1549380323-Statistical Mechanics Theory and Molecular Simulation

Simple examples 385

x

y 0 (x)

x

y 3 (x)

x

y 2 (x)

x

y 1 (x)

Fig. 9.1The first four eigenfunctions of a harmonic oscillator.

ˆa=

√

mω

2 ̄h

ˆx+

i

√

2 m ̄hω

pˆ

ˆa†=

√

mω

2 ̄h

ˆx−

i

√

2 m ̄hω

p ,ˆ (9.3.25)

which can be shown to satisfy the commutation relation

[ˆa,ˆa†] = 1. (9.3.26)

In terms of these operators, the Hamiltonian can be easily derived with the result

Hˆ=

(

ˆa†ˆa+

1

2

)

̄hω. (9.3.27)

The action of ˆaand ˆa†on the eigenfunctions ofHˆ can be worked out using the fact
that

ˆaψn(x) =

[√

α

2

x+

1

√

2 α

d

dx

]

ψn(x) (9.3.28)