If we have BCs such that
f b
ð Þ ¼ f a
ð Þ ¼ 0 and g b
ð Þ
à ¼ g a
ð Þ
à ¼ 0, i:e:, g b
ð Þ ¼ g a
ð Þ ¼ 0,
ð1:131Þ
we get
gjDf
h
i¼ Dgjf
h
i:
ð1:132Þ
In light of (1.120), (1.132) implies that D is Hermitian. In (1.131), notice that the
functions f and g satisfy the same BCs. Normally, for an operator to be Hermitian has
this property. Thus, the Hermiticity of a differential operator is closely related to BCs
of the differential equation.
Next, we consider a following inner product:
f jDf
h
i¼ À
Z b
a
f
à f
00 dx ¼ À f
à f
0
½
Š
b
a þ
Z b
a
f
Ã
0 f
0 dx
¼ À f
à f
0
½
Š
b
a þ
Z b
a
f
0
j j
2 dx:
ð1:133Þ
Note that the definite integral of (1.133) cannot be negative. There are two possibilities for D to be Hermitian according to different BCs.
(i) Dirichlet conditions: f(b) ¼ f(a) ¼ 0. If we could have f
0
¼ 0, h f j Df i would be
zero. But, in that case f should be constant. If so, f(x) 0 according to BCs. We
must exclude this trivial case. Consequently, to avoid this situation we must
have
Z b
a
f
0
j j
2 dx > 0 or f jDf
h
i> 0:
ð1:134Þ
In this case, the operator D is said to be positive definite. Suppose that such a
positive-definite operator has an eigenvalue λ. Then, for a corresponding
eigenfunction y(x) we have
Dy x
ð Þ ¼ λy x
ð Þ:
ð1:135Þ
In this case, we state that y(x) is an eigenfunction or eigenvector that
corresponds (or belongs) to an eigenvalue λ. Taking an inner product of both
sides, we have
26
1 Schrödinger Equation and Its Application