672
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
and a frequency given by
ν
2πE
¯
h
E
h
¯
hκ 2
4π 2 m
hκ 2
2πm
(15.3-34)
A nonzero value of the constant F corresponds to a traveling wave moving to the left.
If D and F are equal, the two traveling waves interfere to produce a standing wave:
ψ(x) D(e
iκx
+ e
−iκx ) 2Dcos(κx)
(15.3-35)
which follows from the identity
cos(α)
1
2
e
iα
+ e
−iα
(15.3-36)
The time-dependent wave function corresponding to Eq. (15.3-35) is
Ψ(x, t) 2Dcos(κx)e
−iEt/¯ h
(15.3-37)
If D −F , a different standing wave results.
If the constants D and F are not equal to each other the complete wave function is
Ψ(x, t) De
i(κx−Et/ ¯
h)
+ Fe
−i(κx+Et/ ¯
h)
(15.3-38)
which represents a combination of traveling waves, one moving to the right and one
moving to the left. A single wave function corresponds to motion in two different directions. This behavior is rather different from that found in classical mechanics, in which
one state always corresponds to only one kind of behavior. The idea that a single particle
can have a single state corresponding to motion in two different directions at the same
time seems unreasonable, but it is permitted in quantum mechanics. A possible interpretation is that since some predictions of quantum mechanics are statistical in nature,
a wave function should be thought of as representing the behavior of a large collection
(an ensemble) of objects, all in the same state but capable of different outcomes of a
particular measurement. We will return to this question in the next chapter.
The Free Particle in Three Dimensions
The time-independent Schrödinger equation for a free particle moving in three dimensions is
∂ 2 ψ
∂x 2 +
∂ 2 ψ
∂y 2 +
∂ 2 ψ
∂z 2
−
2mE
¯
h
2
ψ
(15.3-39)
This is the same as for a particle inside a three-dimensional box, and it can be solved
in the same way by separation of variables. Each factor in the solution can be written
either in the form of Eq. (15.3-6) or in the form of Eq. (15.3-26), with κ replaced by
κ x , κ y , or κ z . For the special case of a traveling wave with definite values of κ x , κ y , and
κ z , we can write a one-term energy eigenfunction as
ψ(x, y, z) De
iκ x x e
iκ y y e
iκ z z
(15.3-40)
The vector κ with components κ x , κ y , and κ z points in the direction in which the
traveling wave moves and is called the wave vector.
The energy eigenvalue is given by the sum of three terms:
E E x + E y + E z
¯
h
2
2m
(κ
2
x + κ
2
y + κ
2
z )
¯
h
2 κ 2
2m
(15.3-41)
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
and a frequency given by
ν
2πE
¯
h
E
h
¯
hκ 2
4π 2 m
hκ 2
2πm
(15.3-34)
A nonzero value of the constant F corresponds to a traveling wave moving to the left.
If D and F are equal, the two traveling waves interfere to produce a standing wave:
ψ(x) D(e
iκx
+ e
−iκx ) 2Dcos(κx)
(15.3-35)
which follows from the identity
cos(α)
1
2
e
iα
+ e
−iα
(15.3-36)
The time-dependent wave function corresponding to Eq. (15.3-35) is
Ψ(x, t) 2Dcos(κx)e
−iEt/¯ h
(15.3-37)
If D −F , a different standing wave results.
If the constants D and F are not equal to each other the complete wave function is
Ψ(x, t) De
i(κx−Et/ ¯
h)
+ Fe
−i(κx+Et/ ¯
h)
(15.3-38)
which represents a combination of traveling waves, one moving to the right and one
moving to the left. A single wave function corresponds to motion in two different directions. This behavior is rather different from that found in classical mechanics, in which
one state always corresponds to only one kind of behavior. The idea that a single particle
can have a single state corresponding to motion in two different directions at the same
time seems unreasonable, but it is permitted in quantum mechanics. A possible interpretation is that since some predictions of quantum mechanics are statistical in nature,
a wave function should be thought of as representing the behavior of a large collection
(an ensemble) of objects, all in the same state but capable of different outcomes of a
particular measurement. We will return to this question in the next chapter.
The Free Particle in Three Dimensions
The time-independent Schrödinger equation for a free particle moving in three dimensions is
∂ 2 ψ
∂x 2 +
∂ 2 ψ
∂y 2 +
∂ 2 ψ
∂z 2
−
2mE
¯
h
2
ψ
(15.3-39)
This is the same as for a particle inside a three-dimensional box, and it can be solved
in the same way by separation of variables. Each factor in the solution can be written
either in the form of Eq. (15.3-6) or in the form of Eq. (15.3-26), with κ replaced by
κ x , κ y , or κ z . For the special case of a traveling wave with definite values of κ x , κ y , and
κ z , we can write a one-term energy eigenfunction as
ψ(x, y, z) De
iκ x x e
iκ y y e
iκ z z
(15.3-40)
The vector κ with components κ x , κ y , and κ z points in the direction in which the
traveling wave moves and is called the wave vector.
The energy eigenvalue is given by the sum of three terms:
E E x + E y + E z
¯
h
2
2m
(κ
2
x + κ
2
y + κ
2
z )
¯
h
2 κ 2
2m
(15.3-41)