15.3 The Particle in a Box and the Free Particle
665
where
κ
2
2mE
¯
h
2
(15.3-5)
Equation (15.3-4) is of the same form as Eq. (14.3-10). Its general solution is
ψ
(II) (x) Bcos(κx) + Csin(κx)
(15.3-6)
We must now make our solution conform to the continuity condition. In order for ψ to
be continuous at x 0 and x a, we must require that
ψ
(II) (0) ψ
(I) (0) 0; ψ
(II) (a) ψ
(III) (a) 0
(15.3-7)
These conditions are similar to the boundary conditions for the vibrations of a string
in Section 14.3. In order for ψ (II) (0) to vanish, the constant B must vanish, because
cos(0) 1. The coordinate wave function (energy eigenfunction) for region II is now
ψ
(II) (x) Csin(κx)
(15.3-8)
The continuity condition has another consequence. The condition that ψ (II) (a) 0
imposes a condition on κ, as in Eq. (14.3-14). The sine function vanishes when its
argument is an integral multiple of π, so that the quantity κ can take on the values
given by
nπ κa or κ nπ/a
(15.3-9)
where n is a quantum number that can take on integral values. The energy eigenfunctions
inside the box are now given by:
ψ n (x) Csin(nπx/a)
(15.3-10)
where we omit the superscript (II). We do not allow n 0, since this would correspond
to ψ 0 and the absence of any wave. If there is no wave there is no particle. We also
ignore negative values of n, since the sine function is an odd function and replacing n
by −n would simply change the sign of the function, corresponding to replacement of
C by −C. The value of C is unimportant for our present purposes since the Schrödinger
equation is satisfied for any value of C and since the energy eigenvalues do not depend
on C.
The energy eigenvalues are quantized, with values determined by the value of n:
E E n
¯
h
2 κ 2
2m
¯
h
2 n 2 π 2
2ma 2
h 2 n 2
8ma 2
(15.3-11)
The quantization of the energy comes from the Schrödinger equation and from the
boundary conditions.
Figure 15.4a represents the energy eigenvalues of a particle in a one-dimensional
box by horizontal line segments at heights proportional to their energy values, and
Figure 15.4b shows the energy eigenfunctions. Each wave function is plotted on an
axis that is placed at a height corresponding to its energy eigenvalue. Note that the
wave functions in Figure 15.4b resemble the standing waves in Figure 14.7.
The energy in Eq. (15.3-11) is all kinetic energy, since we set the potential energy
inside the box equal to zero. Since we do not allow n 0, the minimum possible kinetic
energy is equal to h 2 /8ma 2 , corresponding to n 1. This minimum energy is positive
and is called the zero-point energy. It is not possible for the particle in a box to have
zero kinetic energy. This result is very different from that of classical mechanics, which
always allows a particle to be at rest.
665
where
κ
2
2mE
¯
h
2
(15.3-5)
Equation (15.3-4) is of the same form as Eq. (14.3-10). Its general solution is
ψ
(II) (x) Bcos(κx) + Csin(κx)
(15.3-6)
We must now make our solution conform to the continuity condition. In order for ψ to
be continuous at x 0 and x a, we must require that
ψ
(II) (0) ψ
(I) (0) 0; ψ
(II) (a) ψ
(III) (a) 0
(15.3-7)
These conditions are similar to the boundary conditions for the vibrations of a string
in Section 14.3. In order for ψ (II) (0) to vanish, the constant B must vanish, because
cos(0) 1. The coordinate wave function (energy eigenfunction) for region II is now
ψ
(II) (x) Csin(κx)
(15.3-8)
The continuity condition has another consequence. The condition that ψ (II) (a) 0
imposes a condition on κ, as in Eq. (14.3-14). The sine function vanishes when its
argument is an integral multiple of π, so that the quantity κ can take on the values
given by
nπ κa or κ nπ/a
(15.3-9)
where n is a quantum number that can take on integral values. The energy eigenfunctions
inside the box are now given by:
ψ n (x) Csin(nπx/a)
(15.3-10)
where we omit the superscript (II). We do not allow n 0, since this would correspond
to ψ 0 and the absence of any wave. If there is no wave there is no particle. We also
ignore negative values of n, since the sine function is an odd function and replacing n
by −n would simply change the sign of the function, corresponding to replacement of
C by −C. The value of C is unimportant for our present purposes since the Schrödinger
equation is satisfied for any value of C and since the energy eigenvalues do not depend
on C.
The energy eigenvalues are quantized, with values determined by the value of n:
E E n
¯
h
2 κ 2
2m
¯
h
2 n 2 π 2
2ma 2
h 2 n 2
8ma 2
(15.3-11)
The quantization of the energy comes from the Schrödinger equation and from the
boundary conditions.
Figure 15.4a represents the energy eigenvalues of a particle in a one-dimensional
box by horizontal line segments at heights proportional to their energy values, and
Figure 15.4b shows the energy eigenfunctions. Each wave function is plotted on an
axis that is placed at a height corresponding to its energy eigenvalue. Note that the
wave functions in Figure 15.4b resemble the standing waves in Figure 14.7.
The energy in Eq. (15.3-11) is all kinetic energy, since we set the potential energy
inside the box equal to zero. Since we do not allow n 0, the minimum possible kinetic
energy is equal to h 2 /8ma 2 , corresponding to n 1. This minimum energy is positive
and is called the zero-point energy. It is not possible for the particle in a box to have
zero kinetic energy. This result is very different from that of classical mechanics, which
always allows a particle to be at rest.