PHY662 - Quantum Mechanics II
HWK #11, Due Thursday, Apr. 29,
start
of class
•
Reading: pp. 506 to 521 of Shankar.
1.
Beyond the dipole.
[3 pts] Exercise 18.5.1 in Shankar: (1) Show that the
ei~
k·~r
factor can be kept by replacing
~pf
by
~pf−¯h~
k
in Eq. (18.5.9). (2) Verify
that the nal electron momentum is then biased towards
~
k
, reecting the
¯h~
k
linear momentum imparted to the electron by the electromagnet eld.
2.
Plugging in numbers.
[4 pts]
(a) Carry out this simple practice exercise: that
e2
¯hc
is dimensionless and
calculate its value, using SI values for
¯h
and
c
and converting
e2
to
SI units by tricks we have used in other homework problems.
(b) Exercise 18.5.2 in Shankar, part (1): Estimate the photoelectric cross
section for an electron ejected from a hydrogen atom when the ejected
electron has a kinetic energy of 10 Ry. Compare this cross section
with the atom's geometric cross section
≈πa2
0
.
3.
Playing in oscillator world.
[3 pts; This question is based on Tuesday's
lecture, the bulk of which is posted online already.] Consider a harmonic
oscillator coupled to a bath of harmonic quantum oscillators, as described
in Tuesday's lecture. Let
c
be a constant with the dimensions of velocity.
Explore the consequences of modifying the perturbation discussed in class,
H0=gN −1/2xPixi
, to the slightly more complicated form
H0=gN −1/2X
i
e−ωix/cxxi.
In the limit where the matrix elements of
ωix/c
are small, but not neg-
ligible, compared with
1
, how does this modication aect the rates of
dierent transitions?
1