Mathematical Methods for Physics and Engineering : A Comprehensive Guide

19.2 PHYSICAL EXAMPLES OF OPERATORS

19.2.3 Annihilation and creation operators

As a final illustration of the use of operator methods in physics we consider their

application to the quantum mechanics of a simple harmonic oscillator (s.h.o.).

Although we will start with the conventional description of a one-dimensional

oscillator, using its position and momentum, we will recast the description in

terms of two operators and their commutator and show that many important

conclusions can be reached from studying these alone.

The Hamiltonian for a particle of massmwith momentumpmoving in a

one-dimensional parabolic potentialV(x)=^12 kx^2 is

H=

p^2

2 m

+

1

2

kx^2 =

p^2

2 m

+

1

2

mω^2 x^2 ,

where its classical frequency of oscillationωis given byω^2 =k/m. We recall that

the corresponding operators,pandx, do not commute and that[p, x]=−i
.

In analogy with the ladder operators used when discussing angular momentum,

we define two new operators:

A≡

√

mω

2

x+

ip

√

2 mω

and A†≡

√

mω

2

x−

ip

√

2 mω

.

(19.39)

Since bothxandpare Hermitian,AandA†are Hermitian conjugates, though

neither is Hermitian and they do not represent physical quantities that can be

measured.

Now consider the two productsA†AandAA†:

A†A=

mω

2

x^2 −

ipx

2

+

ixp

2

+

p^2

2 mω

=

H

ω

−

i

2

[p, x]=

H

ω

−

2

,

AA†=

mω

2

x^2 +

ipx

2

−

ixp

2

+

p^2

2 mω

=

H

ω

+

i

2

[p, x]=

H

ω

+

2

.

From these it follows that

H=^12 ω(A†A+AA†) (19.40)

and that

[

A, A†

]

=
. (19.41)

Further,

[H, A]=

[ 1

2 ω(A

†A+AA†),A]

=^12 ω

(

A†0+

[

A†,A

]

A+A

[

A†,A

]

+0A†

)

=^12 ω(−
A−A
)=−
ωA. (19.42)

Similarly,

[

H, A†

]

=
ωA† (19.43).

Before we apply these relationships to the question of the energy spectrum of

the s.h.o., we need to prove one further result. This is that ifBis an Hermitian

operator then〈ψ|B^2 |ψ〉≥0forany|ψ〉. The proof, which involves introducing