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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
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 Spring '09
 Lee
 Physics, mechanics

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