E ¼
p
2
2m
¼
ħ
2
2m
k
2 ,
where
k ¼
2l þ 1
ð
Þπ=2L l ¼ 0, 1, 2, Á Á Á
ð
Þ ,
2nπ=2L
n¼ 1, 2, 3, Á Á Á
ð
Þ :
8
> <
> :
The energy E is a kinetic energy of the particle.
Although in (1.97), ψ(x) 0 trivially holds, such a function may not be regarded
as a solution of the eigenvalue problem. In fact, considering that |ψ(x)|
2 represents
existence probability of a particle, ψ(x) 0 corresponds to a situation where a
particle in question does not exist. Consequently, such a trivial case has physically
no meaning.
1.4 Quantum-Mechanical Operators and Matrices
As represented by (1.55), a quantum-mechanical operator corresponds to a physical
quantity. In (1.55), we connect a Hamiltonian operator to an energy (eigenvalue). Let
us rephrase the situation as follows:
PΨ ¼ pΨ :
ð1:103Þ
In (1.103), we are viewing P as an operation or measurement on a physical system
that is characterized by the quantum state Ψ . Operating P on the physical system
(or state), we obtain a physical quantity p relevant to P as a result of the operation
(or measurement).
A way to effectively achieve the above is to use a matrix and vector to represent
the operation and physical state, respectively. Let us glance a little bit of matrix
calculation to get used to the quantum-mechanical concept and, hence, to obtain
clear understanding about it. In Part III, we will deal with matrix calculation in detail
from a point of view of a general principle. At present, a (2, 2) matrix suffices. Let
A be a (2, 2) matrix expressed as
A ¼
a b
c d
:
ð1:104Þ
Let jψi be a (2, 1) matrix, i.e., a column vector such that
1.4 Quantum-Mechanical Operators and Matrices
21
p
2
2m
¼
ħ
2
2m
k
2 ,
where
k ¼
2l þ 1
ð
Þπ=2L l ¼ 0, 1, 2, Á Á Á
ð
Þ ,
2nπ=2L
n¼ 1, 2, 3, Á Á Á
ð
Þ :
8
> <
> :
The energy E is a kinetic energy of the particle.
Although in (1.97), ψ(x) 0 trivially holds, such a function may not be regarded
as a solution of the eigenvalue problem. In fact, considering that |ψ(x)|
2 represents
existence probability of a particle, ψ(x) 0 corresponds to a situation where a
particle in question does not exist. Consequently, such a trivial case has physically
no meaning.
1.4 Quantum-Mechanical Operators and Matrices
As represented by (1.55), a quantum-mechanical operator corresponds to a physical
quantity. In (1.55), we connect a Hamiltonian operator to an energy (eigenvalue). Let
us rephrase the situation as follows:
PΨ ¼ pΨ :
ð1:103Þ
In (1.103), we are viewing P as an operation or measurement on a physical system
that is characterized by the quantum state Ψ . Operating P on the physical system
(or state), we obtain a physical quantity p relevant to P as a result of the operation
(or measurement).
A way to effectively achieve the above is to use a matrix and vector to represent
the operation and physical state, respectively. Let us glance a little bit of matrix
calculation to get used to the quantum-mechanical concept and, hence, to obtain
clear understanding about it. In Part III, we will deal with matrix calculation in detail
from a point of view of a general principle. At present, a (2, 2) matrix suffices. Let
A be a (2, 2) matrix expressed as
A ¼
a b
c d
:
ð1:104Þ
Let jψi be a (2, 1) matrix, i.e., a column vector such that
1.4 Quantum-Mechanical Operators and Matrices
21