1.
Carefully define or explain:
(1)
(2)
(3)
(4)
Correspondence principle
Uncertainty principle
Probability current density
Pauli’s exclusion principle
(5)
(6)
(7)
(8)
(9)
Zero point energy
Fermi golden rule
Stark effect
The postulate of special relativity
Comparison of the Bose-Einstein, Maxwell-Boltzmann and Fermi-Dirac
distributions
(10) Bohr’s model of the atom
(11) Franck-Hertz experiment
(12) The origin of the fine structure and hyperfine splitting of hydrogen
2. Describe the differences between the Schrödinger’s and Heisenberg’s views of
Quantum Mechanics.
3. (a) Find the eigenfunctions and the energy spectrum of a particle in the potential
well given by V ( x)  0 if
x  a and V ( x)   if
x  a.
(b) Find the coordinate and the momentum matrices in the energy representation
of a particle in the potential well.
4. Show that if the two Hermitian operators A and B satisfy the commutation
relation AB-BA=iC, the following relation will hold:
(A) 2
(B) 2 
C
2
.
5. Prove if two operators A and B commute, they share common eigenstates.
6. Find the wave functions and energy levels of the stationary states of a two
-particle plane rotator with a moment of inertia equal to I=a2, where is the
reduced mass of the this poir of particle and a is their distance apart.
7. 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 the center (but the center
point is fixed). (a) Find the allowed energies of the rigid rotor. (b) What is the
degeneracy of the nth energy level?
8. 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 the possible energies of
this system, and for the isotropic case ( calculate the degeneracy of
the level En.
9. Find the energy spectrum of a system whose Hamiltonian is
H  H 0  H1  
2 d 2 1
 m 2 x 2  ax 3  bx 4 ,
2
2m dx
2
Where a and b are small constants (the “anharmonic oscillator”).
10. (a) Find the bound states energy spectra for an attractive potential
V ( x)  aV0 ( x) .
(b) Scattering from a delta-function well.
A particle is moving along the x-axis. Find the probability of transmission of
the particle through a delta-function potential barrier at the origin
V ( x)  aV0 ( x) .
11.(a) Determine the energy levels and the normalized wave functions of a particle in
a “potential well”. The potential energy V of the particle is
V  ,
V  0,
x0
and
x  a;
0  x  a.
(b) Calculate the expectation values of x, p, and the uncertainty x and p for
ground state.
(c) Show the condition, with which the above result agrees with the corresponding
classical result.
(d) Explain the fact that the ground state energy of a particle in the potential well
is different from zero.
12. Suppose we have two particles, both of mass m, in the previous infinite square
well. Find the ground state and excited state wave functions and the associated
energies for
(a) if the two particles are non-interacting,
(b) if the two particles are identical bosons, and
(c) if the particles are identical fermions.
13. A particle is in the ground state in a box with sides at x=0 ad 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 momentum in the range (p, p+dp)?
14. Let s1 and s2 be the spin operators of two spin-1/2 particles. Find the simultaneous
eigenfunctions of the operators s2 and sz, where s=s1+s2.show that these are also
eigenfunctions of the operators 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 and –1/2?
What is the average value of the component of the spin along the z’ axis?
16. Consider two electrons in a spin singlet state. If a measurement of the spin of one
of the 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 possible energies 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 x) of a particle is of the form
 x,0 
  x2 
1
 2  .
exp
(2 )1 / 4
 2 
Find the change in time of this wave-packet if, for t>0, no force acts on the
particle.
19. Consider a particle subject to a constant force F in one dimension. Solve for the
propagator in coordinate space.
20. A plane rigid rotator having a moment of inertia I and an electric dipole moment
d is placed in a uniform electric field E. By 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 uniform
electric field given by  (t ) 
A
e  ( t /  ) , where A and  are constants. If at t=-,
2

the oscillator is in its ground state, find, to the first order approximation, the
probability that it will be in its first excited state at t=.
22. Consider the three-dimensional infinite cubical well
0, 0  x  a, 0  y  a, and
V ( x, y , z ) 

otherwise
0 za
(a) Find the stationary states.
(b) Find the ground state and the first excited states and their associated
degeneracy.
Now let’s introduce the perturbation
V if 0  x  a / 2 and 0  y  a / 2
H1  0
0
otherwise
(c) Find the first-order correction to the ground state energy, and
(d) Find the first excited state.
23. Use the Gaussian trial function to obtain the upper bound-of the ground-stats
energy of the one-dimensional harmonic oscillator.
24. Find the total cross-section  the  scattering of slow particles by the
spherical potential well V (r )  V0 if r  r0 and V (r )  0 if r  r0 .
25. A charged-particle linear harmonic oscillator is in its ground state when a
time-dependent electric field is suddenly switched on. Calculate the probability of
the excitation of the nth level of the oscillator, assuming that the perturbation
theory is in appropriate in this case.
26. Prove if two operators A and B commute, they share common eigenstates.
27. Describe the differences between the Schrodinger’s and Heisenberg’s views of
Quantum Mechanics.
28. Write down the Hamiltonian in Quantum Mechanics for a charged particle
moving in electromagnetic field.
29. Find the bound states (E < 0) for a delta potential, V=  V ( x ) .
30. A charged particle with mass m is constrained to move on a spherical shell
(r   ) in a weak uniform electric field E. Obtain the energy spectrum to
second order in the field strength.
31. (a). The eigenvalue equation of the harmonic oscillator is given as:
P2 1
(
 m 2 X 2 ) | E   E | E 
2m 2
We usually manage the harmonic problem in the X space with the Hermite
polynomials as its wave functions, formulate this problem in number space instead
using the following lowering and raising operators:
aˆ  (
1
1
m 12
1
m 12
1
) xˆ  i (
) 2 pˆ and aˆ  (
) xˆ  i (
) 2 pˆ
2
2m
2
2m
Find out a | n
and a  | n
with proper coefficients.
(b). Calculate the expectation values 0 | P̂ 2 | 0 , and 3 | ˆ 2 | 2 .
32. (A) Solve the time-independent Schrodinger equation with the time-independent
perturbation method. Find the first- and second-order corrections to the energy
and the first-order correction to the wave function. If the unperturbed states are
two-fold degenerate, find out the first-order correction to the energy.
(B) Consider a quantum system with just three linearly independent states. The
Hamiltonian, in matrix form, is
1  

H  V0  0
 0

0 0

1 
 2 
where 0 is a constant and ε is some small number.
(a) Write down the eigenvectors and eigenvalues of the unperturbed
Hamiltonian(ε=0).
(b) Solve for the exact eigenvalues of H. Expand each of them as a power series
in  , up to the second order.
(c) Use first- and second-order nondegenerate perturbation theory to find the
approximate eigenvalue for the nondegenerate state of unper turbed H.
Compare with the exact result from (b).
(d) Use degenerate perturbation theory to find the first-order correction to the
two degenerate states of the unper turbed H. (Compare the exact results).
33. (A) For the system with a spherically symmertric potential like the Hydrogen
atom, states are with specified quantum numbers n,  , and m. What are the
selection rules for the spontaneous emission?
(B) Derive out these selection rules? Using the commutators of Lz with x, y, and
z, and the following identity
[ L2 , [ L2 , r]]  2 2 (rL2  L2 r)
34. Find the energy spectrum of a system whose Hamiltonian is
2 d 2 1
H  H 0  H1  
 m 2 x 2  ax3  bx 4 ,
2
2m dx
2
Where a and b are small constants (the *anharmonic oscillator*)
35. A particle is moving along the x-axis. Find the probability of transmission of the
particle through a delta-function potential barrier at the origin.
36. A time t=0 a 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.
37. Consider two electrons in a spin singlet state. If a measurement of the spin of one
of the electrons shows that it is in a state with s y=1/2, what is the probability that
a measurement of the x-component of the spin yields sx=-1/2 for the second
electron?
38. Show that if the two Hermitian operators A and B satisfy the commutation
relation AB-BA=iC, the following relation will hold:
C
(A) 2 (B) 2  2
39.



(a) If φi (x) and φj(x) are two different non-degenerate eigenfunctions of the
time independent SchrÖ dinger equation for a potential V(x). Show that
i ( x) j ( x)dx  0
(b) Write down the form of normalized anti-symmetric total eigenfunctions of the
ground states of the lithium atom
40. (a) Show that the concentration of electrons in the conduction band of an
intrinsic semiconductor at temperature T is
nc  2(2me kBT / h2 )3 / 2 e
  E g / k BT
(b) Show that the concentration of holes in the valence band of an intrinsic
semiconductor at temperature T is
np  2(2mp kBT / h2 )3 / 2 e / k BT
where kB=Boltzman constant, me= effective mass of electron, mp= effective
mass of hole, μ=chemical potential, Eg=band gap between conduction band
and valence band. Note that


0
x1 / 2e x dx   / 2 .
41. Consider a central potential
V(r) =
0　for r  R
　for r  R
In this case, we say that we are considering a hard sphere of radius R. If the
incident energy of a particle is low, using the method of partial wave to calculate
the total scattering cross section.
42. Show that for any normalized |Ψ>,<Ψ|H|Ψ≥ E0,where E0 is the ground state
energy (i.e. the lowest eigenvalue). And show that if |δΨ> is a small deviation
from the ground-state |Ψ>, the lowest order of the deviation of <Ψ|H|Ψ>from
E0 is (δΨ)2.
43. If |n> with n=0,1,2,3, .., are the eigenstates of the number operator Nˆ  aˆ  aˆ
of a one-dimensional simple harmonic oscillator, calculate the matrices of the
position operator X̂ and the momentum operator P̂ based on the basis set
of{ |n >}.
44. Find the uncertainty relation between  , the rotation angle about the z-axis, and
Lz, the z component of the angular momentum.
45. A particle is in a potential V(x)=V0sin( 2x / a ), which is invariant under the
transformation x→x+ma, where m is an integer. Is momentum conserved?
Discuss the eigenvalues and eigenstates of the one-dimensional Hamiltonian.
46. 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
lastly R(-  y ,j) is equivalent to R(-  x  y ,k). Then use this identity to show that
[Lx,Ly]=iħLz. (This is called consistent test).
47. Consider a particle in a state described byΨ=N(x+y+2z)exp(-αr), where N is a
normalization factor. Show that the probabilities of finding the Lz eigenstates are
P(1z=0)=2/3, P(1z=+h)=1/6, and P(1z=-h)=1/6.
 
48. Construct the four antisymmetrized wavefunctions  1 ,  2 X  1 , 2 ) of a
two-electron system, whereσ1 andσ2 are the spin states of the two electrons
and X  1 ,  2  * is the total spin state.Ψ(r1,r2) is the orbital part of the


wavefunction. Assume that the two electrons occupy the a r andb r  orbitals.
49. If a proton has a uniform charge distrubution of radius R, the attractive potential
between
the
electron
and
the
proton
will
be
V (r )  
3e2 e2 r 2

2 R 2 R3
e2
for r>R. Calculate the first order shift in the
r
ground-state energy of hydrogen. You may assume R <a0 (the Bohr radius).
50. For the attractive delta function potential V x   aV0 S x use a Gaussian trial
for r ≤ R and V(r)= 
function to calculate the upper bound on E0, the gound state energy.
51. When a particle is scattered from a square well potential of depth V0 and
k

range  0 , show that the s-wave phase shirt is  0  kr0  tan 1  tan k ' r0  ,
 k'

where k and k’ are the wave numbers inside and outside the well, respectively.


52. Find the operator Tˆa of a parallel displacement over a finite vector a in terms
of the momentum operator P̂
 
 
Tˆa (r )   (r  a )
From Taylor expansion,
 2
 

 1
(r  a )  (r )  r  a  
ai a j  ...
2! X i X j 
r

 1 
  (r )  r  a  (a  ) 2  r  ...
2!


a 

 
e

e

e
So Tˆa 
e
i
 a  i 

i
 a  pˆ


 

 
i
a  pˆ

53. A particle of energy E is incident upon a potential barrier of height V0 and width
d (Fig.1). Calculate the transmission probability of the particle through the barrier
as a function of E , where E  V0 . And locate maxima and minimum in the
transmission probability.
V0
d
54. Start from the infinitesimal time evolution operator
m 12
i m( x  x' ) 2
x  x'
) exp{ [
 V (
,0)]}
2i

2
2
Given by the path integral, prove  ( x,  ) satisfying the Schrödinger equation
 ( x,  , x' , o)  (
 i   2  2
 ( x,  )   ( x,0) 
[
 V ( x,0)]( x,0)
 2m x 2
55. Calculate  ' m'[ L , L ] m 
Where L  Lx  iL y and
Lz , m  m , m 
L2 , m  (  1) 2 , m 
a
1 ˆ 1 ˆ
56. A spinor   is rotated by 30 with respect to the rotating axis nˆ 
i
k.
2
2
b
What is the state after the rotation?
57. In the representation of the eigenvectors of S z which is the z component of the

1
 1 0 
spin operation S with S= , S z is written as S z  
. In the same
2
2 0  1

representation, another spin operator  is defined by

=

3

2

2 2
2
i
4
 4
2
2
i

4
4 
3


2
Initially, we have a state which is eight states of Sz with an eight value of

2

Now we make a measurement of  .


?

(b)If we make a measurement of S z right after（a）, what’s the probability of finding the
(a)What’s the probability of getting states ofter  measurement


states?
2
58. A fru particle is restricted to m ove in
particle is
 ( x)  sin
0  x  L if the wavefunction of this
x
L
 2 sin
3x
L
甲、calculate <X>

乙、calculate  H  ,and (c) the probability of finding the particle in the region
L
2
59. A beam of particle is scattered by a potential v(r )  V0 (r0  r ) . Show that
0 x
2
d
2 mv0 r0
2 (sin qr 0  qr0 wqr0 )
 4r0 ( 2 )
d

(qr0 )6
2




where  is the step function and q  K f  K i , K i and K i are the incident and
scattered wavevectors of the particle, respectively.
60. A system has a Hamiltonian of
Pˆ 2 1 ˆ 2
Hˆ 
 KX  Xˆ 3 ,
2m 2
where k and  are constants.
What the following symmetries the system has, translation, parity, time translation
and time reversal? And why? What conserved quantities the symmetries
correspond with?
61. Calculate the following expectation values,




 1 X 1 ,  2 X 2 ,  1 P 1  and  2 P 2 
for a simple homonic oscillator where states l1  and l 2 > are eigenstates of
3
5
 and  respectively.
2
2
Ĥ with eigenvalue
62. The creation and annihilation, â  and â for fermions which operators.
aˆaˆ   aˆ  aˆ  Iˆ ( Iˆ : Identity operator), aa  0 and a  a   0
甲、Show that the fermion number operator Nˆ  aˆ  aˆ




Satisfies aˆ  N  (   Nˆ )aˆ 
aˆ  N  (   Nˆ )aˆ
乙、If one eigenvalue of N̂ is zero, the only other eigenvalue, is=1.