Part I: 50 %
1. [20%]
(a) [3 %] In the x-representation, x = x and p = h dx . Show
d
i
where is a
that the commutation relation [, p] = i 1,
x
h
1
unit operator.
(b) [7 %] Given a dierentiable function F (X) of operator X.
Evaluate the commutators [F (
Quantum Mechanics Qualifying Exam Part I (50 points) 2014/ 10/ 17
1. Prove the following statements:
(a) The trace of a matrix is unaffected by a unitary transformation. (5 points)
(b) If H is a Hermitian operator, U = 3’” is unitary. (5 points)
2. Consid
Part 1: 50%
1. [10%] The Hamiltonian for an electron in an external constant magnetic ﬁeld in the z direction I? : BE is
r q
H:ﬂia.e,
where 6' denotes the Pauli matrices:
_01 _072' M10
“1—10'Uy’i10 '0” 0—1
If the initial state of the electron is
ta).
find
Quantum Mechanics Qualify Exam Part I (50 points)
2013/10/18
m 2 x 2
1. At time t = 0, a particle in the potential V ( x) =
is described by
2
( x)
the wave function ( x, 0) = nn / 2 , where n ( x) are orthonormal
A
2
n
1
eigenstates of the Hamiltonian wi
Qualifying Exam — Quantum Mechanics Part .I
(50 points)
1. [10 points] Use the commutation relation [£15] = iii. to show that:
[562, 332] = 2ih(p"5c" + 3215)
2. [20 points] Consider the scattering of a quantum particle from a one dimensional
step potentia
Oct. 16, 2009, Qualify Exam, Quantum Mechanics l
1.[10 points] Assume that ‘1! is an eigenfunction of the single—particle Schrodinger equation.
Deﬁne the vector 6' as follows:
i{f\p*\1:dv}— ~/ 6 Cat
alt V h V
(a) What is the physical meaning of a ?
(b) As
Quantum Mechanics Qualifying Exam
Part I (50 points)
1. [15 points] An electron beam is prepared by heating a filament, subjecting the
emitted electrons to a potential difference V0, and using a series of electrostatic
lenses to guide the electrons on a t
Quantum Mechanics Qualify Exam Part I (50 points)
2011/10/14
1. Prove the following statements:
(a) The trace of a matrix is unaffected by a unitary transformation.
(b) If H is a Hermitian operator, U = eiH is unitary. (15 points)
(15 points)
2. Use the r
Apr. 1, 2011,
Qualify Exam, Quantum Mechanics, Part I
1.[10 points] Give a brief discussion of the following topics in quantum mechanics:
(a) Physical interpretation of the wave function . (5 points)
(b) Required relationship between any two dynamical qua
Part II (50%)
I. Assume one electron is in p state, the other electron is in d state, and
their total angular momentum is j . Find all the possible values of J2
and J2; (ignore electron’s spin) [15 points].
2. Let Wm be the eigenfunction of the angular mo
Part I (50 points) 2010/3/26
1. Consider the double-slit experiment as shown in the following figure. What is the
pattern on the screen looks like if 10 particles are shot? (10 points)
2. Consider a particle of mass m in one dimension bound by a -function
Quantum Mechanics Qualify Exam Part I (50 points)
2008/10/17
1. If a particle of mass m is constrained to move in the xy plane on a circular orbit of
radius R around the origin O, but is otherwise free, determine the energy
eigenvalues and the eigenfuncti
97 Quantum Mechanics Part III (50 points)
1. [10 points] Show that in the usual stationary state perturbation theory, if the
Hamiltonian can be written H = H0 +H with H00 = E00, then the correction E0
is
E0 0 H 0 .
2. [20 points] A charged particle is bou
1.
Part 1 (50 points)
Consider a quantum mechanical system which contains two complete
|2> . Three
Q, and R are measured in the system. |1>
and orthonormal energy eigenstates, ID and
observables, P, and
|2> may or may not be the eigenstates of P, Q, and R
Part I (50 points) 2006/3/17
1. If U and V are unitary operators and A is an arbitrary operator,
(a) show that UVU , the unitary transformation of V , remains unitary. (10
points)
(b) show that A and its unitary transformation UAU have the same
eigenvalue
94 10 14
Part I
1. Show that if f ( p ) is a differentiable function of p , then
f ( p )e
iqc
e
iqc
(15 points)
f ( p c)
2. Consider the Schrodinger equation in one dimension with the potential
V ( x ) k 2 x 4 . Use the Heisnberg uncertainty relation t
1.If A is any operator, show that
i
d
A A, H
dt
i
A
t
[10 points]
2.If xp , find the minimum energy of simple harmonic oscillator,
2
where ( p) 2 ( p - p ) 2 [10 points]
3.Show that in the stationary states corresponding to the discrete spectrum of t
Part I (50 points)
1. Operators A and B do not commute with each other.
(a) Expand e x ( A B ) about x=0. To what order in x can e x ( A B ) be
approximated by e xAe xB ? (5 points)
(b) Expand e x ( A B ) about x=0. To what order in x can e x ( A B ) be
a
Part I (50 points)
1. If A is Hermitian operator, show that
A2 0 .
[10 points]
2. Consider three observables, A , B , C . If it is known that
B, C iA
A , C iB
show that
( AB ) (C )
Here
1 2
A B2 .
2
O O , (O) O 2 O 2 .
If O1 , O2 = iO
( Hint:
,
O
Electrodynamics Part I (50 points)
1. Consider a potential problem in the half-space defined by z 2 0, with
Dirichlet boundary conditions on the plane z = 0 (and at infinity).
(a) Write down the appropriate Green function G(x, x’).[10pts]
(b) If the poten
102 2013.10.17
Electrodynamics - Part 1 (50 points)
Problem 1: (25 points)
Two infinite parallel planes are maintained at different potential. One is at z = 0 and has the potential
2
( x, y,0) = 5 sin(4 x) cos(3 y ) . The other is at z = 2 / 5 and has th
2013.03.28
Electrodynamics (Part I, 50 points)
1. A point charge q is situated a distance a from the center of a grounded conducting
sphere of radius R.
(a) Find the potential outside the sphere. (10%)
(b) Find the induced surface charge distribution on
101 2012.10.11
Electrodynamics - Part 1 (50 points)
Problem 1: (20 points)
An uniformly polarized dielectric sphere can be viewed as a superposition of two uniformly charged
spheres of radius R with charge densities + & , separarated by a distance d , wit
ﬁlﬁﬁ%¢%fﬁﬁﬁ%ﬁﬁfﬁiw 98 “gig [i§4%§fﬁ] ﬁgﬁbﬁgﬁﬁﬁﬁ
Electrodynamics (Part I, 50 points) 2009-10-15
. Using the method of images, discuss the problem of a point charge q inside a
hollow, grounded, conducting sphere of inner radius a. Find
(a) the potential ins
ElectrodynamicsPartI(50points)
Thetimeaveragedpotentialofaneutralhydrogenatomisgivenby
q e r
r
=
1
4 0 r
2
whereqisthemagnitudeoftheelectroniccharge,and1=a0/2,a0
beingtheBohrradius.
(a)Findthedistributionofcharge(bothcontinuousanddiscrete)that
willgiveth
Electrodynamics (Part II 50 points).
1. As shown in Fig. 1, a spherical shell with uniform surface charge density 0
rotated in a constant angular velocity around a z axis passing through the
spherical center.
v
(a) Find the vector potential A inside and o
=H .33,
% C H ) Electrodynamics Qualify Exam (Part I, 50 points) 201 1-03-31
6701‘
2
I“
( I ) EH B10“ smart 33$ 1. A static charge distribution produces a radial electric ﬁeld EU) :C
P,
where C and D are constants.
a) Find the charge density. (10 point
99-1 Electrodynamics Part II (50 points)
Electrodynamics (Part I, 50 points)
_ - 1. An ionized gas consists of ions (charge Ze, mean concentration No)
1. Briefly explain the fouowmg quesnons [20 pts] and electrons (charge 6, mean concentration no). The co
Electrodynamics, Part I
Qualifying Exam, March 25, Eﬂlﬂ
1. {25 Paints] Calmlate the electrostaﬁx: energy We Bf a 333mm containing a point
chargeandaﬁgidspherendmmifomchm'gedeusitjrpandmdiusﬁmmnf
ﬂnsphereislmtedatthnndﬁnufmgcuordiuatesl'hgpointchmgegisuuts
images? 95 eeererzxeaj reenter ease Haas/$504}
PhD. Qualify Examination. Oct. 12. 3006
Please describeyour aimi'ers as complete as possible
Electrodynamics Part I (50 points)
1. (1|) points) The time-averaged potential of a neutral hydrogen atom given by