15.3 The Particle in a Box and the Free Particle
671
of mass m can move only parallel to the x axis, the time-independent Schrödinger
equation is
−
¯
h
2
2m
d 2 ψ
dx 2 Eψ
(15.3-24)
or
d
2 ψ/dx
2
−κ
2 ψ
(15.3-25)
Equation (15.3-24) is the same as Eq. (15.3-4) for the motion of a particle in a box.
The general solution to Eq. (15.3-25) is the same as that in Eq. (15.3-6), but it can also
be written in a way analogous to Eq. (14.2-22):
ψ(x) De
iκx
+ Fe
−iκx
(15.3-26)
Although the general solution to the Schrödinger equation is the same as for the
particle in a box, the boundary conditions are different. The wave function must be
continuous and finite, but there are now no walls at which the wave function must
vanish. The finiteness condition requires that κ be real. To show this we let
κ a + ib
(15.3-27)
where a and b are real. The solution is now
ψ(x) De
iax e
−bx
+ Fe
−iax e
bx
(15.3-28)
If b is positive the second term grows without bound for large positive values of x.
If b is negative the first term grows without bound if x becomes large and negative.
To keep the wave function finite for all values of x, b must vanish and κ must be real.
The energy eigenvalues are given by Eq. (15.3-5):
E ¯
h
2 κ
2 /2m
(15.3-29)
There is no restriction on the values of the parameter κ except that it must be real.
The energy eigenvalue E, which is equal to the kinetic energy, can take on any real
non-negative value. The energy is not quantized and there is no zero-point energy.
If F vanishes, the time-dependent wave function is
Ψ(x, t) De
iκx−iEt/¯ h
De
i(κx−Et/ ¯
h)
(15.3-30)
where E is given by Eq. (15.3-29). Separating the real and imaginary parts of Ψ by use
of the identity
e
iα
cos(α) + isin(α)
(15.3-31)
we obtain
ψ(x, t) D
cos
κx −
Et
¯
h
+ isin
κx −
Et
¯
h
(15.3-32)
Comparison of this with Eq. (14.3-26) shows both the real and imaginary parts to be
traveling waves moving to the right with a speed given by
c E/κ ¯
h ¯
hκ/2m
(15.3-33)
671
of mass m can move only parallel to the x axis, the time-independent Schrödinger
equation is
−
¯
h
2
2m
d 2 ψ
dx 2 Eψ
(15.3-24)
or
d
2 ψ/dx
2
−κ
2 ψ
(15.3-25)
Equation (15.3-24) is the same as Eq. (15.3-4) for the motion of a particle in a box.
The general solution to Eq. (15.3-25) is the same as that in Eq. (15.3-6), but it can also
be written in a way analogous to Eq. (14.2-22):
ψ(x) De
iκx
+ Fe
−iκx
(15.3-26)
Although the general solution to the Schrödinger equation is the same as for the
particle in a box, the boundary conditions are different. The wave function must be
continuous and finite, but there are now no walls at which the wave function must
vanish. The finiteness condition requires that κ be real. To show this we let
κ a + ib
(15.3-27)
where a and b are real. The solution is now
ψ(x) De
iax e
−bx
+ Fe
−iax e
bx
(15.3-28)
If b is positive the second term grows without bound for large positive values of x.
If b is negative the first term grows without bound if x becomes large and negative.
To keep the wave function finite for all values of x, b must vanish and κ must be real.
The energy eigenvalues are given by Eq. (15.3-5):
E ¯
h
2 κ
2 /2m
(15.3-29)
There is no restriction on the values of the parameter κ except that it must be real.
The energy eigenvalue E, which is equal to the kinetic energy, can take on any real
non-negative value. The energy is not quantized and there is no zero-point energy.
If F vanishes, the time-dependent wave function is
Ψ(x, t) De
iκx−iEt/¯ h
De
i(κx−Et/ ¯
h)
(15.3-30)
where E is given by Eq. (15.3-29). Separating the real and imaginary parts of Ψ by use
of the identity
e
iα
cos(α) + isin(α)
(15.3-31)
we obtain
ψ(x, t) D
cos
κx −
Et
¯
h
+ isin
κx −
Et
¯
h
(15.3-32)
Comparison of this with Eq. (14.3-26) shows both the real and imaginary parts to be
traveling waves moving to the right with a speed given by
c E/κ ¯
h ¯
hκ/2m
(15.3-33)