Harmonic Oscillator:
From
a
and
a
†
to the position representation
Michele Kotiuga
Physics 137A
Spring 2010
In lecture the harmonic oscillator potential was treated using creation and annihilation operators:
a
and
a
†
. In Section 4.7 of Bransden and Joachain the harmonic oscillator potential is solved in the
position representation  i.e. we found the wavefunction as a function of position:
ψ
(
x
). The method is
quite messy and the solution is found from recursion relations:
ψ
n
(
x
) =
N
n
e

α
2
x
2
/
2
H
n
(
αx
)
(1)
Where
N
n
is a constant ensuring that the wavefunction is normalized and
H
n
is a Hermite polynomial.
The Hermite polynomials can be generated from the formula:
H
n
(
ξ
) =
e
ξ
2
/
2
ξ

d
dξ
¶
n
e

ξ
2
/
2
(2)
On the other hand, using creation and annihilation operators, the wavefuction takes the form:
ψ
n
=
(
a
†
)
n
√
n
!

0
>
(3)
Where

0
>
is the ground state.I will show that these two wavefucntions are equivalent, by changing to
the position representation. Writing
a
and
a
†
in terms of
x
and
p
:
a
=
r
mω
2¯
h
x
+
ip
mω
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.
 Spring '09
 Lee
 Physics, mechanics, harmonic oscillator potential, position representation, Michele Kotiuga

Click to edit the document details