Physics 3220 – Quantum Mechanics 1 – Fall 2008
Problem Set #5
Due Wednesday, September 24 at 2pm
Problem 5.1
: Properties of the simple harmonic oscillator. (20 points)
Some of these results are discussed in class or in the book, but it’s quite useful to work
through them.
a) Given the definitions
a
+
≡
1
√
2¯
hmω
(

ip
+
mωx
)
a

≡
1
√
2¯
hmω
(
ip
+
mωx
)
,
(1)
demonstrate that the simple harmonic oscillator Hamiltonian
ˆ
H
=

(¯
h
2
/
2
m
)(
∂
2
/∂x
2
) +
(1
/
2)
mω
2
x
2
can be written in two ways,
ˆ
H
= ¯
hω
a
+
a

+
1
2
= ¯
hω
a

a
+

1
2
.
(2)
b) Use the energy formula
E
n
= ¯
hω
(
n
+ 1
/
2) along with part a) to demonstrate that
a

a
+
u
n
= (
n
+ 1)
u
n
,
a
+
a

u
n
=
nu
n
,
(3)
where
u
n
(
x
) are the normalized stationary state wavefunctions.
c) Use (
??
) and integration by parts to demonstrate that
Z
∞
∞
f
*
(
a
±
g
)
dx
=
Z
∞
∞
(
a
∓
f
)
*
g dx ,
(4)
for any functions
f
(
x
),
g
(
x
) that go to zero at infinity,
f
(
±∞
) =
g
(
±∞
) = 0; for example,
they could be normalizable wavefunctions. (As we will discuss more later, this implies that
a
+
and
a

are
adjoints
or
Hermitian conjugates
of each other.)
d) The raising and lowering operators must take one stationary state to the next, times an
overall constant:
a
+
u
n
=
c
n
u
n
+1
,
a

u
n
=
d
n
u
n

1
,
(5)
where
c
n
and
d
n
are constants to be determined. Consider the expression
R
∞
∞
(
a
+
u
n
)
*
(
a
+
u
n
)
dx
.
Evaluate it two ways, one of them using the results from parts b) and c), to solve for
c
n
.
Now consider
R
∞
∞
(
a

u
n
)
*
(
a

u
n
)
dx
and do something similar to solve for
d
n
.
You should
find:
a
+
u
n
=
√
n
+ 1
u
n
+1
,
a

u
n
=
√
nu
n

1
.
(6)
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
By using the formulas demonstrated in this problem, you will be able to evaluate a lot of
things about the SHO without ever having to “get your hands dirty” with the explicit forms
of the wavefunctions. You will see this in the next couple problems.
Problem 5.2
: Expectation values in the SHO. (20 points)
a) Find an expression for the operators ˆ
x
and ˆ
p
in terms of the raising and lowering operators
a
+
and
a

as well as constants.
b) Calculate
h
x
i
,
h
p
i
,
h
x
2
i
and
h
p
2
i
in the
n
th
stationary state
using the expressions from
part a).
Hint: You don’t ever need to write out the functional form of the
u
n
if you use
results from the previous problem.
c) How must
h
H
i
be related to the expectation values you calculated in the previous part?
Check that this relationship works given what you know
h
H
i
must be for a stationary state.
How much do the kinetic and potential energies each contribute to the total expectation
value
h
H
i
?
d) Calculate the product of uncertainties
σ
x
σ
p
for the
n
th
stationary state and verify the
Heisenberg Uncertainty Principle
σ
x
σ
p
≥
¯
h/
2. For which values of
n
is the minimum possible
uncertainty achieved?
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 STEVEPOLLOCK
 Work, coherent, Simple Harmonic Oscillator, TISE

Click to edit the document details