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Applied Quantum Mechanics
Chapter 11 problems
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Applied quantum mechanics
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Problem 11.1
Derive the following commutation relations:
(a)  L̂ z x̂  = iŷ
(b)  L̂ z ŷ  = – i x̂
(c)  L̂ z ẑ  = 0
2
2
(d)  L̂  x̂  = – 2 x̂ + 2i  L̂ z ŷ – L̂ y ẑ 
2
2 ˆ
2ˆ
2 ˆ 2
(e)  L̂   L̂  r   = 2  r L̂ + L̂ r 
Problem 11.2
The ground state wave function of a hydrogenic atom with nuclear charge Ze is
–r  r
 1  r  = Ae 1 , where r is the distance between the electron and the nucleus and r 1 is
a characteristic length scale. The electron is subject to a radially-symmetric coulomb
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potential given by V  r  = – Ze  4 0  r r .
(a) Find the value of the normalization constant A.
(b) Find the value of r 1 that minimizes the energy expectation value  E 1 .
(c) Use the value of r 1 in (b) to calculate the ground state energy.
(d) Show that  E kinetic = –  E potential  2 (which is a result predicted by the virial
theorem).
(e) Show that the peak in radial probability occurs at r = a B  Z .
(f) Show that the expectation value  r = 3a B  2Z .
(g) Show that the expectation value of momentum  p = 0 .
Problem 11.3
To calculate the spontaneous emission rate
A = 1   sp
for the
2p  1s
( n = 2 l = 1 m  n = 1 l = 0 m = 0 ) transition in hydrogen one usually averages
over the three possible values of the quantum number m so that
2
3
m=1
e 1
- ---  2 1 mr̂ 1 0 0 2
A = ------------------3
3c  0 3 m = –1
2
2
2
2
where  is the energy of the emitted photon. Since r = x + y + z , this equation
can be written as
2
3
m=1
e  - 1--2
2
2
A = ------------------  2 1 mx̂ 1 0 0 + 2 1 mŷ 1 0 0 + 2 1 mẑ 1 0 0 
3
3c  0 3 m = –1
(a) Show that
1
–1
2
x = r sin    cos    = – ------r  Y 1 – Y 1 
3
1
–1
y = r sin    sin    = i 2
------r  Y 1 + Y 1 
3
z = r cos    =
2
0
4
------rY 1
3
and rewrite each matrix element appearing in the expression for spontaneous emission
in terms of a radial integral and an angular integral.

n – x
(b) Use the standard integral  x e
dx = n!
–n–1
to show that the radial integral
0

8
2 3 *
aB
 r R 21  r R 10  r  dr = ----------4
3
0
6
2
1 3  2 r –r  2aB
-------- e
and R 21  r  = ---------  --------
2a B
3  2a B
(c) Show that the angular integrals in (a) are
1
where R 10  r  = 2  -----
 a B
2
– -----3
i 2
-----3
4
-----3
32
e
– r  aB
 =   = 2
– 1 2
m *
1
–1
0
-   m 1 –  m –1 
  Y 1   Y1 – Y 1 Y 0 sin    d d = ---------- ----3

4
=0 =0
 =   = 2
i
m *
1
–1
0
2- 
 m 1 +  m –1 
  Y 1   Y 1 + Y 1 Y 0 sin    d d = ---------- ----3
4
=0 =0

 =   = 2

1 4
m * 0 0
-   m 0 
  Y 1  Y 1 Y 0 sin    d d = ---------- ----3
4
=0 =0
(d) Combine the results of (b) and (c) to show that
m=1
10
3
2
 32- a 2
 2 2
 21mr̂ 100 = 96  --3- aB =  ----27 B
m = –1
and that the spontaneous emission time for the 2p  1s transition in hydrogen is
 sp = 1.6 ns .
Problem 11.4
(a) Show using first-order perturbation theory that the correction to the 1s ground-state
energy of a hydrogen atom subject to a uniform electric field E in the z direction is zero.
(b) Show that the first-order correction to the ground-state wave function is
1
 0  = 1 0 0 – e E
n l mẑ 1 0 0
- n l m
 -------------------------------------E1 – En
n  1 l m
(c) Show to first-order in E that the susceptibility for the 1s state is
1
1
 0 eẑ  0 
- = – 2e 2
 1s = ----------------------------E
2
n l mẑ 1 0 0
 ------------------------------------------E –E
n  1 l m
1
n
Problem 11.5
(a) An electron with zero orbital angular momentum (l = 0 ) moves in a radial potential V  r  = 0 for r  a and V  r  =  for r  a , where a is the radius of a spherical
quantum dot. Use the radial Schrödinger equation to find the eigenenergies and normalized eigenstates of the electron.
(b) Find the eigenenergies and normalized eigenstates of an electron with zero
orbital angular momentum ( l = 0 ) moving in a radial shell potential with V  r  = 0
for a  r  b and V  r  =  elsewhere.
Applied quantum mechanics
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Problem 11.6
Show that the radial expectation value  r for a hydrogen atom in state  nlm is
2
2
a
n a
2
2
2
 r = ----B-  3n – l  l + 1   and that  r  = ----------B-  5n + 1 – 3l  l + 1   , where a B is the
2
2
Bohr radius.
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