Physical Chemistry , 1st ed.

Despite the separated wavefunctions in one dimension each, it is important to

understand that the operator must operate on the entire wavefunction.

Although the entire wavefunction is in three dimensions, the one-dimensional

operator acts only on the part that depends on the coordinate of interest.

Also, it needs to be understood that average values are treated differently in

the 3-D case than in the 1-D case, because of the additional dimensions. Because

each dimension is independent of the other, an integration must be performed

over each dimension independently. This triples the number of integrals to be

evaluated, but since the wavefunction can be separated into x,y, and zparts, the

integrals are straightforward to evaluate. Since this system is three-dimensional,

the dfor the integration must have three infinitesimals:ddx dy dz. For nor-

malized wavefunctions, the average value of an observable is thus given by

A 

x



y



z

*Aˆdx dy dz (10.23)

For wavefunctions and operators that are separable into x,y, and zparts, this

triple integral ultimately separates into the product of three integrals:

A 

x

x*ˆAxxdx 

y

*yˆAyydy 

z

*zˆAzzdz

Each integral has its own limits, depending on the limits of the particular sys-

tem in that dimension. If the operator does not include a certain dimension,

then it has no influence on the integral over that dimension. The following ex-

ample illustrates.

Example 10.14

Although the particle-in-a-box wavefunctions are not eigenfunctions of the

momentum operators, we can determine average or expectation values for

the momentum. Find py for the 3-D wavefunction

(x,y,z) 

a

8

bc

sin^

1

a

x

sin

2

b

y

sin

3

c

z

(This wavefunction has nx1,ny2, and nz3.)

Solution

In order to determine py , the following integral must be evaluated:

py 

x



y



z^

a

8

bc

sin^

1

a

x

sin

2

b

y

sin

3

c

z

i





y

(^) 
a

8

bc

sin^1

a

x sin   2



b

y sin   3



c

z

dx dy dz

Although this looks complicated, it can be simplified into the product of three

integrals, where the normalization constant will be split appropriately and the

operator, affecting only the ypart of, appears only in the integral over y:

py 

a

x 0

2

a

sin^

1

a

x

2

a

sin^

1

a

x

dx

 

b

y 0

2

b

sin^

2

b

y



i





y

2

a

sin^

2

b

y

dy

 

c

z 0

2

c

sin^3

c

z

2

c

sin^3

c

zdz

302 CHAPTER 10 Introduction to Quantum Mechanics