Download CHEM 442 Lecture 15 Problems (see reverse) 15

CHEM 442 Lecture 15 Problems (see reverse)
15-1. Suggest a perturbation operator for a z-polarized photon with electric field
amplitude E and angular frequency w .
¶Y 0
, where
¶t
be the zeroth-order Hamiltonian with
15-2. Consider the time-dependent Schrödinger equation ĤY 0 = i
subscript “0” means the ground state. Let Ĥ (0)
{
-iE
known eigenfunctions Y (0)
e k
k
(0)
t/
} and eigenvalues { E } . Let Ĥ
(0)
k
(1)
be the time-
dependent perturbation of 15-1 and, therefore, Ĥ = Ĥ (0) + l Ĥ (1) with l = 1. We
furthermore expand the wave function in an infinite series as
-iE0( 0)t/
+ lY(1)
+… . Substituting these in the time-dependent Schrödinger
0
¶Y 0
equation ĤY 0 = i
and collecting those terms that are proportional to l , find the
¶t
Y0 = Y(0)
e
0
first-order perturbation equation. What are the unknowns in this equation?
¥
()
15-3. Expand the first-order correction to the wave function as Y (1)
= å Ck t Y (0)
e- iEk
0
k
k=0
(0)
t/
and substitute it into the first-order perturbation equation derived in 15-2. Multiply the
equation by Y (0)*
from the left and integrate the result over the whole space. Simplify the
n
integrals using orthogonality,
òY
(0)*
k
òY
(0)*
k
Y(0)
l dt = 0 (k ¹ l) , and normalization,
(1)
Y(0)
. Show the
k dt = 1 . Substitute the perturbation operator of 15-1 into Ĥ
energy conservation condition between the system and a photon: En(0) - E0(0) = ± w .
15-4. On the basis of 15-3, derive Fermi’s golden rule in the form of
2
(0)
2
wn¬0 µ ò Y (0)*
n ẑY 0 dx E , where wn¬0 is the probability of transition from the state 0 to
state n. What is the name of the integral?
15-5. Consider the particle in a box along x axis, Y (0)
n =
ò
L
0
2
np
sin
x . Evaluate
L
L
(0)
with n = 2, 3, and 4. Discuss optical transitions between state 1 to state n
Y (0)*
n xY1 dx
on this basis. Use
ò
p
0
x sin x sin nx dx = -
{
2n 1+ ( -1)
(n
2
- 1)
2
n
} where n is an integer greater than
1.
15-6. Using the recursion relationship of the Hermite polynomials, show that
a Nn
N
xY n =
Y n+1 + na n Y n-1 , where Y n is the nth harmonic-oscillator wave
2 N n+1
N n-1
function (n = 0, 1, 2, …), N n is the corresponding normalization coefficient, and
a =(
2
/ mk ) .
1/4
15-7. Consider the harmonic oscillator wave functions along x axis, Y (0)
. Evaluate
n
ò
¥
-¥
(0)
with n = 1, 2, and 3. Discuss optical transitions between state 0 to state
Y (0)*
n xY 0 dx
n on this basis. Use 15-6.
15-8. In the spherical coordinates, x = rsinq cosj , y = rsinq sinj , z = r cosq . Some loworder spherical harmonics are Y0,0 =
p
2p
0
0
ò ò
Y1,0* xY0,0 sinq dq dj ,
p
2p
0
0
ò ò
1
3
, Y1,0 =
cosq . Evaluate
4p
4p
Y1,0* yY0,0 sinq dq dj , and
p
2p
0
0
ò ò
Y1,0* zY0,0 sinq dq dj .