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Chapter 39

Particles Behaving as Waves 39.1 Electron Waves Light has a dual nature. Light exhibits both wave and particle characteristics. Louis de Broglie postulated in 1924 that if nature is symmetric, then all forms of matter should exhibit wave properties! If a particle of matter acts like a wave, it should have a frequency and a wavelength associated with it. De Broglie postulated that a free particle with rest mass m, moving with speed v, should have a wavelength λ related to its linear momentum p by

ph

=!

where

!

p = mv (nonrelativistic particle)

!

p = "mv (relativistic particle) Furthermore, in analogy with photons, the frequency f of the waves we associate with a moving particle of mass m is

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hEf =

where E is the energy of the material particle. For his prediction of the wave nature of electrons, Louis de Broglie was awarded the Nobel Prize in physics in 1929. Under the proper conditions, then, every stream of particles will produce an interference or diffraction pattern. Each body – whether an electron, a proton, an atom, a soccer ball, a planet, or star – has a wave associated with it with a wavelength that is related to its momentum by the above formula. A body of large mass and ordinary speed has such a small wavelength that interference and diffraction effects are negligible. But for smaller particles like electrons, diffraction can be appreciable.

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The Electron Microscope An electron microscope makes practical use of the wave nature of electrons. The wavelength of electron beams is typically thousands of times shorter than the wavelength of visible light, so the electron microscope is able to distinguish detail not visible with optical microscopes.

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39.2 The Nuclear Atom and Atomic Spectra

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Rutherford’s Exploration of the Atom

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The Failure of Classical Physics

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39.3 Energy Levels and The Bohr Model of the Atom

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The Bohr Model of the Hydrogen Atom Niels Bohr postulated that each energy level of a hydrogen atom corresponds to a specific stable circular orbit of the electron around the nucleus. He further postulated that an electron in such an orbit does not radiate (contrary to classical physics). Instead, an atom radiates energy only when an electron makes a transition from an orbit of energy Ei to a different orbit with lower energy Ef , emitting a photon of energy hf = Ei – Ef in the process.

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The basic assumptions are: 1. The electron moves in a circular orbit about the proton interacting via the Coulomb force. Use classical mechanics to treat the motion of the electron around the proton:

!

Fc" = mac

!

k(e)(e)r2

= m v2

r

divide by 2 to obtain

!

12mev

2 =ke2

2r

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2. Electromagnetic radiation (or a photon) is emitted when the electron “jumps” from a more energetic initial state (with energy Ei) to a less energetic final state (with energy Ef). The frequency f of the emitted radiation is given by

!

h f = Ei " E f

3. The “allowed” orbits for the electron are those for which the orbital angular momentum of the electron satisfies the relation

!

me v r = n h2"

where n is called the “principal quantum number” and n = 1, 2, 3, 4, … 4. The energy E of the electron when it is in an allowed orbit of radius “r” or principal quantum number “n” is

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!

E = KE +Uelectric =12mev

2 +k(e)("e)

r

Combining the boxed equations above yields the following results for the “allowed orbits” of the electron around the proton, for the “allowed energies” of the electron as it orbits around the proton, AND the speed of an electron around the proton:

!

En = "13.6eVn2

!

rn = ao n2

!

vn =2" k e2

h1n

where

!

ao =h2

4" 2k mee2 (radius of first Bohr orbit)

That is, the values of the radii and of the energy of the electron are “quantized”. The state of the electron corresponding to the lowest allowed energy is called the “ground state”. n = 1: ground state or first Bohr orbit

!

r1 = ao = 0.0529 nm

!

E1 = "13.6 eV “ground state energy”

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n = 2: first excited state or the second Bohr orbit

!

r2 = 4ao = 0.212 nm

!

E2 = "13.6eV4

= "3.40 eV

n = 3: second excited state or third Bohr orbit

!

r3 = 9ao = 0.476 nm

!

E3 = "13.6eV9

= "1.51 eV

and so on… Comments:

1. The negative sign in

!

En = "13.6eVn2

means that the

electron is “bound” to the proton. The electron is not free 2.

!

n =" # E" = 0 , represents the state when the electron is completely removed from the atom. Hence the minimum energy required to ionize the hydrogen atom when it is in the ground state is 13.6 eV and is called the “ionization energy”. 3. When an electron “jumps” (or undergoes a “quantum transition”) from an orbit with higher energy to an orbit with lower energy, the atom emits a photon of frequency f given by

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!

h f = Ei " E f

!

h f ="13.6 eV

ni2

#

$ %

&

' ( "

"13.6 eVn f2

#

$ %

&

' (

!

h f =13.6 eV 1n f2 "

1ni2

#

$ %

&

' (

where ni > nf. if nf = 1: Lyman series ni = 2, 3, 4, 5, … (UV)

!

h f =13.6 eV 112"1ni2

#

$ %

&

' (

if nf = 2: Balmer series ni = 3, 4, 5, … (Visible)

!

h f =13.6 eV 122"1ni2

#

$ %

&

' (

if nf = 3: Paschen series ni = 4, 5, 6, 7, … (IR)

!

h f =13.6 eV 132"1ni2

#

$ %

&

' (

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Explanation of Quantized Energy Levels 1. Why electrons occupy only discrete levels is understood by considering the electron to be a wave, not a particle. 2. A Bohr orbit exists where an electron wave closes on itself constructively. An orbiting electron forms a standing wave only when the circumference of its orbit is equal to a whole-number multiple of wavelengths. 3. The electron wave is spread out in three-dimensions so one speaks of electron clouds”. The Bohr theory of the hydrogen atom was successful when applied to hydrogen-like atoms (atoms containing only one electron). His theory breaks down when applied

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to more complex atoms, and that is where the “theory of quantum mechanics” developed by Schroedinger, Heisenberg, Born, etc does very well in yielding results that agree with experimental data. 39.4 The LASER LASER: This is an acronym meaning Light Amplification by Stimulated Emission of Radiation. The first laser (ruby laser) was built by Maiman in 1960 here in California. Consider the excitation of atoms in a medium (solid, liquid, or gas) by incoming photons of the “right” energy.

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Spontaneous Emission of Radiation This process refers to the case when an “excited atom” (with an electron in an orbit with a higher energy than the level it would ordinarily occupy) is left alone and emits the extra energy via electronic transitions to lower energy levels. That is, If all excited atoms are left alone, they will emit the extra energy in due time spontaneously! Stimulated Emission of Radiation You actually do not have to wait for the excited atom to emit its extra energy. You can stimulate the atom to emit the extra energy by simply shooting a photon at it of the right energy. That is,

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Notice that you send in one photon and one excited atom and you end up with two photons and an atom in its “ground state” (de-excited). This process corresponds to light amplification! All lasers have:

1. an excitation source. This could be a provided by an electric current, a flashlight, or another laser.

2. an active medium where the laser light is generated. The active medium could be a solid, a liquid, or a gas.

3. at least two mirrors to provide the feedback mechanism. One of the mirrors needs to be partially transmitting to let the laser light out.

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39.5 Continuous Spectra - Blackbody Radiation and Planck’s Theory A. Thermal radiation is the radiation emitted by a body as a result of its non-zero temperature on its surface. Emission line spectra come from matter in the gaseous state, in which the atoms are so apart from one another that interactions between them are negligible. By contrast, a heated solid or liquid (in which atoms are close to each other) nearly always emits radiation with a continuous distribution of wavelengths. A blackbody is a body that has a surface that absorbs all radiation incident upon it. All blackbodies at the same temperature T emit thermal radiation with the same spectrum.

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The total energy emitted per unit time, per unit area (intensity), from a blackbody with an absolute temperature T on its surface is given by the Stefan-Boltzmann law:

!

Itotal =2" hc2

#5 ehc /kT $1( )d# =

0

%

& ' T 4

The emitted intensity increases rapidly with increasing temperature. The value in “SI” units of the Stefan-Boltzmann constant ! equals 5.67x10-8 W/m2 K4. The total power emitted by a blackbody with surface area A and surface temperature T is then

4TAePtotal !=

The wavelength max! at which the emitted intensity has a maximum value is related to the absolute temperature T on the surface of the object according to Wien’s displacement law:

KmxT != "3max 10898.2#

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B. Planck’s hypothesis Any physical entity with one degree of freedom whose “coordinates” execute simple harmonic oscillations can possess only total energies ! which satisfy the relation

fhn=!

Where f is the frequency of oscillation and h is a universal constant known as Planck’s constant. The value of Planck’s constant is 6.63x10-34 J s. The energy ! of an entity obeying Planck’s postulate is said to be quantized, the allowed states are called quantum states, and the integer n is called a quantum number. Since n is an integer, that is, n = 0, 1, 2, 3, 4,… then it follows that the allowed values of the total energy ! are

...,4,3,2,,0 fhfhfhfh=! For his theoretical analysis of blackbody radiation, Max Planck was awarded the Nobel Prize in Physics in 1918.