Physical Chemistry , 1st ed.

before quantum mechanics was developed. The polynomial parts of the har-

monic oscillator wavefunctions are called Hermite polynomialsafter Charles

Hermite, the nineteenth-century French mathematician who studied their

properties. For convenience, if we define  1/2x(where is the Greek let-

ter xi, pronounced “zigh”), then the Hermite polynomial whose largest power

ofxis nis labeled Hn(). The first few Hn() polynomials are listed in Table

11.1, and Table 11.2 gives the solutions to an integral involving the Hermite

polynomials. Tables 11.1 and 11.2 should be used with care because of the vari-

able change. The following example illustrates some of the potential pitfalls in

using tabulated Hermite polynomials.

Example 11.5

Using the integrals in Table 11.2, normalize  1 for a quantum-mechanical

harmonic oscillator.

Solution

The integral from Table 11.2 will have to be used with care, because of the

differences in the variables between the equation in the table and the wave-

function  1 .If 1/2x, then d1/2dx, and after substitution for and

dthe integral can be applied directly. The normalization requirement means,

mathematically,







*dx 1

The limits on the integral are and , and the infinitesimal is dxfor the

one-dimensional integrand. For the  1 wavefunction of the harmonic oscil-

lator, it is assumed that the wavefunction is multiplied by some constant N

such that

N^2 





[H 1 (1/2x) ex

(^2) /2
] H 1 (1/2x) ex
(^2) /2
dx 1
Substituting for and d, this is transformed into
N^2 


[H 1 () e
(^2) /2
] H 1 () e
(^2) /2


d
1



/2 1

The complex conjugate does not change the wavefunction and so can be ig-

nored.1/2is a constant and can be moved outside the integral. The func-

tions inside the integral sign are all multiplied together, and so the integral

can be simplified to





N

1/

2

 2 





H 1 () H 1 () e

2

d 1

According to Table 11.2, this integral has a known form and, for n1,

equals 2^1 1!1/2(where! indicates a factorial). Therefore,





N

1/

2

 22

(^1) 1!1/2 1
N^2 
2





1

1

/2

/2

N



2

1



/4

1/4

326 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom

Table 11.1 The first six Hermite

polynomialsa

nHn()

0 1

1 2

2 4^2  2

3 8^3  12 

4 16^4  48 ^2  12

5 32^5  160 ^3  120 

6 64^6  480 ^4  720 ^2  120

aIn the treatment of the harmonic oscillator, note that

1/2x.

Table 11.2 Integral involving Hermite

polynomials







Ha()*Hb()e^2 d 0 if 2 aa!a1/2bifab