d
2 H n ξ
ð Þ
dξ
2
À 2ξ
dH n ξ
ð Þ
dξ
þ 2nH n ξ
ð Þ ¼ 0:
ð2:118Þ
This equation is said to be Hermite differential equation. Using (2.116), (2.118) can
be recast as an eigenvalue equation such that
e
DH n ξ
ð Þ ¼ 2nH n ξ
ð Þ:
ð2:119Þ
Therefore, comparing (2.115) and (2.118) and putting
λ ¼ 2n þ 1,
ð2:120Þ
we get
v ξ
ð Þ ¼ cH n ξ
ð Þ,
ð2:121Þ
where c is an arbitrary constant. Thus, using (2.113), for a solution of (2.109) we get
u n ξ
ð Þ ¼ cH n ξ
ð Þe
Àξ
2 =2 ,
ð2:122Þ
where the solution u(ξ) is indexed with n. From (2.110), as an energy eigenvalue we
have
E n ¼ n þ
1
2
ħω:
Thus, (2.37) is recovered. A normalization constant c of (2.122) can be decided as in
(2.106).
As discussed above, the operator representation and coordinate representation are
fully consistent.
2.5 Variance and Uncertainty Principle
Uncertainty principle is one of most fundamental concepts of quantum mechanics.
To think of this conception on the basis of a quantum harmonic oscillator, let us
introduce a variance operator [4]. Let A be a physical quantity and let hAi be an
expectation value as defined in (1.126). We define a variance operator as
ΔA
ð Þ
2
E
D
,
where we have
2.5 Variance and Uncertainty Principle
51