Physics 486 Homework 8 Due March 18, 2005
1) This is Shankar 7.3.4 (page 196) [5 points].
Use the harmonic oscillator wave functions,
ψ
n(x) (see Shankar, p. 195) to show
nˆ
x m =!
2m
!
"
#
$%
&
'
1
2
n+1
( )
1
2
(
n,m+1+n1
2
(
n,m)1
*
+,
-
nˆ
p m =im
!
!
2
"
#
$%
&
'
1
2
n+1
( )
1
2
(
n,m+1)n1
2
(
n,m)1
*
+,
-
That is,
ˆ
p and ˆ
x
“connect” adjacent energy states.
COMMENT: Why is this result important? Suppose we have solved a problem that
involves the motion of a charged particle and know the
ψ
n. Now, add an electric field. The
potential gains an extra term, V(x) = -eEx, and the matrix that describes the modified
Hamiltonian is no longer diagonal. So, to find the new
ψ
n, we’ll have to rediagonalize the
matrix. This will be one of the first problems we do next semester (P487).
2) This is similar to Shankar 7.3.5 (page 196) [5 points].
Use the result of problem 1to show that:
a) <x> = 0 and <p> = 0 for every energy eigenstate.
b) σxσp = /2 in the ground state (n = 0).
This is an exercise in manipulating infinite dimensional matrices. Writing them out as
* * !
* *
" #
!
"
#
#
$
%
&
&
isn’t practical, unless you have a lot of paper.
(over)
