Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 1

Physics 141.Lecture 12.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 2

Physics 141.Lecture 12.

• Course Information:

• Chapter 8, Energy Quantization:• Quantization of energy.• Emission and absorption spectra.• Vibrational energy levels.• Rotational energy levels.• Example of quantization.• Incoherent and coherent emission of light.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 3

Physics 141.Course Information.

• Homework:• Homework set 5 is due on Friday 10/14 at 12 pm.

• Laboratory:• Laboratory # 3 took place on Monday 10/10.• Laboratory # 3 office hours will be on Monday 10/24.• Laboratory # 3 report will be due on Friday 10/28 at noon.

• Exam # 1:• Exam # 1 was returned during recitations last week.• Any changes in the grading of the exam can only be made by the

instructor (me), not by your TAs.• Any requests to look at grade corrections must be made, in writing,

by Friday 10/14.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 4

Chapter 8.Energy quantization.

• Until the early 20th century, scientists assumed that the totalenergy of a system could have any arbitrary value(constraint only by the potential energy of the system).

• The energy of a planet in a circular orbit around the sun will have anenergy that depends on the radius of the orbit. By changing theradius, we can change the energy of the system.

• The success of the planetary model lead to applications of this modelto the description of atoms where electrons carry out orbital motionaround the nucleus.

• This model predicts that the total energy of the electron is onlyconstraint by the electrostatic potential energy.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 5

Quantization of light.

• Before 1905, light was assumed tobe an electromagnetic wave,characterized by an amplitude and awavelength.

• When the intensity of the lightincreases, the amplitude of theelectromagnetic wave increases.

• Most phenomena could beunderstood in terms of the wavenature of light. Diffraction andinterference provided compellingevidence that the wave picture oflight was correct.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 6

Energy quantization.Quantization of light.

• But …… the photoelectric effect (emission of electronsfrom metals) could not be understood in terms of the wavemodel of light.

• To explain the photoelectric effect, light must be describedin terms of "particle" properties. When the intensity of a"particle" beam increases, the number of particles increasesbut the energy of each particle does not change.

• The particles of light are called photons. The energy of aphoton is determined by its wavelength (the typical energyfor a photon of visible light is around 2 - 3 eV).

• The energy of the photon is hc/λ, where h is Planck'sconstant (= 6.6 × 10-34 Js) and λ is the wavelength of thelight.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 7

Quantization of light.The photoelectric effect.

• Experiments showed that lightcan liberate electrons from amaterial (the photoelectrons).

• The following surprisingobservations were made:

• No electrons are liberated if thewavelength of the light is largerthan some critical value(independent of the intensity ofthe light).

• The maximum energy of theelectrons liberated does notdepend on the intensity of thelight. It only depends on thewavelength of the incident light.

http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html

http://www.mhhe.com/physsci/astronomy/fix/student/chapter6/06f07.html

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 8

Quantization of light.The photoelectric effect.

• Einstein explained this effect byinterpreting light as a collection ofphotons with discreet energies.

• Einstein proposed that the photonenergy is only determined by thewavelength of the light (E = hc/λ).

• Changes in light intensity reflectchanges in the number of photons, notthe energy of the individual photon.

• The resulting wave-particle dualityalso predicted that "particles" canbehave like waves. It was observedthat for example electrons can behavelike waves (electron diffraction).Note: more on this in Phy 143.

http://www.faqs.org/docs/qp/chap03.html

http://www.matter.org.uk/diffraction/electron/electron_diffraction.htm

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 9

3 Minute 40 Second Intermission (42 seconds more than during lecture 11).

• Since paying attention for 1 hourand 15 minutes is hard when thetopic is physics, let’s take a 3minute 40 second intermission.

• You can:• Stretch out.• Talk to your neighbors.• Ask me a quick question.• Enjoy the fantastic music.• Go asleep, as long as you wake up

in 3 minutes and 40 seconds.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 10

Quantization of light.Different wavelengths, different energies.

Figure: Chaisson and McMillan, Astronomy today

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 11

Energy quantization.Energy levels of hydrogen.

• Light is emitted by heated hydrogen gas only at specificwavelengths.

• Excited hydrogen atoms can thus only emit specific amountsof energy.

• This implies that the energy states of hydrogen atoms arequantized.

Emission pattern of Hydrogen."Astronomy! A Brief Edition," J. B. Kaler, Addison-Wesley, 1997.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 12

Energy quantization.Energy levels of hydrogen.

• Experiments examining thedetails of atomic structureshowed that the electrons canonly occupy certain specificenergy levels.

• The measurements show that thetotal energy of the electron, K +U, is equal to -13.6/N2 eV whereN is an integer (N = 1, 2, ….).

• The quantization of the energylevels is a result of quantum-mechanical effects (to bediscussed in more detail inPhysics 143).

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 13

Quantization of light.Emission patterns.

• Light is emitted when an excitedatom makes a transition to alower energy level.

• Since the energy level of atomsare quantized, the light emittedwill have discrete wavelengths(energies).

• The energy levels serve as asignature (finger print) for theatom, and the emission patterncan be used to identify theatom(s).

Eγ = −

13.642 eV

⎛⎝⎜

⎞⎠⎟− −

13.612 eV

⎛⎝⎜

⎞⎠⎟= 12.75 eV

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 14

Quantization of light.Absorption patterns.

• When an atom is in its groundstate, it can only absorb photonsof specific frequencies.

• Only photons with an energy thatexactly match possible transitionsbetween energy levels in the atomare absorbed by the atom.

• The absorption spectrum canalso be used as a signature of theatom.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 15

Quantization of light.Absorption patterns.

• The atoms that absorb thephotons will decay back to theground state.

• When these transitions occur, theemitted photons will have thesame energies as the energies ofthe photons being absorbed.

• However, the reemitted photonsare emitted in random directions,and only a few are emitted in thedirection of the incomingphotons.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 16

Quantization of light.Absorption patterns.

• At room temperature, almost allhydrogen atoms will be in theirground state (E = -13.6 eV).

• The first excited state in hydrogenis at E = -3.4 eV and a 10.2 eVphoton is required to make thetransition.

• Since visible light has energiesaround 2 - 3 eV, the hydrogen willappear to be transparent to visiblelight.

• Note: visible light can be absorbedwhen the atom is already in anexcited state, but this is unlikely atroom temperature.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 17

Emission in the laboratory allows ID via absorption in the universe.

Emission pattern of Hydrogen."Astronomy! A Brief Edition," J. B. Kaler, Addison-Wesley, 1997.

Hydrogen absorption lines."Astronomy! A Brief Edition," J. B. Kaler, Addison-Wesley, 1997.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 18

Absorption patterns. Make sure we consider the velocity of the emitter!

Star moving towards us.

Star moving away from us.

From: Davison E. Soper, Institute of Theoretical Science, University of Oregon, Eugene OR 97403 USA

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 19

Absorption patterns.Temperature dependence.

• The absorption pattern dependson the population of states in thesample.

• Atoms with excited statespopulated are able to absorblower-energy light (longerwavelength) since the energyspacing between atomic levelsdecreases with increasing energy.

• Since the population patternsdepends on temperature, weexpect to see a temperaturedependence of the absorptionspectrum.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 20

Other examples of quantization.The atomic model of matter.

• Although many properties of mattercan be understood based on a simple"spring" model, others, such asthermal conduction, can not.

• In the classical "spring" model, theenergy of the spring-mass systemcan have any value:

E = (1/2)ksA2

• In order to describe the thermalproperties of matter we mustincorporate the quantum nature ofthese oscillators in our model.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 21

Other examples of quantization.The atomic model of matter.

• The effect of the quantum treatmentof the oscillator is that the energy ofthe system is quantized.

• It turns out that in many cases apotential well consistent with thespring force is consistent with theobserved atomic properties.

• The energy levels in this well arequantized and the spacing between thelevels is (h/2π)√(kS/m).

• One important consequence of thisquantization is that atoms can onlytransfer energy to each other indiscrete amounts.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 22

Position of the ground state.

• The ground state of the atom isnot located at the bottom of thewell.

• The energy of the ground stateabove the bottom of the well ishalf the level spacing:

• This is a consequence of theuncertainly principle which willbe discussed in Phy 143. Note: ground

state not at 0!

E0 =

12

ks

m

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 23

Other examples of quantization.The atomic model of matter: dependence on ks.

X

Can be absorbed by atomswith small ks, but not by atomswith large ks.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 24

Example Problem.

• If you have a collection of quantumoscillators that occupy the lowestfour energy levels, you expect to seethe following emission pattern:

• E1 = 0.4 eV (2 to 1, 3 to 2, and 4 to 3).

• E2 = 0.8 eV (3 to 1, and 4 to 2).

• E3 = 1.2 eV (4 to 1).

• The intensity will be a strongfunction of the population pattern.

• The energy of the emitted photonsallows us to probe the energy levelsof the atom.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 25

Other examples of quantization. Molecular vibrational energy.

• When we measure the vibrationalenergy levels for a two-atomicmolecule we find that at lowenergies the vibrational modelworks fine (nearly uniformenergy spacing).

• At higher energies, the potentialwell starts to deviate from aharmonic oscillator well, and thevibrational energy levels are nolonger uniformly spaced.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 26

Other examples of quantization. Molecular rotational energy.

• Molecules can also carryrotational energy.

• The rotational energy of amolecule is also quantized, butthe spacing between levelsincreases with increasingexcitation energy.

• The rotational energy is found tobe equal to

where l is an integer (0, 1, … )and I is the moment of inertia(depends on mass and shape).

E =

12I +1( )2

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 27

Other examples of quantization. Molecular rotational energy.

• When a molecule is in arotational excited state, it willdecay back to the ground state bygradually lowering its excitationenergy.

• The transition energies expectedto be seen are

ΔE→−1 = E − E−1

= 12I +1( )− −1( ){ }2

= I2

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 28

Nuclear energy levels.The rotation of super-deformed nuclei.

http://wnsl.physics.yale.edu/structure/research/highspin/sd.html

The energy of the photonsis consistentwith the rotationof a super-deformed rigid object around its symmetry axis.

Eγ = ΔE→−1 =

I2

ΔEγ = ΔE→−1 − ΔE−1→−2 =

I2 − −1

I2 =

2

I

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 29

Different systems.Different energies.

State Energy level spacing (eV)

Hadronic 100,000,000

Nuclear 1,000,000

Atomic 1

Molecular (vibrational) 0.01

Molecular (rotational) 0.0001

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 12, Page 30

Done for today!