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)