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Physical Chemistry -The Harmonic Oscilator Energies and Wavefunctions via Raising and Lowering Operators_lecture12_sup1

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This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...

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Uploaded on April 25, 2023
Number of pages 7
Written in 2007/2008
Type Class notes
Professor(s) Prof. robert guy griffin
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the harmonic oscilator energies and wavefunctions

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5.61 Fall 2007 12-15 Lecture Supplement Page 1

Harmonic Oscillator Energies and Wavefunctions
via Raising and Lowering Operators

We can rearrange the Schrödinger equation for the HO into an interesting form ...
1 ⎡⎛ ! d ⎞ 2⎤
2
1
⎢⎜ ⎟ + ( mω x ) ⎥ ψ = ⎡ p 2 + ( mω x )2 ⎤ ψ = Eψ
2m ⎢⎣⎝ i dx ⎠ ⎥⎦ 2m ⎣ ⎦

with
1
H= ⎡ p 2 + ( mω x )2 ⎤
2m ⎣ ⎦
which has the same form as
u 2 + v 2 = ( iu + v )( −iu + v ) .

We now define two operators

1
a± ≡ ( ∓ip + mω x )
2!mω

that operate on the test function f(x) to yield

( a a ) f ( x ) = ⎛⎝ ( ip + mω x )( −ip + mω x )⎞ f ( x )
1
− +
2!mω ⎠

1
⎡ p 2 + ( mω x ) − imω (xp − px) ⎤ f (x)
2
=
2!mω ⎣ ⎦

= { 1
2!mω
( i
)
p 2 + ( mω x ) − [ x, p ] f ( x )
2
2! }
a− a+ =
1
2!mω
( 1
p 2 + ( mω x ) + =
2 1
2 !ω
H+ )
1
2

Which leads to a new form of the Schrödinger equation in terms of a+ and a- …

⎛ 1⎞
H ψ = !ω ⎜ a− a+ − ⎟ ψ
⎝ 2⎠

If we reverse the order of the operators-- a− a+ ⇒ a+ a− -- we obtain …

, 5.61 Fall 2007 12-15 Lecture Supplement Page 2

⎛ 1⎞
H ψ = !ω ⎜ a+ a− + ⎟ ψ
⎝ 2⎠
or
⎛ 1⎞
!ω ⎜ a± a∓ ± ⎟ ψ = Eψ
⎝ 2⎠

and the interesting relation
a− a+ − a+ a = [ a a ] = 1
− − +

A CLAIM: If ψ satisfies the Schrödinger equation with energy E, then a+ψ
satisfies it with energy (E+!ω ) !
⎛ 1⎞ ⎛ 1 ⎞
H ( a + ψ ) = ! ω ⎜ a + a − + ⎟ ( a + ψ ) = !ω ⎜ a + a − a + + a + ⎟ ψ
⎝ 2⎠ ⎝ 2 ⎠
⎛ 1⎞ ⎧ ⎛ 1⎞ ⎫ ⎧ ⎛ 1⎞ ⎫
= !ω a+ ⎜ a− a+ + ⎟ ψ = a+ ⎨!ω ⎜ a+ a− + 1 + ⎟⎠ ψ ⎬ = a+ ⎨!ω ⎜⎝ a+ a− + ⎟⎠ + !ω ⎬ψ
⎝ 2⎠ ⎩ ⎝ 2 ⎭ ⎩ 2 ⎭
= a+ ( H + !ω )ψ = ( E + !ω )( a+ψ )

H ( a+ψ ) = (E + !ω ) ( a+ψ )

Likewise, a-ψ satisfies the Schrödinger equation with energy (E-!ω ) …
⎛ 1⎞ ⎛ 1 ⎞ ⎛ 1⎞
H ( a−ψ ) = !ω ⎜ a− a+ − ⎟ ( a−ψ ) = !ω ⎜ a− a+ a− − a− ⎟ ψ = a !ω ⎜ a+ a− − ⎟ ψ
⎝ 2⎠ ⎝ 2 ⎠ ⎝ −
2⎠
⎧ ⎛ 1⎞ ⎫
= a− ⎨!ω ⎜ a− a+ − 1 − ⎟ ψ ⎬ = a− ( H − !ω )ψ = a− ( E − !ω )ψ
⎩ ⎝ 2⎠ ⎭

H ( a−ψ ) = ( E − !ω ) ( a−ψ )

So, these are operators connecting states and if we can find one state then we can
use them to generate other wavefunctions and energies. In the parlance of the
trade the a± are known as LADDER operators or

a+ = RAISING and a- = LOWERING operators.

We know there is a bottom rung on the ladder ψ0 so that

a−ψ 0 = 0

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Physical Chemistry -The Harmonic Oscilator Energies and Wavefunctions via Raising and Lowering Operators_lecture12_sup1

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