical shutters. Toady I will discuss two more following techniques of Q-switching:

b) The rotating reflector method. In this method of Q- switching, one of the end mirrors of the cavity is replaced with a

total reflection prism which spins rapidly around its axis set at right angle to theresonator axis. As the prism revolves, it faces the cavity with its reflecting side andmakes the laser cavity quality factor Q high for a short time. When the prism is out of this position, the Q value drops. As it revolves on further rotation, Q value drops tominimum.



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Figure 20.

Thus, it shows that laser action can occur only when the prism is brought into alignmentwith the laser cavity. If the rotating reflector rotates at 1000 revolutions per second, thenthe time in which the Q value of cavity switches from its maximum to minimum value isabout 10-7 seconds.

Figure 21

In place of total reflecting prism, the total reflecting mirror of optical resonator can bemade to rotate with the help of motor. The speed required for a optical resonator of length 50 cm is of the order of 30,000 revolutions per minute.

c) Passive shutter.

Passive shutters use saturable dyes.



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As a laser oscillates in large number of modes and the modes do not oscillate at the sametime and their phases have random values, thus output is in the form of irregular spikes.Now if modes are forced to oscillate with approximate same amplitude and with theirphases maintained constant, then this operation is called mode locked operation and thisprocess of obtaining ultrashort pulses of high peak power is called mode locking.

Methods of Mode locking :The most commonly used methods of mode locking are :1. Active mode locking2. Passive mode locking.Let us discuss the active mode locking:

a) Active mode locking: In this case, mode locking is achieved by forcing thelongitudinal modes to maintain fixed phase relationships. This can be done bymodulating the loss or gain of the laser cavity at a frequency equal to the intermodefrequency separation

∆υ = c/2LWher c is the speed of light and L is the length of the laser cavity.Let us discuss, how it happens :For example, a loss modulator like a polished quartz piece is placed inside the resonator.Let us assume that the loss of the laser cavity is varied at a frequency equal to theintermode separation that is ∆υ = c/2L. As the laser is switched ON, the mode that liesnearest to the line center (say υo ) would start oscillating first. As the loss is modulated atfrequency ∆υ, the amplitude of this mode would also oscillate at frequency ∆υ.

Thus, this modulation converts a mode at frequency υo to the three synchronously phasedmodes at υo + ∆υ, υo and υo – ∆υ.The modes at υ

o+ ∆υ, and υ

o– ∆υ coincide with the modes that have been adjacent to

the basic mode at υo. As a result, the oscillating field at frequencies υo + ∆υ, and υo – ∆υ forces the modes corresponding to these frequencies into oscillation and therefore, thesenew modes have a perfect phase relationship with the mode υo. The amplitudes of these modes are also modulated at frequency ∆υ and they createadditional frequencies in addition to already present frequencies.Thus all the modes are forced into oscillations in a definite phase and this will result inmode locking.

b) Passive mode locking: In passive mode locking, saturable absorbers like organic dyesare placed into the laser cavity to mode lock. These are those absorbers which absorblight at the laser wavelength but is saturated very soon and become transparent for thelight as if bleached and is, thus, also called bleachable cell.Thus, these are the dyes whose absorption decreases with increasing intensity. Let usdiscuss, how passive mode locking happens :The laser medium emits radiations due to spontaneous emission and this give rise toincoherent fluctuations in the energy density with in the cavity. Some of these radiationsor fluctuations may be amplified by the laser medium and grow in intensity to such anextent that peak power of fluctuation is transmitted by the absorber with little attenuation.



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The low power parts of fluctuations are much strongly attenuated and thus a high powerpulse can grow within the cavity.Because of this non-linear bleaching mechanism, the shortest and most intensefluctuations grow at the expense of the weaker ones.With use of such bleachable dyes with careful concentrations, with in the cavity, an initial

fluctuation can grow into a narrow pulse bouncing back and forth within the cavityproducing a periodic train of active mode locked pulse.

Chapter-2

The Theory of Relativity

1.1 The concept of Relativity and frames of reference

The term relativity comes from the word “relative that is in relation”. In this universe,everything is relative, there is nothing like absolute rest. If you think that you are sittingsomewhere and you are in the state of rest, then my dear you are wrong. You are also inthe state of motion, because if someone is able to see you from outside the earth (supposefrom moon), then you are also moving. Therefore, everything is relative.Here question arises with respect to which we can study the motion of another object?This problem is solved with the concept of frame of reference.Definition: The frame of reference is defined as any coordinate system with respect towhich one can study the motion of the another object. For example, a bus can be a frameof reference with respect to which one can study the motion of another object.1.2 Types of the frame of referencea) Inertial frame of reference:

[This you can understand only if you know the Newton’s first law or law of inertia. Sowhat is that? Remember of forgot! Let us revise it. Newton’s first law is that a body atrest will remain at rest; a body in uniform motion will remain in uniform motion, untiland unless some external force is applied on it.Here a body at rest will remain at rest, until and unless some external force is applied onit is also called law of inertia of rest.A body in uniform motion will remain in uniform motion until and unless some external

force is applied on it is also called law of inertia of uniform motion.]Thus inertial frame of reference is that frame in which the law of inertia is obeyed. Inother words, the non-accelerated frame of reference is called inertial frame of reference.b) Non - inertial frame of reference:[It should be simple for you know if you understood the inertial frame of reference].Non-inertial frame of reference is that frame of reference in which the law of inertia isnot obeyed. In other words, the accelerated frame of reference is called inertial frame of reference.



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Brain Teaser: Can you give an example of inertial frame of reference that is non-

accelerated one?Answer: The answer is nothing can be perfect inertial frame. But we consider earth as aninertial frame because acceleration of the earth is negligible.1.3 Types of theory of relativity

On the basis of this, theory of relativity is divided into two types:1. Special theory of relativity: This theory deals with inertial frames of reference.2. General theory of relativity: This theory deals with non-inertial frames of reference.

2.1 Galilean transformation equations for space and time

Figure 1Let there are two inertial frames of references S and S’. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, theyare at the same position that is Observers O and O’ coincides. After that S’ frame startsmoving with a uniform velocity v along x axis.Let an event happen at position P in the frame S’. The coordinate of the P will be x’according to the observer in S’ and it will be x according to O in S.The frame S’ has moved a distance “vt” in time t (refer figure 1).What should be the relation between x and x’. As we can see from the figure thatx = x’ + vt’But here the t = t’ thusx = x’ + vt (1)Where t and t’ are the time measured from S and S’ frames respectively.But what should be x’ = ?Yes you are right, it is x’ = x - vt (2)It can be achieved by just exchanging the sides of the equation (1).But there is more to it. It is just not by exchanging the sides.If we see equation 1, we will find that it is the position measured by O when S’ is movingwith +v velocity. But if the same thing is measured by O’ then velocity of S should be –v.(For example, when we travel in a train, then according to the outside observers, we aretravelling in x direction (suppose), but the outside objects, according to me travel in theopposite direction with the same but negative velocity).What should be the relation of y with y’?



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Figure 2

Derivation and Discussion of Michelson-Morley experiment:

Figure 3

(Special Note to students: The Michelson and Morley started the experiment in his

laboratory and they themselves tried to derive the results theoretically using certain

physics laws. If the theory and experiments results matched, then it is alright. If it

does not, then either theory is wrong or experiment.So students, now assume you are Michelson and Morley and you are doing the

theoretical calculations in your respective copies. Then we will match the result with

the experiment. Therefore, Let us start and wait what will happen?As the observer is assumed to be on ether and he is studying the motion of the earth witrespect to ether. Thus the light beam when incident on P and reflected towards M1. It willcatch the M1 at new position M’1 at B (according to Michelson-Morley). Then the lightreflected back to P at C.



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Therefore the total path difference will bex = x2 - x1x = 2 lv2 /c2 (5)Then the number of fringe shift is calculated by following relation (fringes are the patternobtained by interference of two more rays that is by constructive and destructive

interference):N = path difference/wavelength of lightN = x/ λ Put equation (5)N = 2 lv2 /c2 λ (6)Then N is calculated by putting l = 11m, v = 3 x 10 4m/s, c = 3 x 108m/s and λ = 5800angstromsThus N = 0.37 fringesBut experimentally N = 0Thus the theory and experiment results are not matched.But the experimental were right. So there was some problem in theory calculation.

Different scientists tried to explain these negative results of Michelson-Morleyexperiment.The explanation of negative results of Michelson-Morley experiment is given in separatearticle here.(The main reason lies in equation (2), when c – v and c + v is done. Can you add anyvelocity in c or subtract any velocity from c? The answer is given in another article in theexplanation of negative results of Michelson-Morley experiment.)3.2 Explanation of the negative results of the Michelson-Morley

experimentAs the theoretical results of the M-M experiment did not match with the experimentalresults (Theoretically fringe shift result was not zero, it was about 0.4 but the

experimentally it was 0). Even then the scientists of that time had conviction that etherexist and earth moves with respect to it. Thus there was lot of debate and discussionregarding this as why there were negative results (that is why the theory and experimentresult did not match?).Following are the reasons given by the scientists:1. Ether Drag Hypothesis: The then scientists assumed that ether exits and it is attachedwith the earth. Thus it will be dragged with the earth so the relative velocity of ether andearth will be zero. If this is taken then theory and experimental will get matched. But thishypothesis is discarded as there was no proof for this.2. Lorentz-Fitzgerald Hypothesis: Lorentz told that the length of the arm (distancebetween the pale and the mirror M2) towards the transmitted side should be L(√1 – v2 /c2)

but not L. If this is taken then theory and experimental will get matched. But thishypothesis is discarded as there was no proof for this.3.3 Einstein explanation of negative results of M-M experiment

(Einstein’s postulates of theory of relativity)Then Einstein gave his postulates by suggesting that there is no medium like ether. Thereare two postulates of the theory of relativity given by Einstein:1. Einstein’s First Postulate of theory of relativity:



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or vt’ = x/k – kx + kvtor t’ = x/kv – kx + kvtor t’ = kt – kx (1 – 1/k2)/v (3)Similarly from frame S, time t will bet = kt’ + kx’ (1 – 1/k2)/v (4)

(This equation can be derived by putting equation 2 in 1 and then solving.)Calculation of k:Let us suppose a flash of light is emitted from the common origin of S and S’ at timet=t’=0. From Einstein’s 2nd second postulate, the flash of light travels with the velocity of light c and which remains same in both the frames.After sometime, the position of the flash of the light as seen from observer O will bex = ctAnd as seen from O’ will bex = ct’ (Here the form of Physics law is same that is position = (velocity)(time) fromEinstein 1st postulate)Put these two values in equation (1) and (2) respectively, we get

ct’ = k (ct – vt) = kt (c –v)and ct = kt’ (c + v)Multiply above two equationsc2tt’ = k2tt’(c2 – v2)or k2 = c2 /(c2 – v2)or k2 = 1/ (1- v2 /c2) (5)or k = 1/ √(1- v2 /c2) (6)The k is known as relativistic factor.Substitute equation (6) in (1), we getx’ = (x – vt)/(√1 – v2 /c2) (7)As it is assumed that frame S’ is moving only along x direction, therefore along y and zdirectiony’ = y (8)And z’ = z (9)Equations 7-9 are known as Lorentz transformation equations for space.Let us derive Lorentz transformation equation for time:Cross-multiply equation (5)1/k2 = 1 – v2 /c2 Or 1 – 1/k2 = v2 /c2

Put the above equation in equation (3)t’ = kt – kx(v2 /c2)/vor t’ = k (t – kxv/c2)Put value of k from equation 5 in above equation, we gett’ = (t – kxv/c2)/ (√1 – v2 /c2) (10)Equation (10) is Lorentz transformation equation for time.

Equations 7 -10 are known as Lorentz transformation equations for space and time.

These are again rewritten below: x’ = (x – vt)/(√1 – v2 /c2)y’ = yz’ = z



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t’ = (t – xv/c2)/(√1 – v2 /c2)If the frame is changed (that is from S), then the equations are known as Lorentz inverse

transformation equations for space and time. These are given as:

x = (x’+ vt’)/(√1 – v2 /c2)y = y’

z = z’t = (t’ + x’v/c2)/(√1 – v2 /c2)Special case:If v <<< cThen Lorentz equations will become Galilean by neglecting v2 /c2 or v/c2 wherevernecessary as shown below:x’ = x – vty’ = yz’ = zt’ = t

5.1 Length Contraction

Figure 5Let there are two inertial frames of references S and S’. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, theyare at the same position that is Observers O and O’ coincides. After that S’ frame startsmoving with a uniform velocity v along x axis.Let an object is placed in the frame S’. The coordinate of the initial point (A) of theobject will be x1 (see the second line till A from S in figure) according to the observer inS and the coordinate of the final point will be will be x2 according to same observer.The coordinate of the initial point (A) of the object will be x’1 (see the second last linetill A from S’ in figure) according to the observer in S’ and the coordinate of the finalpoint will be will be x’2 according to same observer.Therefore the length of the object as seen by observer O’ in s’ will beL’ = x’2 – x’1 (1)The length L’ is called the proper length of the object. Proper length is defined as thelength of the object measured by the observer which is in the same frame in which theobject is placed.



44

The apparent length of the object from frame S at any time t will beL = x2 – x1 (2)Now use Lorentz transformation equations for space, that isx’1 = (x1 – vt)/(√1 – v2 /c2) (3)x’ 2= (x2– vt)/(√1 – v2 /c2) (4)

By putting equations (3) and (4) in equation (1) and solving, we getL’ = (x2 – x1)/ (√1 – v2 /c2)Substitute equation (2) in above equation,L’ = L/(√1 – v2 /c2)Or Apparent length that is the length from frame S will beL = L’(√1 – v2 /c2) (5)This is the relation of the length contraction.Why it known as length contraction, if we solve (√1 – v2 /c2), then the value will alwaysbe in decimal (except when v = c).If we multiply L’ with a decimal value, then L will be lesser than L’. This is the reasonthat it is called length contraction.

For example let us discuss a numerical: If an object is moving with speed 0.8c and itslength is 1 meter, then what will be its apparent length?Solution:Given v = 0.8cProper length L’ = 1 mTo calculate LAs L = L’(√1 – v2 /c2),If we substitute the given values, we getL = 0.6 mThus L < L’Special Case:

If v <<< c, then v2 /c2 will be negligible in L’(√1 – v2 /c2) and it can be neglectedThen equation (5) becomesL = L6.1 Time Dilation

Figure 6



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Let there are two inertial frames of references S and S1. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, theyare at the same position that is Observers O and O’ coincides. After that S’ frame startsmoving with a uniform velocity v along x axis.Let a clock is placed in the frame S’. The time coordinate of the initial time of the clock

will be t1 according to the observer in S and the time coordinate of the final tick (time )will be will be t2 according to same observer.The time coordinate of the initial time of the clock will be t’1 according to the observer inS’ and the time coordinate of the final tick (time ) will be will be t’2 according to sameobserver.Therefore the time of the object as seen by observer O’ in S’ at the position x’ will bet’ = t’2 – t’1 (1)The time t’ is called the proper time of the event. Proper time is defined as the time of anevent measured by the observer which is in the same frame in which the event isoccurred.The apparent time of the same event from frame S at the same position x will be

t = t2 – t1 (2)Now use Lorentz inverse transformation equations for time, that ist1 = (t’1 + x’v/c2) /(√1 – v2 /c2) (3)t2 = (t’2 + x’v/c2) /(√1 – v2 /c2) (4)By putting equations (3) and (4) in equation (2) and solving, we gett = (t’2 – t’1)/ (√1 – v2 /c2)Substitute equation (1) in above equation,t = t’/(√1 – v2 /c2)This is the relation of the time dilation.Why it known as time dilation, if we solve (√1 – v2 /c2), then the value will always be indecimal (except when v = c).If we divide t’ with a decimal value, then t will be more than t’. This is the reason that itis called time dilation as dilation means to get increase. Thus the clock will appear to beslower now.For example let us discuss a numerical: If an object is moving with speed 0.8c and itskeeping the time 1 second, then what will be its apparent time?Solution:Given v = 0.8cProper length t’ = 1 secTo calculate tAs t = t’/(√1 – v2 /c2)If we substitute the given values, we gett = 1.67 secThus t > t’Special Case:If v <<< c, then v2 /c2 will be negligible in t’/(√1 – v2 /c2) and it can be neglectedThen t = t’6.2 Real example of time dilationAs we have already discussed the concept of time dilation. Let us discuss its example:Decay of µ- mesons:



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µ- mesons are the particles formed in the earth atmosphere. The half life time of µ-mesons is 3.1 microseconds.They travel with the speed of 0.9c.where c is the speed of the light.So they must covered the distance d = vt

d = 3.1 x 10

-6

x 0.9c = 840mIt means there population should become half after this distance. But this does nothappen. Population remains much higher than the half value.Why this happened? This is because the time here should be dilated time and 3.1microseconds should be the proper time. This is because the µ- mesons are travellingwith speed comparable to the speed of the light.So t = t’/(1 – v2 /c2)Here t’ = 3.1 microsecondsAfter solving, we gett = 7.2 microsecondsthus distance traveled by µ- mesons will be

d = vtor d = 7.2 x 10-6 x 0.9cor d = 1920 mNow when the population is measured after this distance it was approximately half.It proves that the time dilation is a real effect.

6.3 Twin paradox Paradox means confusion and meaning of twins you know. This is related with theconcept of time dilation in relativity. Suppose there are twins A and B.In the twin paradox, one of the twins say A was sent to space in a spaceship which istraveling with a speed comparative to the speed of the light. B remains at earth.

So according to the time dilation, time should be dilated (increased) or in other wordsclock should be moving slower. But out of A and B, whom clock should be slower? Inother words, whose age will be different after certain time?According to B (which remains at earth), age of A will be different as he is travellingwith relativistic speed. But according to A, age of B will be different as he is travellingwith relativistic speed in opposite direction.So who is speaking truth? Both are right at their places. This is called twin paradox.What is your view on this?

7.1 Simultaneity in relativity



47

Figure 7Let there are two inertial frames of references S and S’. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, theyare at the same position that is Observers O and O’ coincides. After that S’ frame startsmoving with a uniform velocity v along x axis.Let two events in frame S occur simultaneously at positions P1 and P2. The coordinatesof the P1 will be (x1,y1,z1,t1) and of P2 will be (x2,y2,z2,t2). The events will besimultaneous (occur at the same time) according to the observer in frame S. Thereforet1 = t2 (1)The question arises here will the event be simultaneous from frame S’? Let us discuss it?From Lorentz transformation equations of time:

t’1 = (t1 – x1v/c2

)/(√1 – v2

/c2

) (2)and t’2 = (t2 – x2v/c2)/(√1 – v2 /c2) (3)Subtracting equation (2) from equation (3), we gett’2 – t’1 = (t2 – x2v/c2)/(√1 – v2 /c2) - (t1 – x1v/c2)/(√1 – v2 /c2)or t’2 – t’1 = (t2 – t1)/ )/(√1 – v2 /c2) - v/c2(x2 – x1)/(√1 – v2 /c2)Put equation (1) in above equation, we gett’2 – t’1 = - v/c2(x2 – x1)/(√1 – v2 /c2) (4)As the two events occur at different positions, that isx2 ≠x1Therefore L.H.S of equation (4) will not be zero, thust’2 – t’1 ≠ 0

or t’1 ≠ t’2It proves that the same events will not be simultaneous from frame S’, that is it will notappear to occur at the same time from S’.8.1 Addition of velocityLet there are two inertial frames of references S and S’. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, theyare at the same position that is Observers O and O’ coincides. After that S’ frame startsmoving with a uniform velocity v along x axis.



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or m2 [(c2-v2)/c2] = m’or m2c2 – m2v2 = m’2c2 Differentiating, we getc2(2mdm) – m2(2vdv) – v2(2mdm)or v2dm + mvdv = c2dm (3)

Comparing equations (2) and (3), we getdw = c2dm (4)The total amount of work done by the applied force in order to change its velocity from 0to v (or mass from m’ to m) is achieved by integrating the L.H.S of the followingequation with limits 0 to W and R.H.S. from m’ to m (because when work is 0 then bodyhas rest mass m’ and when work W is done then body has variable mass m).∫dw = c2∫dmOr W = c2(m – m’) (5)As this work W is done to give motion to the object. Therefore, W will appear in the formof kinetic energy acquired by the body, Thus relativistic kinetic energy will beK = = c2(m – m’) (6)

By definition of potential energy or the rest mass energy, it is equal to the internal energyof the body. It is also equal to the work done to bring all the particles which make theobject of rest mass m’. Thus the rest mass energy of the body is derived as by integratingthe L.H.S of the following equation with limits 0 to W and R.H.S. from 0 to m (becausewhen work is 0 then body has rest mass does not exist and when work W is done then allthe particles make an object of rest mass m’).∫dw = c2∫dmThus W = m’c2

Therefore, W will appear in the form of rest mass energy of the body, Thus rest massenergy will beR = m’c2 (7)The total energy of the object will beE = kinetic energy + rest mass energyPut equations (6) and (7) in this equation, we getE = c2(m – m’) + m’c2

Or E = mc2

This is the famous Einstein mass-energy equivalence relation.Significance:This equation represents that energy can neither be created nor be destroyed, but it canchange its form.Example:Pair Annihilation:In pair annihilation, electron and positron reacts to release photons.e- + e+ → γ As electron and positron have mass but photon has energy but not mass. Therefore, heremass is changed into energy.The opposite of this reaction is called pair production.

11.1 Relativistic energy momentum relationFrom Einstein mass energy relation



53

E = mc2 (1)Also from variation of mass with velocity relationm = m0 /(1 – v2 /c2)1/2 (2)Where m0 is the rest mass of the objectPut value of m in equation (1) and then square both sides, we get

E

2

= m0

2

c

4

/(1 – v

2

/c

2

) (3)As momentum is given byp = mvPut equation (2) and squarep2 = m0

2v2 /(1 – v2 /c2)

Multiply both sides by c2

p2c2 = m02v2 c2 /(1 – v2 /c2) (4)

Subtract equation (4) from (3) and solve, we getE2 - p2c2 = m0

2c4 Or E = (p2c2 + m0

2c4)This is Relativistic energy momentum relation.

Chapter 4

Superconductors1. Definition Superconductors:

Superconductors are the materials whose conductivity tends to infinite as resistivity tendsto zero at critical temperature (transition temperature).2. Critical temperature (Tc): The temperature at which a conductor becomes asuperconductor is known as critical temperature.3. Critical Magnetic Field (Hc): The magnetic field required to convert thesuperconductor into a conductor is known as critical magnetic field.4. Critical magnetic field is related with critical temperature as:

Hc(T) = Hc(0)[1 – T2 /Tc2]

5. Meissner Effect:

Suppose there is a conductor placed in a magnetic field at temperature T (refer figure).Then the temperature is decreased till the critical temperature. See what happened(figure). Lines of force are expelled from the superconductor. This is called Meissnereffect.

Definition Meissner Effect:

The expulsion of magnetic lines of force from a superconducting specimen when it iscooled below the critical temperature is called Meissner effect.5.1. To prove that superconductors are diamagnetic by nature:



55

(i) The magnetic properties in superconductors undergo change. In the puresuperconducting state practically no magnetic flux is able to enter the metalwhich thus behaves as if it had zero permeability or strong diamagneticsusceptibility. This effect is called Meissner effect.

(ii) The specific heat changes discontinuously at the transition temperature. There is

small change of volume at transition in the presence of magnetic field.(iii)All the thermoelectric effects disappear in the superconducting state.(iv) The thermal conductivity changes discontinuously when the superconductivity is

destroyed in magnetic field. It is lower in the superconducting state for puremetal but higher for certain alloys.

(v) The entropy in the superconducting state is lesser comparative to the normal state,that is the superconductive state is more ordered state.

8. Type I and Type II superconductors

Depending upon their behavior in an external magnetic field, superconductors are dividedinto two types:a) Type I superconductors and b) Type II superconductorsLet us discuss them one by one:1) Type I superconductors:a). Type I superconductors are those superconductors which loose their superconductivityvery easily or abruptly when placed in the external magnetic field. As you can see fromthe graph of intensity of magnetization (M) versus applied magnetic field (H), when theType I superconductor is placed in the magnetic field, it suddenly or easily looses itssuperconductivity at critical magnetic field (Hc) (point A).

After Hc, the Type I superconductor will become conductor.b). Type I superconductors are also known as soft superconductors because of thisreason that is they loose their superconductivity easily.c) Type I superconductors perfectly obey Meissner effect.



56

d) Example of Type I superconductors: Aluminum (Hc = 0.0105 Tesla), Zinc (Hc =0.0054)2) Type II superconductors:a). Type II superconductors are those superconductors which loose theirsuperconductivity gradually but not easily or abruptly when placed in the external

magnetic field. As you can see from the graph of intensity of magnetization (M) versusapplied magnetic field (H), when the Type II superconductor is placed in the magneticfield, it gradually looses its superconductivity. Type II superconductors start to loose theirsuperconductivity at lower critical magnetic field (Hc1) and completely loose theirsuperconductivity at upper critical magnetic field (Hc2).

b) The state between the lower critical magnetic field (Hc1) and upper critical magneticfield (Hc2) is known as vortex state or intermediate state.

After Hc2, the Type II superconductor will become conductor.c). Type I superconductors are also known as hard superconductors because of thisreason that is they loose their superconductivity gradually but not easily.c) Type I superconductors obey Meissner effect but not completely.d) Example of Type I superconductors: NbN (Hc = 8 x 106 Tesla), Babi3 (Hc = 59 x 103 Tesla)e) Application of Type II superconductors: Type II superconductors are used for strongfield superconducting magnets.9. London EquationsAs discussed in the Meissner effect that one of the conditions of the superconductingstate is that Magnetic flux density (B) = 0 inside the superconductors that is the magnetic

flux cannot penetrate inside the superconductor. But experimentally it is not so. Themagnetic flux does not suddenly drop to zero inside the surface. The phenomenon of fluxpenetration inside the superconductors was explained by H. London and F. London.Derivation of London first equation:Let ns and vs be the number density (number/volume) and velocity of superconductingelectrons respectively. The equation of motion or acceleration of electrons in thesuperconducting state is given bym(dvs /dt) = -eE



57

or dvs /dt = -eE/m (1)where m is the mass of electrons and e is the charge on the electrons.Also the current density is given byJs = -nsevs Differentiate it with respect to time,

dJs /dt = -nse(dvs /dt)Put equation (1) in above equation, we getdJs /dt = (nse

2 E)/m (2)Equation (2) is known as London’s first equationDerivation of London second equation:

Take curl (that is cross or vector product of del operator with a vector) of London’s firstequation, we getdel operator x dJs /dt = [(nse

2 )del operator x E]/m (3)By differential form of Faraday’s law of electromagnetic induction (or Maxwell’s thirdequation)del x E = -dB/dt

Put this in equation (3), we getdel x dJs /dt = -[(nse2(dB/dt)/m)Integrate both sides with respect to time, we getdel x Js = -[(nse

2(B)/m] (4)This is known as London’s second equation.Explanation of flux penetration (Meissner effect) from London equations:To explain Meissner effect from London equations consider the differential form of Ampere’s circuital law:del x B = µ oJs

where B is magnetic flux density and Js is current densityTake curl on both sides of above equationdel x (del x B) = µ

o(del x J

s) (5)

As del x (del x B)= del(del.B) - del2BPut above equation and London second equation (equation 4 is derived in last article) inequation (5), we getdel(del.B) - del2B = -[( µ o nse

2(B)/m]But del.B = 0 (Maxwell’s second equation or Gauss law for magnetism)Therefore above equation becomesdel2B = [( µ o nse

2(B)/m] (6)del2B = B/ λl

2 (7)where λl

2 = m/ µ o nse2

or λl = (m/ µ o nse2)1/2

where λl is known as London’s penetration depth and it has units of length.The solution of differential equation (7) isB = B(0)e-x/ λ

l (8)Where B(0) is the field at the surface and x is the depth inside the superconductor.The equation (8) shows that a uniform magnetic field equal to zero can not exist in asuperconductor, which is Meissner effect. In the pure superconducting state the only fieldallowed in the exponentially decreasing field as the flux penetrated from external surfaceand it is given by equation (8) (Refer figure).



59

.The superconducting wires used in the above applications are produced from thesolid solution alloys such as niobium-zirconium,niobium or niobium – titanium-zirconium.. High temperature superconducting wires may soon replace the wires operatingat liquid- Helium temperatures.

12. Particle accelerators:.Superconductors has very poor mechanical strength.. Hence superconducting solenoid is already in use to provide the high magneticfields needed for the large particle accelerators using Nb3Sn

Magnetic levitation:. The only alternatives to airplanes, cars, buses, ships and convential trains are just too slow for today‘s fast –paced society . However, there is a form of transportation that could revolutionize transportation of the 21istcentury the wayairplanes did in the 20th century.

. A few countries are using powerful superconducting electromagnets todevelophigh-speed trains,called maglev trains. Maglev is short for magnetic levitation,

which means that these trains will float over a guideway using the basic

principle of magnets to replace the old steel wheel and track

trains.Electromagnetic propulsion is used to provide highspeed to maglev trains..Electromagnetic suspension: is used to reduce friction between the train and itstracks.There are three components to magnetic train:. A large electrical power source. Metal coils lining a guideway or track

. Large guidance superconducting magnets attached to the underside of the train.Superconducting magnetic coils produce the magnetic repulsion required tolevitate the train .Maglev trains will not slide over the rails but will float on an aircushion over a magnetized track. There in a 1 cm air gap between the railwaytrack and superconducting magnets attached underside of the train. This processvirtually eliminating friction between the train and its tracks, so that speeds upto500 km/hr can be achieved easily.

The big difference between a maglev train and a conventional train is thatmaglev trains do not have an engine at least not the kind of engine used to pulltypical train, cars along steel tracks.The engine for Maglev trains is rather inconspicuous ,Instead of using fossil fuels,the magnetic field created by the electrified coils in the guideway walls and thetrack combine to propel the train As iron is not required for the production of magnetic field ,so maglev train could be much lighter in weight,

The world’s first MAGLEV train to be adapted in to commercial service, ashuttle in Birmingham England shut down in 1997 after operating for 11 years. A sino-



61

which can either be observed directly or can be made observable by instruments likemicroscope. But, the classical mechanics can not explain the mechanics of subatomicparticles like electron, proton, neutron etc. Then there comes in picture the quantummechanics, which explain the mechanics of these subatomic particles successfully.Following examples will show that classical mechanics was inadequate to give

explanation of observed facts:(a) Photoelectric effect: when a photon of light falls on the metal surface and if thefrequency of light is more than the certain minimum frequency (called thresholdfrequency), then electrons are ejected from metal surface. This effect is known asphotoelectric effect.

According to classical theory of light, there should be nothreshold frequency, the photoelectric current should increase with increase infrequency of light and kinetic energy of ejected electrons should increase withincrease in intensity of light. There must also be a finite delay between theincidence of photons and ejection of electrons from metal surface.

But experimental results were totally opposite ,like, photoelectric effect is instantaneous,and other observations were contradicted to aforementioned classical theory basedobservations.

Therefore classical mechanics failed to explain the photoelectric effect. Theseobservations successfully later on by Einstein’s theory of photoelectric emission, which isbased on Planck’s Quantum theory of Radiation.

(b) Stability of atom: Atomic model prepared by Rutherford was based on classicalphysics. This model assumes the atom to be consisting of a positively chargednucleus in the centre and negatively charged electrons revolved the nucleus in thecircular orbit. Classical theory says that whenever a charged particle undergoes

accelerated motion, it emits electromagnetic radiation. Therefore, an electronmust emit energy continuously as its motion is accelerated. Thus, the orbitalradius should decrease continuously and ultimately electron would fall in tonucleus. Therefore, an atom would collapse at the end.

But, this does not happen. As atom is stable and stability of atom could not beestablished by classical physics .This stability of atom was explained by Bohr’squantum theory of atom.Conclusion

Similarly, there were a lot of other observations like black bodyradiation, optical spectra, Compton Effect etc. which were not explained byclassical mechanics. Therefore, it can be concluded that classical mechanics isinadequate to discuss the subatomic phenomenon. Hence, there is a need tointroduce a new theory, which can deal with mechanics or motion of thesubatomic or microscopic particles. Thus, the quantum hypothesis originated andthe branch of physics dealing with motion of subatomic particles is calledquantum mechanics.

2. MATTER WAVES : dE-BROGLIE CONCEPT



62

In 1924, Lewis de-Broglie proposed that matter has dual characteristic just like radiation.His concept about the dual nature of matter was based on the following observations:-

(a) The whole universe is composed of matter and electromagnetic radiations. Sinceboth are forms of energy so can be transformed into each other.

(b) The matter loves symmetry. As the radiation has dual nature, matter should also

possess dual character.According to his hypothesis, a moving particle (e,p,n) has wave propertyassociated with it. The waves associated with moving particles are matter wavesor de-Broglie waves.

2.1 WAVELENGTH OF DE-BROGLIE WAVES

Consider a photon whose energy is given byE=hυ=hc/ λ - - (1)

If a photon possesses mass (rest mass is zero), then according to the theory of relatively, its energy is given by

E=mc2 - - (2)From (1) and (2) ,we haveMass of photon m= h/cλ Therefore Momentum of photon

P=mc=hc/cλ=h/ λ - - (3)Or λ = h/pIf instead of a photon, we consider a material particle of mass m moving withvelocity v,then the momentum of the particle ,p=mv. Therefore, the wavelength of

the wave associated with this moving particle is given by:h/mv -

Or λ = h/p (But here p = mv) (4)This wavelength is called DE-Broglie wavelength.

Special Cases:

a). de-Broglie wavelength for material particle:

If E is the kinetic energy of the material particle of mass m moving with velocityv,then

E=1/2 mv2=1/2 m2v2=p2 /2mOr p=√2mE

Therefore the by putting above equation in equation (4), we get de-Brogliewavelength equation for material particle as:

λ = h/ √2mE - - (5)b). de-Broglie wavelength for particle in gaseous state:

According to kinetic theory of gases , the average kinetic energy of the materialparticle is given by



63

E=(3/2) kTWhere k=1.38 x 10-23 J/K is the Boltzmann’s constant and T is the absolutetemperature of the particle.Also E = p2 /2mComparing above two equations, we get:

p2 /2m = (3/2) kTor p = / √3mKTTherefore Equation (4) becomes

λ=h/ √3mKTThis is the de-Broglie wavelength for particle in gaseous state:

c). de-Broglie wavelength for an accelerated electron:

Suppose an electron accelerates through a potential difference of V volt. Thework done by electric field on the electron appears as the gain in its kinetic energy

That is E = eVAlso E = p2 /2mWhere e is the charge on the electron, m is the mass of electron and v is thevelocity of electron, thenComparing above two equations, we get:eV= p2 /2mor p = √2meVThus by putting this equation in equation (4), we get the the de-Brogliewavelength of the electron as

λ = h/ √2meV 6.63 x 10-34 / √2 x 9.1 x 10-31 x1.6 x 10-19 V

λ=12.27/ √V ÅThis is the de-Broglie wavelength for electron moving in a potential difference of V volt.

3. Heisenberg’s Uncertainty PrincipleStatement: According to Heisenberg uncertainty principle, it is impossible to measurethe exact position and momentum of a particle simultaneously within the wave packet.

We know, group velocity of the wave packet is given byvg =ω / kWhere ω is the angular frequency and k is the propagation constant or wave numberBut vg is equal to the particle velocity vThus vg = v = ω / k (1)

But ω=2пf Where f is the frequency

Therefore ω = 2п f (2)



64

Also k=2 п / λ Since de-Broglie wavelength λ=h/pBy putting this value in equation of k, we getk=2пp/ λ Therefore k=2пp / λ (3)

Put equations (2) and (3) in equation (1), we getv= 2пhf/2пp =hf / (4)Let the particle covers distance x in time t, then particle velocity is given by

v = x/ t (5)Compare equations (4) and (5), we get

x/ t=hf/ pOr x.p=hf t (6)The frequency f is related to t by relation

t≥ 1/ f (7)

Hence equations (6) becomes

x.p≥ hA more sophisticated derivation of Heisenberg’s uncertainty principle givesx.p=h/2п (8)

This is the expression of the Heisenberg uncertainty principle.As the particle is moving along x-axis, therefore, the momentum in equation (8) of Heisenberg’s uncertainty principle should be the component of the momentum in the x-direction, thus equation Heisenberg’s uncertainty principle can be written as,

x.px=h/2п (9)Note: There can not be any uncertainty if momentum is along y direction.

3.1 Applications of Heisenberg’s Uncertainty principle

The Heisenberg uncertainty principle based on quantum physics explains a number of facts which could not be explained by classical physics. One of the applications is toprove that electron can not exist inside the nucleus. It is as follows:-

a) Non-existence of electrons in the nucleus

In this article, we will prove that electrons cannot exist inside the nucleus.But to prove it, let us assume that electrons exist in the nucleus. As the radius of the nucleus in approximately 10-14 m. If electron is to exist inside the nucleus,then uncertainty in the position of the electron is given by

x= 10-14 m

According to uncertainty principle,xpx =h/2∏ Thus px=h/2∏xOr px =6.62 x10-34 /2 x 3.14 x 10-14

px=1.05 x 10-20 kg m/ secIf this is p the uncertainty in the momentum of electron ,then the momentum of electron should be at least of this order, that is p=1.05*10-20 kg m/sec.



65

An electron having this much high momentum must have a velocity comparableto the velocity of light. Thus, its energy should be calculated by the followingrelativistic formula

E= √ m20 c4 + p2c2

E = √(9.1*10-31)2 (3*108)4 + (1.05*10-20)2(3*108)2

= √(6707.61*10-30) +(9.92*10-24)=(0.006707*10-24) +(9.92*10-24)= √9.9267*10-24

E= 3.15*10-12 JOr E=3.15*10-12 /1.6*10-19 eV

E= 19.6* 106 eVOr E= 19.6 MeVTherefore, if the electron exists in the nucleus, it should have an energy of theorder of 19.6 MeV. However, it is observed that b-particles (electrons) ejected

from the nucleus during b –decay have energies of approximately 3 Me V, whichis quite different from the calculated value of 19.6 MeV. Second reason thatelectron can not exist inside the nucleus is that experimental results show that noelectron or particle in the atom possess energy greater than 4 MeV.Therefore, it is confirmed that electrons do not exist inside the nucleus.

b) Applications of the Heisenberg Uncertainty Principle: The Radius of Bohr’s First

OrbitIn one of my earlier articles, I have discussed the one the applications of the Heisenberguncertainty principle that is non-existence of electron in the nucleus. Let us discuss todaythe one more application of the Heisenberg uncertainty principle that is the determinationof the radius of the Bohr’s first orbit. Let us start:If x and px are the uncertainties in the simultaneous measurements of position andmomentum of the electron in the first orbit, then from uncertainty principle

xpx = Ћ Where Ћ = h/2∏ Or px = Ћ / x (1)As kinetic energy is given as

K = p2 /2mThen uncertainty in K.E is

K =p2x/2m

Put equation (i) in above equationK= Ћ2 /2m(x)2 (2)

As potential energy is given byV= -1/4∏ε0 Ze2 / x (3)

The uncertainty in total energy is given by adding equations (2) and (3), that isE= K+V

= Ћ2 /2∏(x)2 –Ze2 /4∏ε0x



66

If x =˘ r= radius of Bohr’s orbit, thenE= Ћ2 /2mr2 –Ze2 /4∏ε0r (4)

The Uncertainty in total energy will be minimum if d(E)/dr=0 and d2((E)/dr2 is positiveDifferentiating equation (4) w.r.t. r, we get

d(E)/dr=0= - Ћ

2

/mr

3

+Ze

2

/4π ε0r

2

(5)For minimum value of Ed(E)/dr=0= - Ћ 2 /mr2+Ze2 /4π ε0r

2 or Ze2 /4π ε0r

2= Ћ 2 /mr3 Or r=4π ε0 Ћ

2 /me2 (6)Further differentiating equation (5), we getd2(E)/dr2=3 Ћ 2 /mr4-2Ze2 /4π ε0r

3 By putting value of r from equation (6) in above equation, we get positive value of d2(E)/dr2 Therefore equation (4) represents the condition of minimum in the first orbit.Hence, the radius of first orbit is given by

r=4π ε0Ћ

2

/me

2

=0.53 A (For H atom Z=1)Put value of r in equation (4), we getEmin= -13.6 e V

This value is same as determined by using Bohr’s theory.Therefore, with the help of Heisenberg’s uncertainty principle, one can determine theradius of the Bohr’s first orbit.4. WAVE FUNCTION

If there is a wave associated with a particle, then there must be a function to represent it.This function is called wave function.Wave function is defined as that quantity whose variations make up matter waves.It is represented by Greek symbol ψ(psi), ψ consists of real and imaginary parts.

Ψ=A+iB5. PHYSICAL SIGNIFICANCE OF WAVE FUNCTIONS (BORN’S INTER

PRETATION)

Born’s interpretationThe wave function ψ itself has no physical significance but the square of its

absolute magnitude |ψ2| has significance when evaluated at a particular point and at aparticular time |ψ2| gives the probability of finding the particle there at that time.The wave function ψ(x,t) is a quantity such that the product

P(x,t)=ψ*(x,t)ψ(x,t)Is the probability per unit length of finding the particle at the position x at time t.P(x,t) is the probability density and ψ*(x,t) is complex conjugate of ψ(x,t)Hence the probability of finding the particle is large wherever ψ is large and vice-versa.

6. NORMALIZATION CONDITIONThe probability per unit length of finding the particle at position x at time t is

P=ψ*(x,t)ψ(x,t)So, probability of finding the particle in the length dx is

Pdx=ψ*(x,t)ψ(x,t)dxTotal probability of finding the particle somewhere along x-axis is



68

Also on differentiating (2) wr.t. t ,we getdΨ(x,t)/dt= A exp [-i/ Ћ(Et-px)] (-i/ Ћ E)

dΨ /dt =-(i/ Ћ) E Ψ or E Ψ=(i Ћ) Ψ / t (4)

When the particle is acted upon by a force then its total energy is the sum of Kinetic and

potential energies i.e.Total energy = Kinetic energy + potential energyE=p2 /2m +VE Ψ= p2 /2m Ψ+ V Ψ (5)

Putting equations (3) and (4) in equations (5), we get-Ћ /t dΨ /dt=-h2 /2m d2Ψ /dx2+V Ψ

i Ћ dΨ /dt= - Ћ2 /2m( Ψ /x2) + V Ψ (6)Which is time dependent form of Schroedinger wave equationIn three –dimensional form

i Ћ Ψ /t =- Ћ 2 /2m(d2Ψ /dx2+ d2Ψ /dy2+ dΨ /dz2)+ V Ψ where the particle potential V is a function of x,y,z and t . Any restriction on the particle

motion will effect the potential energy V. once V is known, Schroedinger equation maybe solved for the wave function Ψ of the particle form where Ψ2 may be determined.8.2 Time Independent Schrödinger Wave EquationAs discussed in the article of time dependent Schrodinger wave equation:

V=A exp[-i/ Ћ(Et-px]= A exp(-i/ Ћ Et) exp(i/ Ћ)Ψ=ψ’ exp(-iEt/ Ћ) (1)

Where Ћ = h/2π So, ψ is a product of a time dependent function exp(-i/ Ћ Et) and a position dependentfunction

Ψ’= A exp(-i/ Ћ px)Differentiating equation (1) w.r.t.x, We have

dψ /dx = exp(-i/ ЋEt) dψ’ /dxand d2ψ /dx2= exp(-i/ Ћ Et) d2ψ’ /dx2 (2)

Also on differentiating ψ w.r.t. t, we havedψ /dt=ψ’ exp (-iEt/ Ћ) (I E/ Ћ)dψ /dt=-(iE/ Ћ)ψ’ exp(-I Et/ Ћ) (3)

Put equations [1-3] in time dependent Schrodinger wave equation (discussed earlier),iЋ[-iE/ Ћψ’ exp(-iEt/ Ћ)]= -Ћ2 /2m[exp(i/ ЋEt) d2ψ’ /dx2] +V ψ’ exp(iEt/ Ћ)Eψ’ exp(iEt/ Ћ) = -Ћ2 /2m exp(i/ Ћ Et) d2ψ’ /dx2 + V ψ’exp(iEt/ Ћ)Dividing throughout by expression (i/ Ћ Et) we have

Eψ’= (-Ћ2 /2π) d2ψ’ /dx2+V ψ’ Or (E-V)ψ’=-Ћ2 /2m dψ’ /dx2

d2Ψ’ /dx2 + (2m/ Ћ2)(E-V)ψ’ (4)This is time independent form of Schrödinger wave equation in one dimension.In three-dimensional form:

d2 Ψ’ /dx2+ d2ψ’ /dy2+ d2Ψ’/ d2x2+2m/ Ћ2(E-V)ψ’=0In this equation, ψ’ equation, ψ’(x) is also called the wave function. The potentialV(x) does not contain the time explicitly and E, the total energy of the particle is aconstant.



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The particle will always remain inside the box because of infinite potential barrier at the walls .So the probability of finding the particleoutside the box is zero i.e.ψx=0 outside the box.

We know that the wave function must be continuous at the boundaries of potential well at x=0 and x=L, i.e.

Ψ(x)=0 at x=0 (5)Ψ(x)=0 at x= L (6)

These equations are known as Boundary conditions.(iii) Determination of Energy of Particle

Apply Boundary condition of eq.(5) to eq.(4)0=A sin (X*0) +B cos (K*0)0= 0+B*1B=0 (7)

Therefore eq.(4) becomes

Ψ(x) = A sin Kx (8)Applying the boundary condition of eq.(6) to eq.(8) ,we have0=A sin KL

Sin KL=0 KL=nπ K=nπ /L (9)

Where n= 1, 2, 3 - - - ACannot be zero in eq. (9) because then both A and B would be zero. Thiswill give a zero wave function every where which means particle is notinside the box.

Put this value of K in eq. (3)nπ /L = 2m E/ Ћ2

Squaring both sidesn2π2 /L2=2mE/ Ћ2 E=n2π2Ћ2 /2mL2

Where n= 1, 2, 3… Is called the Quantum numberAs E depends on n, we shall denote the energy of particle ar En. Thus

En= n2π2Ћ2 /2mL2 (10)(iv) Energy level diagram

When n=1E1= (1)2n2Ћ2 /2mL2=π2Ћ2 /2mL2

It is called the Ground state energy of the particle. When n=1, the state of the particle is called Ground state. When n= 2, 3, 4 … then correspondingstates of the particle are called Excited states. The energies of excitedstates are

E2= (2)2n2Ћ2 /2mL2 =4π2Ћ2 /2mL2=4E1



E3 =(3)2 π2Ћ2 /2mL2=9π2Ћ2 /2mL2=9E1

The energy of the particle in the box is quantized because a particle insidethe infinite potential well cannot have any arbitrary energy but onlydiscrete energy values given by E1,E2,E3,….

The energy separation between successive levels is not uniform.Zero point energy . The minimum allowed energy to the particle ininfinite potential well is called Zero point Energy.

E1=π2Ћ2 /2mL2(zero point energy) (11)

Wave functions. Substitute the value of K from eq. (9) in eq. (8) to getΨ(x)=A sin(nπ /Lx)As the wave function depends on quantum number π so we write it ψn.ThusΨn=A sin (nπx/L)0<x<L

(12)Ψn=0 outside the box(v) Determination of Normalization constant

The Normalization condition is given by∫-∞

4-∞ ψnψn* dx=1 (13)

As the wave function of particle exists in the region 0 to L and zerooutside.Thus ∫L0 ψnψ

*ndx=1

∫L0 A sin (nπx/L) A sin (nπx/L)dx=1

A2∫L0sin2(nπx/L)dx=1A2∫L0

(1-cos2(nπx/L)/2)dx=1A2 /2 ∫L0 (1-cos 2nπx/L)dx=1A2 /2 [∫L0 dx- ∫

L0 cos 2nπx/L dx]=1

A2 /2[(x)0L [sin 2nπx/L/(2nπ /L)])=1A2/ /2 [(L-0) – L/2nπ(sin 2nπL/L-sin 0)]=1A2 /2 [L-L/2nπ(sin2nπ-sin0]=0A2 /2[L-0]=0

A2=2/L A=2/L=(2/L)1/2

Substitute in eq (12) we have

Physics-Lasers and Relativity and Superconductor and Quantum

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