Physical Chemistry , 1st ed.

Example 11.25

a.What is the average value of the total angular momentum for  3 pfor the

hydrogen atom?

b.Is there an easier way to determine this value?

Solution

a.The squareof the total angular momentum is defined, so we will assume

that the average angular momentum is the square root of the squared angu-

lar momentum. Therefore, we need to find

L L^2 

To do this, we need to determine the average value L^2 . This is done with the

following expression:

L^2 * 3 pLˆ^2  3 pd

It might seem at first that we may have to use the long, complete form of 3 p

and the long, complete form ofˆL^2 ; we don’t, though. Since  3 pis an eigen-

function ofˆL^2 , we can substitute the eigenvalue for the operator in the inte-

gral above. Since the eigenvalue is (1)^2 , the integral above becomes

L^2 * 3 p[(1)^2 ]  3 pd

where the constants are multiplied with the wavefunctions instead of any

function-changing operation occurring. Multiplicative constants are moved

outside the integral sign, so the above expression becomes

L^2 (1)^2 * 3 p 3 pd

The wavefunction is normalized, so the integral is simply 1. Therefore,

L^2 (1)^2

and using the value of1 for a porbital, we can determine the average

value of the total angular momentum Las

L L^2   (1) ^2  2 1.491 

10 ^34 J s

b.The easier way is to realize that L^2 is a quantized observable for the 3p

wavefunction of a hydrogen atom. The average value is equal to the quantized

value. This will not always be the case for average values (see exercises 11.58

or 11.60, for example).

11.12 Summary

With the solution of the hydrogen atom, the list of analytically solvable sys-
tems to be considered here is complete. Planck’s quantum theory of light de-
scribed blackbody radiation, and now the simplicity of the spectrum of the hy-
drogen (and hydrogen-like ions) is adequately explained by quantum
mechanics. We will find in the next chapter that although an exact analyticun-
derstanding of the behavior of electrons in larger atoms is not forthcoming,
quantum mechanics does provide the tools for a numerical solution to larger

11.12 Summary 365