QMGreensite_merged

9.1. RAISINGANDLOWERINGOPERATORS 149

= (a†a+1)(a†)n−^1

= a†a(a†)n−^1 +(a†)n−^1

= a†(a†a+1)(a†)n−^2 +(a†)n−^1

= (a†)^2 (a†a+1)(a†)n−^3 +2(a†)n−^1

= (a†)^3 (a†a+1)(a†)n−^4 +3(a†)n−^1

.

.

= (a†)na+n(a†)n−^1 (9.55)

Equation(9.54)becomes

1 =c^2 n<(a†)n−^1 φ 0 |[(a†)na+n(a†)n−^1 ]|φ 0 > (9.56)

Usinga|φ 0 >=0,

1 = nc^2 n<(a†)n−^1 φ 0 |(a†)n−^1 |φ 0 >

= n

c^2 n

c^2 n− 1

<φn− 1 |φn− 1 >

= n

c^2 n

c^2 n− 1

(9.57)

sothat
cn=

cn− 1

√

n

(9.58)

Usingthisequation,andc 0 =1,weget

c 1 = 1

c 2 =

1

√

2 · 1

c 3 =

1

√

3 · 2 · 1

(9.59)

or,ingeneral

cn=

1

√

n!

(9.60)

We now have the general solution for the energy eigenstates of the harmonic
oscillator:

φn(x) =

1

√

n!

(a†)nφ 0 (x)

=

(

mω

πh ̄

) 1 / 4

1

√

n!



√^1

2

(

√

mω

̄h

x−

√

̄h

mω

∂

∂x

)





n

e−mωx

(^2) /2 ̄h
(9.61)