3
3.1
Other Early Evidence for
Quantum Behavior
Photoelectric effect
Effect: blue light shining on metal observed to kick out electrons, longer
wavelength light doesn’t, independent of how intense it is!
Einstein (1905) proposed Planck idea might explain this. Light (like SHO
energies) might come in “lumps” or quanta of energy hν. Suppose electrons in metal have binding energy Φ, then energy of e− knocked out
should be
E ≤ hν − Φ
(1)
i.e., “photons” with frequency < ν don’t carry enough energy.
Richardson & Compton (1912) verified Einstein relation and showed value
of h agreed with Planck!
3.2
Atomic spectroscopy and Ritz principle
Spectrum of radiation from hot gas known since late 19th cent. to consist
of discrete “lines” at frequencies νi
Ritz (1908?) noticed the set of νi can be written as the set of differences
among smaller set of frequencies:
νi = fα − fβ ∀ α, β
(2)
Agrees with Einstein’s concept if atoms have discrete set of energy levels
1
Eα , and light is produced by transitions among them, of freq.
ν = (Eα − Eβ )/h
(3)
Atomic hydrogen
• Balmer Series (1885): set of optical spectroscopic lines of H at the
set of wavelengths (λ = c/ν) associated with freqs.
1
1
ν = R( − 2 ),
4 n
n = 3, 4, 5...
(4)
n = 3, λ = 6363A (Hα )
n = 4, λ = 4861A (Hβ )
n = 5, λ = 4340A (Hγ )
⇓
• Lyman Series (1914): set of ultraviolet spectroscopic lines of H at
ν = R(1 −
1
),
n2
n = 2, λ = 1215A (Lα )
n = 3, λ = 1025A (Lβ )
n = 4, λ = 972A (Lγ )
2
n = 2, 3, 4...
(5)
⇓
• Paschen series (1908)
1
1
ν = R( − 2 ),
9 n
n = 4, 5, 6...
(6)
1
1
− 2 ), n = 5, 6, 7...
16 n
(7)
• Brackett series (1922)
ν = R(
Clearly H-atom energy levels vary as ∼ 1/n2. Here’s a Maple plot of
the Balmer and Lyman series.
6000
5000
4000
3000
2000
1000
0
0.2
0.4
0.6
0.8
1
x
3.3
Bohr model
Bohr quantized the classical model of the H atom, namely an electron
orbiting a fixed proton in a circle of radius a. Did so using criterion
3
suggested by Ehrenfest, that angular momentum be quantized:
L=
nh
≡ nh̄,
2π
n = 1, 2, ...
(8)
Classically,
mv 2
e2
centripetal force
= Coulomb force (SI units)
,
a
4π²0a2
(9)
and the L = nh̄ gives
4π²0n2h̄2
a=
≡ n2 a 0 ,
2
me
(10)
where the smallest allowed radius, the Bohr radius, is given by
4π²0h̄2
a0 =
' 0.5A,
me2
(11)
and the energy is
1 2 e2 −e2
E = mv − =
2
a
2a

2
Ã
!
m  e2  1


= −
2h̄2 4π²0 n2
(12)
(13)
explains observed spectra remarkably accurately.
3.4
de Broglie wavelength
Is light a particle (Einstein photoelectric effect picture) or a wave (classical
optics)? Prince Louis de B. said in his 3-page PhD thesis:
1. wave phenomena characterized by wavelength λ = distance between
fronts of constant phase φ.
4
Plane wave E ∼ exp i(k · r − ωt):
φ = k · r − ωt = phase of wave
λ = 2π/k = distance to make φ change by 2π
c = ω/k = phase velocity of wave
(14)
(15)
(16)
2. particle phenomena photons have energy
E = h̄ω
(17)
and momentum (relativity!)
E
= h̄ω/c = h̄k
(18)
c
Since photon travels in direction of k, take p = h̄k in fact. Compton
(1922) found electrons recoil off x-rays (photons) as if x-rays were particles,
energy h̄ω, momentum h̄k.
de Broglie: suppose electrons and other particles have a wave nature
like photons, with
p=
E = h̄ω, p = h̄k.
(19)
Assume E = p2/2m for nonrelativistic particles, get “dispersion relation”
(relation between ω and k)
h̄2ω 2
h̄ω =
=⇒ ω = h̄k 2/2m
2m
5
(20)
Success of de Broglie picture: “justification” of Bohr-Ehrenfest angular
momentum quantization. Suppose electron viewed as wave moving in
circular orbit around proton. Integral number of wavelengths fit into circle!
But wavelength is length for wave to change phase by 2π, so λ = 2π/k =
2πh̄/p,
λ = h/p de Broglie wavelength
(21)
and angular momentum L = ap = nh/2π, so
L = nh̄
3.5
(22)
Motion of Wave Packet
de Broglie noted plane waves ψ(r, t) ∼ exp i(k · r − ωt) can be linearly
superposed to obtain more general “wave function” representing electron
somehow. 1D example:
ψ(x, t) =
Z ∞
i(kx−ωt)
f
(k)e
dk
−∞
h̄k 2
ω =
2m
Choose f (k) roughly as shown, centered around k0.
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(23)
When is ψ(x, t) large? Consider phase φ = kx − ω(k)t. For most
values of x and t, phase is pure oscillatory as fctn. of k =⇒ ψ small. If
however
∂φ
= 0 at k = k0,
∂k
(24)
then exp iφ has a max/min near k0 and ψ will be large!
So condition for ψ to be large is
∂
(kx − ω(k)t), or
∂k
∂ω
x =
t,
∂k
0 =
7
(25)
(26)
so the peak (pt. of large ψ) advances with speed ∂ω/∂k ≡ group velocity vg = h̄k/m. Compare to phase velocity Eq. (16), vp ≡ ω/k =
h̄k/2m.
Q: What type of wave equation do such strange de Broglie packets
satisfy?
3.6
Expts. Suggesting Wave Nature of Electrons
1. W. Elasser (1925): predicted electrons scattered off a crystal should
show interference pattern, like optical diffraction grating. Observed
by Davisson & Germer.
2. Ramsauer effect. Beam of slow electrons in noble gas has mean free
path À expected from known sizes of atoms.
Classical. If σ is cross-sectional area of atom, line of length L hits all
atoms within cylindrical volume σL around it. Contains on avg. σLn
atoms, so mean free path (average distance between collisions) is (σn)−1.
Quantum. Elasser pointed out electrons would be scattered less if they
really behaved as de Broglie waves, provided λdB À size of atom. Electrons of suff. small momentum don’t “see” atoms at all!
Classical analog. Q: how can we see through air?
Crude estimates:
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• What is n, # of air molecules/ vol? Suppose ideal gas, p = nkB T .
Atmos. pressure is ∼ 106dynes/cm2, T ∼ 300K yields
106
n ∼ −16
∼ 3 × 1019atoms/cm3
10 · 300
(27)
• Geometric size of atom: σ ∼ π(10−8cm)2 ∼ 3 × 10−16cm2.
• Mean free path ` ∼ 1/(σn) ∼ 10−4 cm =⇒ photons should be
scattered and attenuated over tiny distance. Must be that wavelength
of light ∼ 104A, is so large that atoms don’t influence it–doesn’t “see”
atoms.
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