Physics 486 Discussion Week 9
Solution
1) You saw in lecture that the parity and position operators,
ˆ
P
and
ˆ
x
, don’t commute:
ˆ
P,ˆ
x
!
"#
$=%2ˆ
xˆ
P
. One can use it to derive a very useful relationship among the harmonic
oscillator energy states. You’ll see later that the same analysis can be applied to hydrogen
atom states.
a) What is the commutator of
ˆ
P
and
ˆ
p
(the momentum operator)? This is most easily done
in the position basis (similar to what was done in lecture).
Use
ˆ
px=!i!"
"x
to obtain
ˆ
P,ˆ
p
!
"#
$=%2ˆ
pˆ
P
.
Now consider the 3-D isotropic harmonic oscillator.
b) What is the parity of its ground state?
It’s just a 3-D gaussian,
e!r2
, so the parity is even.
c) The x-raising operator is
ˆ
ax
†=c1ˆ
x!ic2ˆ
px
. We don’t care about the constants, so I’ve
lumped them into the c’s. Calculate the commutator,
ˆ
P,ˆ
ax
†
!
"#
$
. The y and z operators
behave similarly.
ˆ
P,ˆ
ax
†
!
"#
$=%2ˆ
ax
†ˆ
P
d) Use this result to calculate the parity of the first excited state of the harmonic oscillator.
Does it matter whether
nx,ny,nz
( )
=1, 0, 0
( )
, 0,1, 0
( )
, or 0, 0,1
( )
?
We want to evaluate
ˆ
Pˆ
ax
†0
( )
. From the commutator, this is
!ˆ
ax
†ˆ
P0
( )
=!ˆ
ax
†0
.
Thus, the parity is odd. This is the case for all of the raising operators.
e) Write a general formula for the parity of an arbitrary state,
nx,ny,nz
( )
.
ˆ
P nxnynz=!1
( )
nx+ny+nz
