量子力學
1. Carefully define or explain:
(1) Correspondence principle
(2) Uncertainty principle
(3) Probability current density
(4) Pauli’s exclusion principle
(5) Zero point energy
(6) Fermi golden rule
(7) Stark effect
(8) Describe Bose-Einstein, Maxwell-Boltzmann and Fermi-Dirac
distributions
(9) Franck-Hertz experiment
(10) The origin of the fine structure and hyperfine splitting of hydrogen
2. Describe Schrödinger’s and Heisenberg’s pictures of quantum mechanics.
3. (a) Find eigenenergies and corresponding normalized eigenfunctions of a particle
in the potential well given by V(x)=0 if |x|<a and V(x)=∞ if |x|>a.
(b) Find the position and momentum matrices in the basis set composed of the
eigenfunctions of a particle in the potential well given in (a).
4. Show that if the two hermitian poerators A and B satisfy the commutation
relation [A,B]=iC, the following relation will hold:
(A) 2 (B) 2 
5.
C
2
Prove if two operators A and B commute, they share common eigenstates.
6. Two particles of mass m are attached to the ends of a massless rigid rod of length
a. The system is free to rotate in three dimensions about a fixed center
(a) Find eigenenergies of this rigid rotor. (b) What is the degeneracy of the nth
energy level?
7. A system described by the Hamiltonian
2 2 m 2 2
H 
  (1 x   22 y 2  32 z 2 )
2m
2
is called an “anisotropic harmonic oscillator’’. Determine eigenenergies of this
system, and for the isotropic case (ω1=ω2=ω3=ω) calculate the degeneracy of the
eigenenergy En   (n  3 2 ) , where n is a positive integer.
8. Find eigenenergies of an anharmonic oscillator with a Hamiltonian given by
2 d 2 1
 m 2 x 2  ax 3  bx 4 ,
2m dx 2 2
where a and b are small real constants.
H 
9. Find bound-state eigenenergies of a particle in an attractive potential,
V(x)=–aVo  (x).
10. A particle is moving along the x-axis. Find the transmission probability of this
particle through a delta-function potential barrier at the origin V(x)= aVo  (x).
11. (a) Determine eigenenergies and corresponding normalized eigenfunctions of a
particle in a potential well
V =  , x < 0 and x > a ;
V = 0,
0< x < a .
(b) Calculate the expectation values of x and p and uncertainties  x and  p in the
ground state.
(c) Explain why the ground-state energy of a particle in the potential given in (a)
is different from zero.
12. Suppose we have two particles, both of mass m, confined in 0 < x < a described
by a potential V = 0 for 0 < x < a and V =  for x < 0 and x > a. Assume that
these two particles are not interacting with each other. Find the ground-state and
first excited-state eigenenergies and eigenfunctions, if they are (a) identical
fermions and (b) identical bosons.
13. A particle is in the ground state in a box with sides at x=0 and x=a. Suddenly the
walls of the box are moved to   , so that the particle is free. What is the
probability that the particle has a momentum in the range (p,p+dp)?
14. Let S1 and S2 be the spin operators of two spin-1/2 particles. Find the common
eigenstates of the operators S2 and Sz, where S=S1+S2. Show that they are also
eigenstates of the operator S1•S2.
15. Suppose an electron is in a state, in which the component of its spin along the
z-axis is +1/2. What is the probability that the component of the spin along an axis
z’(which makes an angle  with the z-axis) will have the value +1/2 or -1/2?
16. Consider two electrons in a singlet spin state. If a measurement of the spin of one
of these two electrons shows that it is in a state with Sy=1/2, what is the
probability that a measurement of the x-component of the spin yields Sx=–1/2 for
the second electron?
17. Find the ground-state energy of a particle in the spherical potential well given by
V(r)=– V0 if r < a, and V(r)=0 if r > a.
18. Assume that at time t=0 the wave function of a particle is of the form
 ( x,0) 
1
 
2 1/ 4
  x2 
exp  2  . Find  ( x, t ) .
 2 
19. A particle is subject to a constant force F in one dimension. Solve for the
propagator in the coordinate space.
20. A plane rigid rotator with a moment of inertia of I and an electric dipole moment
of d is placed in a homogeneous electric field E. Considering the electric field as a
perturbation, determine the first non-vanishing correction to the energy levels of the
rotator.
21. A charged-particle linear harmonic oscillator is in a time-dependent homogeneous
A

2

, where A and  are constants. If at
2

t=–  , the oscillator is in its ground state, find to the first order approximation the
probability that it will be in the first excited state at t=  .
electric field given by E (t ) 
exp  t
22. A particle is confined in a cubic box and its potential can be described by
 0 0  x  a,0  y  a, and 0  z  a
V ( x, y , z )  
otherwise

(a) Find eigenenergies and corresponding eigenfunctions.
(b) If there is a perturbation described by
V 0  x  a / 2,0  y  a / 2, and 0  z  a / 2
H1   0
,
otherwise
0
where V0 is a small constant potential. Find the first-order correction to the
ground-state and first excited-state energies.
23. Using a Gaussian trial function to obtain the lowest upper bound of the
ground-state energy of the one-dimensional harmonic oscillator.
24. Find the total cross-section for the scattering of slow particles by a spherical
potential well V(r) = - V0 if r < r0 and V(r) =0 if r > r0 .
25. Write the Hamiltonian of a charged particle moving in the electromagnetic field.
26. A charged particle with mass m is constrained to move on a spherical surface (r
=) in a weak uniform electric field E . Obtain the approximate eigenenergies to the
second order in the field strength.
27. The Hamiltonian operator of a harmonic oscillator is given as
P2 1
(
 m 2 X 2 )
2m 2
Using the lowering and raising operators,
a
m
1
X i
P
2
2m
m
1
X i
P
2
2m
to formulate the new Hamiltonian in terms of these two operators and using the
commutation relations to find out eigenenergies.
a 
28. Derive the first- and second-order corrections to the energy and the first-order
correction to the wave function in the perturbation theory. If two unperturbed states
are degenerate, find out corrections to the energies of these two states.
29. Consider a quantum system with just three linearly independent states. The
Hamiltonian, in matrix form, is given by
1  

H= V0  0

 0
0
0 
1  ,
 2 
where V0 is a constant and  is a small real number.
(a) Find exact eigenvalues and corresponding eigenvectors of this Hamiltonian.
(b) Use the first- and second-order perturbation theory to find the approximate
eigenvalues.
30. For a system with a spherically symmetric potential like a hydrogen atom,
eigenfunctions are specified by quantum numbers n, l, and m. Find the selection rule
for the dipole transition involved in absorption and emission of a photon.
31. A time t=0 s spin 1/2 particle with spin in the x-direction enters a region of space
in which there is a uniform magnetic field H in the z-direction. Find the probability
that at time t the spin is still in the x-direction.
32. If  i (x) and  j (x) are two different non-degenerate eigenfunctions of the time
independent Schrödinger equation with a potential V(x), show that



i ( x) j ( x)dx  0 if i  j .
33. The wavefunction of a multi-electron system has to be anti-symmetric with
respect to permutation of any pair of electrons. Write down the normalized
anti-symmetric three-electron wavefuncion of the lithium atom in the ground state.
0 r  R
34. Consider a central potential V (r )  
corresponding to a hard sphere,
 r  R
using the method of partial wave to calculate the total scattering cross section of a
low-energy incoming particle.
35. Show that for any normalized | >, < |H|  E0 , where E0 is the ground-state
energy (i.e. the lowest eigenvalue). And show that if |   is a small deviation from
the ground-state | 0  , the lowest order of the deviation of   |H|  from E0 is
proportional to ( ) 2 .
36. If |n> with n=0,1,2,3….., are the eigenstates of the number operator N= a  a of a
one-dimensional simple harmonic oscillator, calculate the matrices of the position
operator X and the momentum operator P based on the basis set of |n>.
37. Find the uncertainty relation between  , the rotation angle about the z-axis, and
Lz , the z component of the angular momentum.
38. A particle is in a potential V(x)= V0 sin (2 x / a) , which is invariant under the
transformation x  x  ma , where m is an integer. Is the momentum conserved?
Discuss the eigenvalues and eigenstates of the one-dimensional Hamiltonian.
39. Let R(  ,n) be the operator that rotates a vector by  about the axis n. Show that
the four successive infinitesimal rotations, R(x,i), R(y,j), R(-x,i) and R(-y,j) is
equivalent to R(-xy,k). Then use this identity to show that [ Lx , Ly ]  ihLz .(This is
called consistent test).
40. Consider a particle in a state described by   N ( x  y  2 z ) exp( r ), where N
is a normalization factor, Show that the probabilities of finding the Lz eigenstates
are P( lz =0)=2/3, P( lz =+h)=1/6, and P( lz =–h)=1/6.
 
41. Construct the four antisymmetrized wavefunctions (r1 , r2 )  ( 1 , 2 ) of a
two-electron system, where  1 and  2 are the spin states of the two electrons and
 
 ( 1 ,  2 ) is the total spin state. (r1 , r2 ) is the orbital part of the wavefunction.
42. If a proton has a uniformly dense charge distribution of radius R, the attractive
potential between the electron and the proton will be V (r )  
3e2 e2 r 2

for
2 R 2 R3
e2
for r > R. Calculate the first order shift in the ground-state
r
energy of a hydrogen. You may assume R<< a0 (the Bohr radius).
r  R and V (r )  
43. For the attractive delta function potential V ( x)  aV0 ( x) , use a Gaussian trial
function to calculate the upper bound on E0 , the ground-state energy.
44. When a particle is scattered from a square well potential of depth V0 and range
r0 , show that the s-wave phase shift is  0  kr0  tan 1 (
k
tan k ' r0 ) , where k and k’
k'
are the wave numbers inside and outside the well, respectively/

45. Find the operator Tˆa of a parallel displacement over a finite vector a , defined by


 
Tˆa (r )   (r  a ) , in terms of the momentum operator p .
46. A particle with an energy E is incident upon a potential barrier of height V0 and
width d. Calculate the transmission probability of this particle through the barrier as a
function of E, where E  V0 .
47. Calculate  ' m ' | [ L , L ] | m  , where L  Lx  iL y . Note
Lz | , m  m h| , mand L2 | , m  (  1)h2 | , m  .
a 
48. A spinor   is rotated by 300 with respect to the rotating axis,
b 
nˆ 
1
2
iˆ 
1
2
kˆ . What is the state after the rotation?
49. In the representation, in which the basis states are eigenvectors of Sz, Sz is written
 1 0 
as S z  
. In the same representation, another spin operator ̂ is
2 0  1

3


2
ˆ  
defined as 
2 2
2
i
4
 4
of S z yields
2
2
i

4
4  . If in the initial state, a measurement
3


2


. (a) What’s the probability of getting
for a subsequent
2
2
measurement of ̂ ? (b) What’s the probability of getting 

for a subsequent
2
measurement of Sz instead of ̂ ?
50. A free particle is restricted to move within 0  x  L and its wavefunction is
x
 ( x ) s i n 
L
3 x
2 sin
L
Calculate (a) <x>, (b) <H> and (c) the probability of finding the particle in the region
0 x
L
2
51. A beam of particles is scattered by a potential V (r )  V0 (r0  r ) . Show that
2
2
qr0 
d
2  mV0 r0  s i nqr0  qr0 c o s

,
 4r0 
2

d
qr0 6
  
2
where  is the step function and q | K f  Ki | ; K i and Kｆ are incident and
scattered wavevectors of the particles.
52. For an infinitesimal rotation,  , about the unit vector nˆ  (n x , n y , n z ) , prove that
the rotation matrix R can be represented as R = exp ( nˆ  X ) , where
0 0 0 
 0 0 1
 0 1 0




X x  0 0 1 , X y   0 0 0 and X z   1 0 0
 1 0 0
0  1 0
 0 0 0
are three antisymmetric matrices. Find their commutation relations and compare them
with the commutation relations of the components of angular momentum.
53. A particle of mass m confined within a one-dimensional box with 0  x  a is
initially in the ground state E1 . At time t=0, this particle is subject to a perturbation
potential V(x)= V0 if 0  x  a / 2 , V(x)=0 if a / 2  x  a , and V(x)=  ,
otherwise, where V0  E1 . After a time T, the perturbation potential is removed.
After T, what is the probability of this particle in the first excited state E2 ?
54. Calculate the total cross-section of a particle scattering from a Yukawa potential in
the Born approximation. Express your answer as a function of energy E.