Physical Chemistry Third Edition

670 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

whereCis a constant and where the quantum numbersnx,ny, andnzare 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.

Enznynz

h^2

8 m

(

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 inxyzorder inside parentheses, as for example (1,2,3).

Ifabc(a cubical box) the energy eigenvalue is

Enznynz

h^2

8 ma^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 eigen-

value. A set of states with the same energy eigenvalue is called anenergy level, and the

number of states making up the energy level is called thedegeneracyof the level. The

lowest-energy state (“the ground state”) of this system isnondegenerate, because only

one state corresponds to its energy eigenvalue.

EXAMPLE15.4

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 

14 h^2

8 ma^2



(14)(6. 6261 × 10 −^34 Js)^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 degen-

eracy is 6.

Exercise 15.8

a.For an electron in the cubical box of Example 15.4 find the energy eigenvalues and degen-

eracies of all energy levels of lower energy than that in Example 15.4. Express the energies

asEdivided byh^2 / 8 ma^2.

b.A certain box isabyaby 4a. Find the energy levels and degeneracies (if any) for states up

to the energy corresponding to (3,3,3). Express the energies asEdivided byh^2 / 8 ma^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