670
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
where C is a constant and where the quantum numbers n x , n y , and n z are positive
integers, not necessarily equal to each other. The energy eigenvalue is the sum of three
terms, each of which is an energy eigenvalue for a particle in a one-dimensional box.
E n z n y n z
h 2
8m
n 2
x
a 2 +
n 2
y
b 2 +
n 2
z
c 2
(15.3-22)
A particular energy state is specified by giving the values of the three quantum numbers,
which we write in xyz order inside parentheses, as for example (1,2,3).
If a b c (a cubical box) the energy eigenvalue is
E n z n y n z
h 2
8ma 2
n
2
x + n
2
y + n
2
z
(15.3-23)
For a cubical box there can be several states that correspond to the same energy eigenvalue. A set of states with the same energy eigenvalue is called an energy level, and the
number of states making up the energy level is called the degeneracy of the level. The
lowest-energy state (“the ground state”) of this system is nondegenerate, because only
one state corresponds to its energy eigenvalue.
E X A M P L E 15.4
For an electron in a cubical box of side 1.00 × 10 −9 m, find the energy eigenvalue and the
degeneracy of the level in which the state corresponding to (1,2,3) occurs.
Solution
The energy eigenvalue is
E 123
14h 2
8ma 2
(14)(6.6261 × 10 −34 J s) 2
(8)(9.109 × 10 −31 kg)(1.00 × 10 −9 m) 2
8.43 × 10 −19 J
There are six permutations (different orders) of the three distinct numbers: (1,2,3), (2,3,1),
(3,1,2), (3,2,1), (1,3,2), and (2,1,3), each of which corresponds to a different state. The degeneracy is 6.
Exercise 15.8
a. For an electron in the cubical box of Example 15.4 find the energy eigenvalues and degeneracies of all energy levels of lower energy than that in Example 15.4. Express the energies
as E divided by h 2 /8ma 2 .
b. A certain box is a by a by 4a. Find the energy levels and degeneracies (if any) for states up
to the energy corresponding to (3,3,3). Express the energies as E divided by h 2 /8ma 2 .
The Free Particle in One Dimension
A free particle has no forces acting on it and can move in all of space. The potential
energy of the particle is equal to a constant, which we set equal to zero. If a free particle
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
where C is a constant and where the quantum numbers n x , n y , and n z are positive
integers, not necessarily equal to each other. The energy eigenvalue is the sum of three
terms, each of which is an energy eigenvalue for a particle in a one-dimensional box.
E n z n y n z
h 2
8m
n 2
x
a 2 +
n 2
y
b 2 +
n 2
z
c 2
(15.3-22)
A particular energy state is specified by giving the values of the three quantum numbers,
which we write in xyz order inside parentheses, as for example (1,2,3).
If a b c (a cubical box) the energy eigenvalue is
E n z n y n z
h 2
8ma 2
n
2
x + n
2
y + n
2
z
(15.3-23)
For a cubical box there can be several states that correspond to the same energy eigenvalue. A set of states with the same energy eigenvalue is called an energy level, and the
number of states making up the energy level is called the degeneracy of the level. The
lowest-energy state (“the ground state”) of this system is nondegenerate, because only
one state corresponds to its energy eigenvalue.
E X A M P L E 15.4
For an electron in a cubical box of side 1.00 × 10 −9 m, find the energy eigenvalue and the
degeneracy of the level in which the state corresponding to (1,2,3) occurs.
Solution
The energy eigenvalue is
E 123
14h 2
8ma 2
(14)(6.6261 × 10 −34 J s) 2
(8)(9.109 × 10 −31 kg)(1.00 × 10 −9 m) 2
8.43 × 10 −19 J
There are six permutations (different orders) of the three distinct numbers: (1,2,3), (2,3,1),
(3,1,2), (3,2,1), (1,3,2), and (2,1,3), each of which corresponds to a different state. The degeneracy is 6.
Exercise 15.8
a. For an electron in the cubical box of Example 15.4 find the energy eigenvalues and degeneracies of all energy levels of lower energy than that in Example 15.4. Express the energies
as E divided by h 2 /8ma 2 .
b. A certain box is a by a by 4a. Find the energy levels and degeneracies (if any) for states up
to the energy corresponding to (3,3,3). Express the energies as E divided by h 2 /8ma 2 .
The Free Particle in One Dimension
A free particle has no forces acting on it and can move in all of space. The potential
energy of the particle is equal to a constant, which we set equal to zero. If a free particle