Physical Chemistry , 1st ed.

differential terms in the Hamiltonian contain , so no simplification is gained.

(The third term is not a differential in terms of, but it does have the sin^2 term

in the denominator.) The mixing of variables in the third term introduces an-

other twist. The ultimate solution for () will also depend on the quantum

number m. In addition, we will find that the limitations imposed on the accept-

able wavefunctions (that is, they must be bounded) will generate a relationship

between the quantum number mand whatever new quantum number arises.

The part of the differential in equation 11.46 does have a known solution.

The solution is a set of functions known as associated Legendre polynomials.(As

with the Hermite polynomials, differential equations of the form in equation

11.46 had been previously studied, by the French mathematician Adrien

Legendre, but for different reasons.) These polynomials, listed in Table 11.3,

are functions ofonly, but have two indices labeling the functions. One of the

indices, an integer denoted , indicates the maximum power, or order,of

terms. (It also indicates the total order of the combination of cos and sin 

terms.) The second index,m, specifies which particular combination of

sin and cos terms are in the Legendre polynomial of that particular order.

For associated Legendre polynomials, the absolute value ofmyields the same

polynomial. The possible combinations are limited to those where the absolute

value ofmalways has a value less than or equal to . That is, because of the re-

quirements of the associated Legendre polynomials, there is a new quantum

numberwhose value must be some nonnegative integer:

0,1,2,... (11.47)

and the only possible values of the mquantum number associated with any

particular quantum number are those integers whose absolute value is less

than or equal to :

m (11.48)

These constraints are imposed by the forms of the polynomials, which must be

acceptable wavefunctions and eigenfunctions of the Schrödinger equation.Because

of the limit that puts on m, it is common to use the symbol mas the label

for this quantum number.

Example 11.15

List the possible values ofmfor the first five possible values of.

Solution

The quantum number can have integer values starting from zero. Therefore,

the first five possible values ofare 0, 1, 2, 3, and 4. For 0,mcan only

be 0. For 1, the absolute value ofmmust be less than or equal to 1, so

for integers the only possibilities are 0, 1, and 1 (otherwise listed as 1, 0,

1). For 2, the possible integer values ofmare 2,1, 0, 1, and 2. For

3, the possible mvalues are 3,2,1, 0, 1, 2, and 3. For 4, the

possible values ofmare 4,3,2,1, 0, 1, 2, 3, and 4.

Because the quantum number mcan have integer values from to in-

cluding 0, there are 21 possible values ofmfor each value of.

Table 11.3 lists several of the associated Legendre polynomials. They are rep-

resented here as ,m. It doesn’t matter whether mis positive or negative; its

magnitude determines which polynomial is needed.

11.7 Three-Dimensional Rotations 343

Table 11.3 The associated Legendre
polynomials ,m
 m ,m
0 0 ^12  2
1 0 ^12  6 cos 
1  1 ^12  3 sin 
2 0 ^14  10 (cos^2 1)
2  1 ^12  15 sin cos 
2  2 ^14  15 sin^2 
3 0 ^34  14 (^53 cos^3 cos )
3  1 ^18  42 sin (5 cos^2 12)
3  2 ^14  105 sin^2 cos 
3  3 ^18  70 sin^3