University of Colorado, Department of Physics
PHYS3220, Fall 09, HW#13
due Wed, Dec 2, 2PM at start of class
1. Matrix representation of the eigenvalue problem (Total: 20 pts)
Suppose there are two observables
A
and
B
with corresponding Hermitian operators
ˆ
A
and
ˆ
B
. The eigenfunctions to
ˆ
A
as well as to
ˆ
B
form a complete set of basis functions,
and satisfy the eigenvalue equations:
ˆ
A

a
n
>
=
a
n

a
n
>
and
ˆ
B

b
n
>
=
b
n

b
n
> .
(Here we are adopting a common practice of labeling an eigenstate by its eigenvalues, so

a
n
>
is the eigenvector of
ˆ
A
with eigenvalue
a
n
. Do not confuse the eigenvalues, which
are numbers, with their eigenvectors, which are vectors in Hilbert space).
a) Since the eigenfunctions form a complete set, any wavefunction

Ψ(
t
)
>
can be written
as an expansion in either basis (we assume for simplicity a discrete basis):

Ψ(
t
)
>
=
X
n
c
n
(
t
)

a
n
>
=
X
n
d
n
(
t
)

b
n
>
Derive a formula that expresses any particular
d
n
in terms of the set of
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 STEVEPOLLOCK
 Physics, Uncertainty Principle, complete set, Matrix representation, angular momentum operator, eigenvalue equation Aan

Click to edit the document details