Physical Chemistry Third Edition

676 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

relation pick the other values, then Eq. (15.4-4) gives a solution to the Schrödinger

equation.

We must now apply the appropriate boundary conditions to our solution. The solution

must be finite all values ofx, including very large values of|x|. If the series is permitted

to have infinitely many terms, it is found that the series becomes large very rapidly for

large values of|x|and the wave function fails to remain finite.^5 Therefore, the series

cannot have infinitely many terms. It must be a polynomial.

We cannot simply require all coefficients past a certain point in the series to vanish if

this violates the recursion relation. The function would then fail to satisfy the Hermite

equation. Let us assume thatcv+ 2 is a vanishing coefficient and thatcvdoes not

vanish. The numerator in the right-hand side of Eq. (15.4-6) must then vanish for

nv:

2 av+a−b 2 av+a− 2 mE/h ̄^2  2 a(v+ 1 / 2 )− 2 mE/h ̄^2  0 (15.4-7)

We can solve this equation for the energy eigenvalues:

EEv

h ̄^2

2 m

2 a

(

v+

1

2

)



h

2 π

√

k

m

(

v+

1

2

)

hνclass

(

v+

1

2

)

(15.4-8)

whereνclassis the frequency of the oscillator as predicted by classical mechanics,

given in Eq. (14.2-29), and where the quantum numberνequals 0, 1, 2, 3, and so on.

The energy is quantized and there is a zero-point energy:

E 0 

1

2

hνclass (zero-point energy) (15.4-9)

The recursion relation relates the coefficients of even powers to each other, and the

coefficients of the odd powers to each other. Since there is only one recursion relation,

each polynomial can contain only even powers or only odd powers. If the power series

contained both even and odd powers, the recursion relation would cause only the even

powers to terminate or the odd powers to terminate, and the other powers would not

terminate, violating the finiteness condition. The series in our solution now represents

a polynomial that contains only even powers ofxor only odd powers ofx. These

polynomials are calledHermite polynomials.

In the case of the particle in a box, the quantization was produced by the condition

that the wave function must be continuous at the ends of the box. In the case of the

harmonic oscillator, the energy quantization is produced by the condition that the wave

function must be finite for all values ofx, which causes the series to terminate.

EXAMPLE15.5

If a harmonic oscillator has an electric charge, it can absorb or emit electromagnetic radiation.

a.Find a formula for the frequency of a photon with energy equal to the difference in energy

between thev0 state and thev1 state.

b.How does the frequency of the photon compare with the classical frequency of the oscil-

lator? How do you interpret this comparison?

(^5) This assertion is not obvious. See I. N. Levine,Quantum Chemistry, 5th ed., Prentice-Hall, Englewood
Cliffs, NJ, 2000, pp. 67–68.