H is called an Hermitian matrix. Then, applying (1.112) to the Hermitian matrix H,
we have
H
{
ψjξi
¼ ψjHξi
h
¼ ψjH
{
ξ
or Hψjξi
h
¼ ψjH
{
ξ
¼ ψjHξi
h
:
ð1:120Þ
Also let us introduce a norm of a vector jψi such that
jψ
j jj ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
ψjψi
h
p
:
ð1:121Þ
A norm is a natural extension for a notion of a “length” of a vector. The norm j|ψ|j is
zero, if and only if jψi ¼ 0 (zero vector). Then, from (1.105) and (1.107), we have
ψjψi
h
¼ e
j j
2 þ f
j j
2 :
Therefore, hψ| ψi ¼ 0 ⟺ e ¼ f ¼ 0, i. e. , j ψi ¼ 0.
Let us further consider an eigenvalue problem represented by our newly introduced notation. The eigenvalue equation is symbolically written as
H j ψi ¼ λ j ψi,
ð1:122Þ
where H represents an Hermitian operator and jψi is an eigenfunction that belongs to
an eigenvalue λ. Operating hψ| on (1.122) from the left, we have
ψj
h H ψi ¼ ψj
h λ
j
j ψi ¼ λ ψj
h ψi ¼ λ,
ð1:123Þ
where we assume that jψi is normalized, namely hψ| ψi ¼ 1 or j|ψ| j ¼ 1. Notice that
the symbol “j” in an inner product is of secondary importance. We may disregard
this notation as in the case where a product notation “” is omitted by denoting ab
instead of a  b.
Taking a complex conjugate of (1.123), we have
ψjH
h
ψi
à ¼ λ
Ã
:
ð1:124Þ
Using (1.116) and (1.124), we have
λ
Ã
¼ ψjH
h
ψi
à ¼ ψjH
{
ψ
¼ ψjHψi
h
¼ λ,
ð1:125Þ
where with the third equality we used the definition (1.119). The relation λ
Ã
¼ λ
obviously shows that any eigenvalue λ is real, if H is Hermitian. The relation (1.125)
immediately tells us that even though jψi is not an eigenfunction, hψ| Hψi is real as
well, if H is Hermitian. The quantity hψ| Hψi is said to be an expectation value. This
value is interpreted as the most probable or averaged value of H obtained as a result
of operation of H on a physical state jψi. We sometimes denote the expectation value
as
24
1 Schrödinger Equation and Its Application
we have
H
{
ψjξi
¼ ψjHξi
h
¼ ψjH
{
ξ
or Hψjξi
h
¼ ψjH
{
ξ
¼ ψjHξi
h
:
ð1:120Þ
Also let us introduce a norm of a vector jψi such that
jψ
j jj ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
ψjψi
h
p
:
ð1:121Þ
A norm is a natural extension for a notion of a “length” of a vector. The norm j|ψ|j is
zero, if and only if jψi ¼ 0 (zero vector). Then, from (1.105) and (1.107), we have
ψjψi
h
¼ e
j j
2 þ f
j j
2 :
Therefore, hψ| ψi ¼ 0 ⟺ e ¼ f ¼ 0, i. e. , j ψi ¼ 0.
Let us further consider an eigenvalue problem represented by our newly introduced notation. The eigenvalue equation is symbolically written as
H j ψi ¼ λ j ψi,
ð1:122Þ
where H represents an Hermitian operator and jψi is an eigenfunction that belongs to
an eigenvalue λ. Operating hψ| on (1.122) from the left, we have
ψj
h H ψi ¼ ψj
h λ
j
j ψi ¼ λ ψj
h ψi ¼ λ,
ð1:123Þ
where we assume that jψi is normalized, namely hψ| ψi ¼ 1 or j|ψ| j ¼ 1. Notice that
the symbol “j” in an inner product is of secondary importance. We may disregard
this notation as in the case where a product notation “” is omitted by denoting ab
instead of a  b.
Taking a complex conjugate of (1.123), we have
ψjH
h
ψi
à ¼ λ
Ã
:
ð1:124Þ
Using (1.116) and (1.124), we have
λ
Ã
¼ ψjH
h
ψi
à ¼ ψjH
{
ψ
¼ ψjHψi
h
¼ λ,
ð1:125Þ
where with the third equality we used the definition (1.119). The relation λ
Ã
¼ λ
obviously shows that any eigenvalue λ is real, if H is Hermitian. The relation (1.125)
immediately tells us that even though jψi is not an eigenfunction, hψ| Hψi is real as
well, if H is Hermitian. The quantity hψ| Hψi is said to be an expectation value. This
value is interpreted as the most probable or averaged value of H obtained as a result
of operation of H on a physical state jψi. We sometimes denote the expectation value
as
24
1 Schrödinger Equation and Its Application