Midterm Exam
30.Oct.2001
Quantum Mechanics
Intructions:
Please read all questions carefully before attempting to answer them. Write your
name and number on top of every answersheet. No books, calculators or
materials other than blank paper and pencils/pens are allowed on the desks. If
you need a formula or integral from the book to solve an exercise, which you
cannot remember please ask and it will be given to you (for a possible small
point penalty). You have until 19:00 to answer all questions in Swedish or
English. The exam counts for 30% of your total grade (1 point=0.5%).
1.) (15 points) Consider a one-dimensional simple harmonic oscillator. Do the
following algebraically in the energy basis (i.e. without using wavefunctions).
Hint:
ω
ω
m
iP
X
m
a
2
/
)
(
+
=
a.) Construct a linear combination of
0
and
1
such that
X
is as large as
possible.
b.) Suppose at t=0 the oscillator is in the state contructed in a.). What is the
state vector for t>0 in the Schrödinger picture?
c.) Evaluate the expectation values
X
and
2
X
as a function of time t>0.
2.) (18 points) Consider the angular momentum operators with the commutation
relations
[
]
,
,
k
ijk
j
i
J
i
J
J
ε
=
where i,j,k can be any of the directions x,y,z.
a.) Show that the z-component of the angular momnetum
z
J
and the angular
momentum squared
2
2
2
z
y
x
J
J
J
+
+
commute with each other and therefore
have simultaneaus eigenvalues m and j(j+1). Show from the
commutation relations of the “ladder” operators
y
x
iJ
J
J
±
=
±
that these
can be used to raise and lower the z-component of the angular
momentum. Derive which are the allowed values for m and j(j+1).
b.) Derive the constants
±
jm
c
in the equations
1
,
,
±
=
±
±
m
j
c
m
j
J
jm
. (Result:
)
1
)(
(
+
±
=
±
m
j
m
j
c
jm
).
c.) Use the result of b.) to explicitly construct the matrix
m
j
J
m
j
y
,
'
,
'
,
for
j’=j=1 and m,m’=-1,0,1. Find the eigenvalues and eigenvectors of
y
J
in
the basis where
z
J
is diagonal.
