Quantum Control midterm Examination
(Introduction of Quantum Mechanics)
◆ Chapter 1 –
1. Consider the Gaussian distribution
 ( x)  Ae  ( xa ) ,
2
where A , λ, and a are constants.
(a) Use
(b) Find



 ( x)dx  1 to determine A
x ,
x 2 , and 
(c) sketch the graph of  (x )
2. At time t=0, a particle is represented by the wave function.
 Ax / a,

 ( x,0)   A(b  x) /(b  a),
 0

if 0  x  a
if a  x  b
otherwise
where A, a , and b are constants.
(a) Normalize 
(b) Sketch  (x,0) as a function of x
(c) where is the particle most likely to be found, at time t=0 ?
(d) what is the probability of finding the particle to the left of a ? Check your result in the
limiting cases b  a
(e) what is the expection value of x
3. Consider the wave function
( x, t )  Ae  | x | e i t ,
where A , λ, and  are positive real constants.
(a) Normalize 
(b) Determine the expectation values of x and x2.
(c) Find the standard deviation of x. Sketch the graph of |  | 2 , as a function of x, and mark
the points (〈x〉+σ ) and (〈x〉-σ ) to illustrate the sense in which σ represents the
“spread” in x. What is the probability that the particle would be found outside this range?
d  p
V
 
.
dt
x
(This is known as Ehrenfest’s theorem; it tells us that expectation values obey Newton’s
second law.)
4. Calculate
5. A particle of mass m is in the state
( x, t )  Aea[(mx / )it ]
2
where A and a are positive real constants.
(a) Find A.
(b) For what potential energy function V(x) does  satisfy the Schrödinger equation?
(c) Calculate the expectation values of x , x2, p, and p2 .
(d) Find σx and σp . Is their product consistent with the uncertainty principle?
◆ Chapter 2 –
6. A particle in the infinite square well has the initial wave function
 ( x,0)  Ax(a  x)
(a) Normalize  (x,0) and graph it.
(b)Compute
x ,
H
p , and
at
p
(note: this time you can not get
t=0
by differentiating
x , because you only know
x
at one instant of time)
7. Consider the double delta-function potential
V ( x)    ( x  a)   ( x  a)
where and are positive constants.
(a) Sketch this potential
2
(b) How many bound states does it possess? Find the allowed energies, for  
and
ma
2
for  
,and sketch the wave functions.
4ma
8. A particle in the infinite square well has the initial wave function
 x
 ( x,0)  A s i n3 (
Find
a
)
x as a function of time.
9. Solve the time-independent Schrödinger Equation for an infinite square well
with a delta-function barrier at the center:
V ( x) 
 ( x),

f or a  x  a
for x a
Treat the even and odd wave functions separately.
Find the allowed energies and wave functions. Comment on the limiting cases    and
  0?
◆ Chapter 3 –
10. A Hermitian linear transformation must satisfy  Tˆ   Tˆ 
 . Prove that it is sufficient that  Tˆ   Tˆ 
en Tˆ en  Tˆ en en
could show that
for all vectors 
and
for all vectors  . Suppose you
for every member of an orthonormal basis.
 1 1 i

11. T  
1  i 0 
(a)
(b)
(c)
(d)
Verify that T is Hermitian
Find its eigenvalues ( note that they are real)
Find and normalize the eigenvectors (note that they are orthogonal)
Construct the unitary diagonalizing matrix S, and check explicitly that it diagonalizes
T
12. A unitary linear transformation is one for which Uˆ Uˆ  1
(a) Show that unitary transformation preserve inner products, in the sense that
Uˆ Uˆ    
for all vectors 
and 
(b) Show that the eigenvectors of a unitary transformation belonging to distinct
eigenvalues are orthogonal
13. Imagine a system in which there are just two linearly independent states:
1
1   
0
0
2   
1
and
The most general state is a normalized linear combination:
a
  a 1  b 2   
b
with
a  b 1
2
2
Suppose the Hamiltonian matrix is
h
H  
g
equation says
g

h 
where g and h are real constants. The (time-dependent) Schrödinger
d

dt
(a) Find the eigenvalues and (normalized) eigenvectors of this Hamiltonian
H   i
(b) Suppose the system starts out (at t = 0) in stste 1 . What is the state at time t?
◆ Chapter 4 –
14. Derive a free particle in Heisenberg picture
d
pˆ
rˆ(t ) 
dt
m
15. Derive 1-dimensional Harmonic oscillator in Schrödinger picture
p
(a) x(t )  x cos wt  x sin wt
mw
(b) p x (t )  p x cos wt  mwx sin wt
16. Derive 1-dimensional Harmonic oscillator in Heisenberg picture
i
sin w(t 2  t1 )
(a) x(t1 ), x(t 2 ) 
mw
(b)  p(t1 ), p(t 2 )  imw sin w(t 2  t1 )
(c) x(t1 ), p(t 2 )  i cos w(t 2  t1 )
◆
17. Consider a quantum system with M external control-linear forces.
Assume M =1 and control forces u1 (t )  A sin wt , where A and w are
constants. How to represent time-independent Schrödinger equation?
(Hint : total Hamiltonian of the system is given by: Hˆ (t )  Hˆ 0  u1 (t ) Hˆ 1 )
18. Schrödinger equation is a first order differential equation in time, the wave function
 (t ) is uniquely determined by the value of  (0) , if we introduce the time evolution
operator Sˆ (t ) as defined
 (t )  Sˆ (t ) (0) .
(a). Show that Sˆ (t ) is unitary and satisfies iSˆ (t )  Hˆ Sˆ (t ) .
(b).Show that if hamiltonian Ĥ does not depend on time, Sˆ (t ) is of the form
Sˆ (t )  e x p(iHˆ t / ) .