where H n (x) is a n-th order polynomial. We wish to show the following relation on
the basis of mathematical induction:
f
ψ n ξ
ð Þ ¼ f
N n H n ξ
ð Þe
À
1
2 ξ
2 :
ð2:101Þ
Comparing (2.87), (2.98), and (2.99), we make sure that (2.101) holds with n ¼ 0.
When n ¼ 1, from (2.99) we have
f
ψ 1 ξ
ð Þ ¼ f
N 1 ξ À
∂
∂ξ
e
À
1
2 ξ
2 ¼ f
N 1 ξe
À
1
2 ξ
2 À Àξ
ð Þe
À
1
2 ξ
2
h
i
¼ f
N 1 Á 2ξe
À
1
2 ξ
2
¼ f
N 1 À1
ð Þ
1 e
ξ
2 d
dξ
e
Àξ
2
!
e
À
1
2 ξ
2 ¼ f
N 1 H 1 ξ
ð Þe
À
1
2 ξ
2 :
ð2:102Þ
Then, (2.101) holds with n ¼ 1 as well.
Next, from supposition of mathematical induction we assume that (2.101) holds
with n. Then, we have
g
ψ nþ1 ξ
ð Þ ¼ g
N nþ1 ξ À
∂
∂ξ
nþ1
e
À
1
2 ξ
2 ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 n þ 1
ð
Þ
p
ξ À
∂
∂ξ
f
N n ξ À
∂
∂ξ
n
e
À
1
2 ξ
2
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 n þ 1
ð
Þ
p
ξ À
∂
∂ξ
f
N n H n x
ð Þe
À
1
2 ξ
2
h
i
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 n þ 1
ð
Þ
p
f
N n ξ À
∂
∂ξ
À1
ð Þ
n e
ξ
2 d
n
dξ
n e
Àξ
2
!
e
À
1
2 ξ
2
&
'
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 n þ 1
ð
Þ
p
f
N n À1
ð Þ
n ξ À
∂
∂ξ
e
1
2 ξ
2 d
n
dξ
n e
Àξ
2
!
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 n þ 1
ð
Þ
p
f
N n À1
ð Þ
n ξe
1
2 ξ
2 d
n
dξ
n e
Àξ
2
À
∂
∂ξ
e
1
2 ξ
2 d
n
dξ
n e
Àξ
2
!
&
'
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 n þ 1
ð
Þ
p
f
N n À1
ð Þ
n ξe
1
2 ξ
2 d
n
dξ
n e
Àξ
2
À ξe
1
2 ξ
2 d
n
dξ
n e
Àξ
2
À e
1
2 ξ
2 d
nþ1
dξ
nþ1
e
Àξ
2
&
'
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 n þ 1
ð
Þ
p
f
N n À1
ð Þ
nþ1 e
1
2 ξ
2 d
nþ1
dξ
nþ1
e
Àξ
2
¼ g
N nþ1 À1
ð Þ
nþ1 e
ξ
2 d
nþ1
dξ
nþ1
e
Àξ
2
!
e
À
1
2 ξ
2 ¼ g
N nþ1 H nþ1 x
ð Þe
À
1
2 ξ
2 :
ð2:103Þ
This means that (2.101) holds with n + 1 as well. Thus, it follows that (2.101) is true
of n that is zero or any positive integer.
Orthogonal relation reads as
48
2 Quantum-Mechanical Harmonic Oscillator