Hi
h ψjHψi
h
,
ð1:126Þ
where jψi is normalized. Unless jψi is normalized, it can be normalized on the basis
of (1.121) by choosing jΦi such that
j Φi ¼j ψi= j ψ
j jj :
ð1:127Þ
Thus, we have an important consequence; if an Hermitian operator has an eigenvalue, it must be real. An expectation value of an Hermitian operator is real as well.
The real eigenvalue and expectation value are a prerequisite for a physical quantity.
As discussed above, the Hermitian matrices play a central role in quantum
physics. Taking a further step, let us extend the notion of Hermiticity to a function
space.
In Example 1.1, we have remarked that we have finally reached a solution where λ
is a real (and positive) number, even though at the beginning we set no restriction on
λ. This is because the SOLDE form (1.61) accompanied by BCs (1.62) is Hermitian,
and so eigenvalues λ are real.
In this context, we give a little bit of further consideration. We define an inner
product between two functions as follows:
gjf
h i
Z b
a
g x
ð Þ
à f x
ð Þdx,
ð1:128Þ
where g(x)
à is a complex conjugate of g(x); x is a real variable and an integration
range can be either bounded or unbounded. If a and b are real definite numbers, [a, b]
is the bounded case. With the unbounded case, we have, e.g., (À1, 1), (À1, c),
and (c, 1), etc. where c is a definite number. This notation will appear again in
Chap. 10. In (1.128) we view functions f and g as vectors in a function space, often
referred to as a Hilbert space. We assume that any function f is square-integrable; i.e.,
|f |
2 is finite. That is,
Z b
a
f x
ð Þ
j
j
2 dx < 1:
ð1:129Þ
Using the above definition, let us calculate hg| Df i, where D was defined in
(1.63). Then, using the integration by parts, we have
g
h j Df i ¼
Z b
a
g x
ð Þ
à d
2 f x
ð Þ
dx
2
!
dx ¼ À
Â
g
à f
0
à b
a
þ
Z b
a
g
Ã
0 f
0 dx
¼ À g
à f
0
b
a þ
Â
Â
g
Ã0 f
b
a À
Z b
a
g
Ã
00 fdx ¼ g
Ã0 f À g
à f
0
½
b
a þ
Z b
a
Àg
Ã
00 f
dx
¼ g
Ã0 f À g
à f
0
½
b
a þ Dg
h
j f i:
ð1:130Þ
1.4 Quantum-Mechanical Operators and Matrices
25
h ψjHψi
h
,
ð1:126Þ
where jψi is normalized. Unless jψi is normalized, it can be normalized on the basis
of (1.121) by choosing jΦi such that
j Φi ¼j ψi= j ψ
j jj :
ð1:127Þ
Thus, we have an important consequence; if an Hermitian operator has an eigenvalue, it must be real. An expectation value of an Hermitian operator is real as well.
The real eigenvalue and expectation value are a prerequisite for a physical quantity.
As discussed above, the Hermitian matrices play a central role in quantum
physics. Taking a further step, let us extend the notion of Hermiticity to a function
space.
In Example 1.1, we have remarked that we have finally reached a solution where λ
is a real (and positive) number, even though at the beginning we set no restriction on
λ. This is because the SOLDE form (1.61) accompanied by BCs (1.62) is Hermitian,
and so eigenvalues λ are real.
In this context, we give a little bit of further consideration. We define an inner
product between two functions as follows:
gjf
h i
Z b
a
g x
ð Þ
à f x
ð Þdx,
ð1:128Þ
where g(x)
à is a complex conjugate of g(x); x is a real variable and an integration
range can be either bounded or unbounded. If a and b are real definite numbers, [a, b]
is the bounded case. With the unbounded case, we have, e.g., (À1, 1), (À1, c),
and (c, 1), etc. where c is a definite number. This notation will appear again in
Chap. 10. In (1.128) we view functions f and g as vectors in a function space, often
referred to as a Hilbert space. We assume that any function f is square-integrable; i.e.,
|f |
2 is finite. That is,
Z b
a
f x
ð Þ
j
j
2 dx < 1:
ð1:129Þ
Using the above definition, let us calculate hg| Df i, where D was defined in
(1.63). Then, using the integration by parts, we have
g
h j Df i ¼
Z b
a
g x
ð Þ
à d
2 f x
ð Þ
dx
2
!
dx ¼ À
Â
g
à f
0
à b
a
þ
Z b
a
g
Ã
0 f
0 dx
¼ À g
à f
0
b
a þ
Â
Â
g
Ã0 f
b
a À
Z b
a
g
Ã
00 fdx ¼ g
Ã0 f À g
à f
0
½
b
a þ
Z b
a
Àg
Ã
00 f
dx
¼ g
Ã0 f À g
à f
0
½
b
a þ Dg
h
j f i:
ð1:130Þ
1.4 Quantum-Mechanical Operators and Matrices
25