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PHY662, Spring 2004, Apr. 13, 2004
15th April 2004
1 Miscellaneous
- Reading: Continue Shankar Ch. 18 for electromagnetism, also Griffiths Ch. 9 (though it is a bit simplified, especially as it sticks to two-level systems at first). Skip higher order perturbation theory in Shankar and read up to “Field Quantiza- tion” (p. 506 in the second edition) by Tuesday, April 13. Much of what follows in today’s notes is derived from Shankar and Ch. 13 of G. Bayms’ Lectures on Quantum Mechanics.
- Office hours are back to 3:30-5:00 on Monday. This week’s homework will be in your mailboxes later this afternoon and is due Tuesday, April 20.
2 Electromagnetic waves
Remember the wave equation for the vector potential:
∇^2 A~ −
c^2
∂^2 A~
∂t^2
These equations give that waves in A~ travel at speed c and that plane wave solutions for A~ are of the form A~ = A~ 0 cos(~k · ~r − ωt)
with k^2 c^2 = ω^2 and with the important resulting condition from the Coulomb gauge, ∇ · A~ = 0 (transverse waves) ~k · A~ 0 = 0.
2.1 Aharonov-Bohm effect
Clarification from last lecture: one can set A~ = 0 over any region where B~ = 0 by carrying out a gauge transformation with ∇Λ = − A~. This gauge change modifies the wave function at ~r 1 by a well-defined phase change,
ψ(~r 1 ) → e
−i(q/¯hc)
∫ (^) ~r 1 ~r 0 A^ ~·d~s ψ(~r 1 ).
There are no observable effects within this region from magnetic fields. But relative changes in phases along two paths can lead to changes in interference: a quantum particle can explore more than one path simultaneously, so changes in A~ can lead to observable effects! If we compare the phase change due to A~ for two paths of a particle about a solenoid, we can see this interference effect, whenever the flux is not a multiple of the flux quantum hcq. This interference effect has applications in imaging. Tonomura’s group at Hitachi lab has imaged vortices of magnetic flux in superconductors using this effect. Here is an example - electrons that pass through a superconductor (niobium) subject to a magnetic field interfere with electrons that pass by the superconductor. This gives an interference pattern / hologram that can be used to infer the magnetic field in the superconductor. Note that the field is not uniform!
This sensitivity to magnetic fields is used in SQUID (Superconducting Quantum Intef- erence Device) or even for electrons in mesoscopic devices.
2.2 Electromagnetic modes
We can simply express solutions to the wave equation for A~ using plane waves in a box of volume V as
A^ ~(~r, t) =
k~λ
V
[
Ak~λ~λ(~k)ei( ~k·~r−ωt)
- A∗ kλ~λ∗(~k)e−i( ~k·~r−ωt)] ,
where the second term is the c.c. of the first to ensure that A~ is real (which it must be, in order for particle conservation to hold and for B~ to be real) and the V −^1 /^2 factor is a convenient normalization. The Coulomb gauge condition ∇ ·~ A~ = 0 is satisfied iff ~k · ~λ = 0. The polarization vectors ~λ can be complex, but one often chooses two plane
polarizations, with λ~ 1 , 2 ⊥ ~k, ~λ 1 ⊥ ~λ 2. The scalar A~kλ gives the amplitude and phase of the wave.
we get
Γ 0 →n =
2 πe^2 ¯hc^2 (2πc)^3
dωdΩω^4 |A~k~λ|^2 |λ · 〈n|~r| 0 〉|^2 δ(En − E 0 − ¯hω)
2 πe^2 ω^4 ¯h^2 c^2 (2πc)^3
dΩ|A|~k|=ω/c,~λ|^2 |λ · 〈n|~r| 0 〉|^2.
Consider radiation incident upon the atom from an incoherent polarized source with an intensity measured in energy per unit solid angle per frequency interval, I(ω). It turns out that
I(ω) =
dΩω^4 |Akλ|^2 (2πc)^4
Substituting this in gives
Γ 0 →n =
2 πe^2 ω^4 ¯hc^2 (2πc)^3
(2πc)^4 ω−^4 I(ω)|λ · 〈n|~r| 0 〉|^2.
What can we infer from this form of the rate? ∑ [Can work with total position operator r = R. Note that [Lz , Rz ] = 0, [Lz , Rx] = i¯hRy , to get selection rules for m′, m and can also get l′^ = l ± 1 .]
[Note: this type of calculation is from Baym. Shankar considers the photoelectric effect, where the final state is a plane wave state. There, you also have to be careful in the initial state - it is easiest for an s-state. In this case, the matrix element is
( (^) e
2 mc
) (^1
2 π¯h
(πa^30 )−^1 /^2
d^3 ~r e−i~pf^ ·~r/¯h^ A~ 0 · (−i¯h∇)e−r/a^0 = N A~ 0 · ~pf
d^3 ~r e−i~pf^ ·~r−r/a^0
8 πa 0 N ( A~ 0 · ~pf ) [(1/a 0 )^2 + (pf /¯h)^2 ]^2
and inserting into the golden rule gives
2 π ¯h
N 2 ( A~ 0 · ~pf )^2 64 π^2 a^60 [1 + (pf a 0 /¯h)^2 ]^4
δ(Ef − Ei − ¯hω)
and the transition rate into the angle dΩ of
Γi→dΩ = 4 a^30 e^2 pf | A~ 0 · ~pf |^2 mπ¯h^4 c^2 [1 + (pf a 0 /¯h)^2 ]^4
dΩ.
This is the transition rate for the photoelectric effect.
