Physical Chemistry Third Edition

15.4 The Quantum Harmonic Oscillator 675

Eq. (14.2-32). If we set the constantV 0 equal to zero, the time-independent Schrödinger

equation is

Hψ̂ −h ̄

2

2 m

d^2 ψ

dx^2

+

1

2

kx^2 ψEψ (15.4-1)

We define the constants

b

2 mE

h ̄^2

, a

√

km

h ̄

(15.4-2)

so that the Schrödinger equation can be written

d^2 ψ/dx^2 +(b−a^2 x^2 )ψ 0 (15.4-3)

This differential equation is the same as a famous equation known as theHermite

equation(see Appendix F). Hermite solved this equation by assuming that the solution

was of the form

ψ(x)e−ax

(^2) / 2
S(x) (15.4-4)
whereS(x) is a power series
S(x)c 0 +c 1 x+c 2 x^2 +c 3 x^3 +...

∑∞

n 0

cnxn (15.4-5)

The Hermite equation is named for
Charles Hermite, 1822–1901, a great
French mathematician who made many
contributions to mathematics, including
the proof that e (2.71828...)isa
transcendental irrational number.

with constant coefficientsc 1 ,c 2 ,c 3 ,...When Eqs. (15.4-4) and (15.4-5) are substituted

into the Hermite equation and the exponential factor is canceled, an equation with power

series on both sides results. If two power series are equal to each other for all values

of the independent variable, the coefficients of the same power in the two series must

be equal to each other. From this fact Hermite obtained the relation

cn+ 2 

2 an+a−b

(n+2)(n+1)

cn (n0, 1, 2,...) (15.4-6)

Exercise 15.9

Assume that the following equation is valid for any value ofx:

a 0 +a 1 x+a 2 x^2 +...b 0 +b 1 x+b 2 x^2 +...

Show thata 0 b 0 by lettingx0. Differentiate both sides of the equation with respect tox,

and show thata 1 b 1 by lettingx0 in the formula for the derivative. Continue to show that

anbnfor any value ofn.

Equation (15.4-6) is called arecursion relation.Given a value ofcn, it provides a value

forcn+ 2. If we pick any value forc 0 and any other value forc 1 and let the recursion