660
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
We substitute (15.2-11) into Eq. (15.2-9) and divide by ψ(x)η(t), obtaining
1
ψ
Hψ
i ¯
h
η
dη
dt
(15.2-12)
The variables x and t are now separated. Since x and t are independent variables, each
can be held fixed while the other varies. Each side of the equation must be a constant
function of its argument and must be equal to the same constant, which we denote by E:
1
ψ
Hψ E
(15.2-13)
and
i ¯
h
η
dη
dt
E
(15.2-14)
Multiplication of the first equation by ψ and of the second equation by η/i ¯
h gives
Hψ Eψ
(15.2-15)
and
dη
dt
E
i ¯
h
η −
iE
¯
h
η
(15.2-16)
Equation (15.2-15) is the same as the time-independent Schrödinger equation,
Eq. (15.2-8), so ψ is the coordinate wave function that satisfies that equation and
E is the constant energy of the system.
Equation (15.2-16) has the solution
η(t) Ce
−iEt/¯ h
(15.2-17)
where C is a constant. Since the Eq. (15.2-16) is satisfied for any value of C its value
is unimportant. We let C 1 and write the complete wave function as
Ψ(x, t) ψ(x)e
−iEt/¯ h
(15.2-18)
We will see that a solution to the time-independent Schrödinger equation provides
both a coordinate wave function ψ and an energy value E. We can immediately write a
solution to the time-dependent equation by multiplying a coordinate wave function by
the time factor e −iEt/¯ h . This type of solution, with the coordinate and time dependence
in separate factors, corresponds to a standing wave, because any nodes are stationary.
There are also solutions of the time-dependent Schrödinger equation that are not products of a coordinate factor and a time factor. These solutions can correspond to traveling
waves.
The coordinate wave function can in many cases be chosen to be a real function.
The function η is always complex, and can be written as a real part plus an imaginary
part by use of a mathematical identity (see Appendix B):
η(t) e
−iEt/¯ h
cos(−Et/ ¯
h) + isin(−Et/ ¯
h) cos(Et/ ¯
h) − isin(Et/ ¯
h) (15.2-19)
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
We substitute (15.2-11) into Eq. (15.2-9) and divide by ψ(x)η(t), obtaining
1
ψ
Hψ
i ¯
h
η
dη
dt
(15.2-12)
The variables x and t are now separated. Since x and t are independent variables, each
can be held fixed while the other varies. Each side of the equation must be a constant
function of its argument and must be equal to the same constant, which we denote by E:
1
ψ
Hψ E
(15.2-13)
and
i ¯
h
η
dη
dt
E
(15.2-14)
Multiplication of the first equation by ψ and of the second equation by η/i ¯
h gives
Hψ Eψ
(15.2-15)
and
dη
dt
E
i ¯
h
η −
iE
¯
h
η
(15.2-16)
Equation (15.2-15) is the same as the time-independent Schrödinger equation,
Eq. (15.2-8), so ψ is the coordinate wave function that satisfies that equation and
E is the constant energy of the system.
Equation (15.2-16) has the solution
η(t) Ce
−iEt/¯ h
(15.2-17)
where C is a constant. Since the Eq. (15.2-16) is satisfied for any value of C its value
is unimportant. We let C 1 and write the complete wave function as
Ψ(x, t) ψ(x)e
−iEt/¯ h
(15.2-18)
We will see that a solution to the time-independent Schrödinger equation provides
both a coordinate wave function ψ and an energy value E. We can immediately write a
solution to the time-dependent equation by multiplying a coordinate wave function by
the time factor e −iEt/¯ h . This type of solution, with the coordinate and time dependence
in separate factors, corresponds to a standing wave, because any nodes are stationary.
There are also solutions of the time-dependent Schrödinger equation that are not products of a coordinate factor and a time factor. These solutions can correspond to traveling
waves.
The coordinate wave function can in many cases be chosen to be a real function.
The function η is always complex, and can be written as a real part plus an imaginary
part by use of a mathematical identity (see Appendix B):
η(t) e
−iEt/¯ h
cos(−Et/ ¯
h) + isin(−Et/ ¯
h) cos(Et/ ¯
h) − isin(Et/ ¯
h) (15.2-19)