N n
ffiffiffiffiffiffiffiffi ffi
1
2
n n!
r
mω
πħ
1=4 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
β
π 1=2 2
n n!
r
,
ð2:93Þ
we get
ψ n ξ=β
ð Þ ¼ N n ξ À
∂
∂ξ
n
e
À
1
2 ξ
2 :
ð2:94Þ
We have to normalize (2.94) with respect to a variable ξ. Since ψ n (q) has already
been normalized as in (2.53), we have
Z 1
À1
ψ n q
ð Þ
j
j
2 dq ¼ 1:
ð2:95Þ
Changing a variable q to ξ, we have
1
β
Z 1
À1
ψ n ξ=β
ð Þ
j
j
2 dξ ¼ 1:
ð2:96Þ
Let us define f
ψ n ξ
ð Þas being normalized with ξ. In other words, ψ n (q) is converted
to f
ψ n ξ
ð Þ by means of variable transformation and concomitant change in normalization condition. Then, we have
Z 1
À1
f
ψ n ξ
ð Þ
j
j
2 dξ ¼ 1:
ð2:97Þ
Comparing (2.96) and (2.97), if we define f
ψ n ξ
ð Þ as
f
ψ n ξ
ð Þ
ffiffi ffi
1
β
r
ψ n ξ=β
ð Þ,
ð2:98Þ
f
ψ n ξ
ð Þ should be a proper normalized function. Thus, we get
f
ψ n ξ
ð Þ ¼ f
N n ξ À
∂
∂ξ
n
e
À
1
2 ξ
2
with f
N n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
π 1=2 2
n n!
r
:
ð2:99Þ
Meanwhile, according to a theory of classical orthogonal polynomial, the Hermite
polynomials H n (x) are defined as [3]
H n x
ð Þ À1
ð Þ
n e
x
2 d
n
dx n e
Àx
2
n ! 0
ð
Þ,
ð2:100Þ
2.4 Coordinate Representation of Schrödinger Equation
47