Physical Chemistry , 1st ed.

 511 

a

8

    3 sin^

5

a

x sin   1



a

y sin   1



a

z

 151 

a

8

    3 sin^

1

a

x

sin

5

a

y

sin

1

a

z

 115 

a

8

    3 sin^

1

a

x

sin

1

a

y

sin

5

a

z

(Here we are writing the quantum number 1 to illustrate the point; typically,

1 values aren’t written explicitly.) It should be obvious that these four wave-

functions are all different, having integer quantum numbers that are either

different or in different parts of the wavefunction.

10.13 Orthogonality

One other major property of wavefunctions needs to be introduced. We should

recognize by now that a system has not just a single wavefunction but many

possible wavefunctions, each of which has an energy (obtained using an eigen-

value equation) and perhaps other eigenvalue observables. We can summarize

the multiple solutions to the Schrödinger equation by writing it as

HˆnEnn n1,2,3,... (10.26)

Equation 10.26, when satisfied, usually yields not just a single wavefunction

but a set of them (perhaps even an infinite number), like those for the parti-

cle-in-a-box. Mathematically, this set of equations has a very useful property.

The wavefunctions must be normalized, for every n:



all

*nnd 1

space

This is the expression that defines normalization. If, on the other hand, two dif-

ferentwavefunctions were used in the above expression, the different wave-

functions mand nhave a property that requires that the integral be exactly

zero:



all

*mnd 0 mn (10.27)

space

It does not matter in what order the wavefunctions are multiplied together.

The integral will still be identically zero. This property is called orthogonality;

the wavefunctions are orthogonalto each other. Orthogonality is useful be-

cause, once we know that all wavefunctions of a system are orthogonal to each

other, many integrals become identically zero. We need only recognize that the

wavefunctions inside an integral are different and we can apply the orthogo-

nality property: that integral equals zero. Both wavefunctions must be for the

same system, they must have different eigenvalues,†and there must be no op-

erator in the integral sign (there may be a constant operator, but constants

can be removed from inside the integral and what remains must satisfy equa-

tion 10.27).

306 CHAPTER 10 Introduction to Quantum Mechanics

†Equation 10.26 does not apply if the two wavefunctions mand nhave the same

energy eigenvalue (that is, if they are degenerate). Other considerations are necessary to

circumvent this, but we will not discuss that here.