130_notes.dvi

Because the lowering must stop at a ground state with positive energy, we can show that the allowed
energies are

En=

(

n+

1

2

)

̄hω.

The actual wavefunctions (See section 10.5) can be deduced by using the differential operators for
AandA†, but often it is more useful to define thentheigenstate in terms of the ground state and
raising operators.

un=

1

√

n!

(A†)nu 0

Almostany calculationof interest can be done without actual functions since we can express the
operators for position and momentum.

x =

√

̄h

2 mω

(A+A†)

p = −i

√

m ̄hω

2

(A−A†)

10.1 IntroducingAandA†

The Hamiltonian for the1D Harmonic Oscillator

H=

p^2

2 m

+

1

2

mω^2 x^2

can be rewritten in terms ofthe operatorA

A≡

(√

mω

2 ̄h

x+i

p

√

2 m ̄hω

)

and its Hermitian conjugate

A†=

(√

mω

2 ̄h

x−i

p

√

2 m ̄hω

)

Both terms in the Harmonic Oscillator Hamiltonian are squares of operators. Note thatAis chosen
so thatA†Ais close to the Hamiltonian. First just compute the quantity

A†A =

mω

2 ̄h

x^2 +

p^2

2 m ̄hω

+

i

2 ̄h

(xp−px)

A†A =

mω

2 ̄h

x^2 +

p^2

2 m ̄hω

−

i

2 ̄h

[p,x]

̄hω(A†A) =

p^2

2 m

+

1

2

mω^2 x^2 −

1

2

̄hω.

From this we can see that theHamiltonian can be written in terms ofA†Aand some constants.

H= ̄hω

(

A†A+

1

2

)

.