Physical Chemistry Third Edition

17.2 The Relative Schrödinger Equation. Angular Momentum 735

TheYlmfunctions and the possible values ofL^2 andLzare the same for any

central-force system, no matter what the potential energy functionV(r) is. In a later

chapter we will study a model for the rotation of a diatomic molecule that is a central-

force system. The angular momentum of this model takes on the same values as that

of the hydrogen atom.

EXAMPLE17.2

Show that̂Hrel,̂L^2 , and̂Lzcommute with each other.

Solution

Ĥrel−h ̄

2

2 μr^2

∂

∂r

[

r^2

∂

∂r

]

+

1

2 μr^2

̂L^2 +V(r)

̂L^2 −h ̄^2

[

1

sin(θ)

∂

∂θ

[

sin(θ)

∂

∂θ

]

+

1

sin^2 (θ)

∂^2

∂φ^2

]

̂Lzh ̄

i

∂

∂φ

Ĥrelcommutes witĥL^2 becausêL^2 commutes with itself and contains nordependence

while the other terms in̂Hrelcontain noθorφdependence.̂Lzcommutes witĥL^2 because

the first term in̂L^2 contains noφdependence, and

∂

∂φ

commutes with

1

sin^2 (θ)

∂^2

∂φ^2

.̂Hrel

commutes witĥLzif̂L^2 commutes witĥLz, since the first term inĤrcontains noφ

dependence.

SinceĤrel,̂Lz, and̂L^2 commute with each other, they can possess a common set

of eigenfunctions. That is, the energy eigenfunctions can also be eigenfunctions of

thêLzand̂L^2 operators. The magnitude of the angular momentum and itszcom-

ponent will simultaneously have predictable values if the wave function is an eigen-

function of botĥL^2 and̂Lz. The operatorŝLx,̂Ly, and̂Lzdo not commute with

each other, so these three operators cannot have a full set of common eigenfunctions.

Only statistical predictions can be made aboutLxandLyif the wave function is an

eigenfunction of̂Lz.

Figure 17.4 depicts the possible angular momentum values for the case thatl2,

for which√ mcan take on the values 2, 1, 0,−1, and−2. The magnitude ofLis

6 h ̄ 2. 4495 h ̄, and the possible values ofLzare 2h ̄,h ̄,0,−h ̄, and− 2 h ̄. If the angular

momentum vector is measured without experimental error, it can point in any direction

on any one of the five cones drawn in the figure. Notice the similarity between each

cone on Figure 17.4 and the cone of directions around which a gyroscope axis pre-

cesses, as shown in Figure E.3 of Appendix E. For a given value ofl, there are 2l+ 1

cones, one for each possible value ofm. If the wave function is known to correspond

to a particular value ofland a particular value ofm, it is known which cone applies,

but the direction on that cone is not known.

z

x

y

m 52

m 51

m 50

m 521

m 522

Figure 17.4 Cones of Possible

Angular Momentum Directions for

l2.These cones represent possible

directions for the angular momentum

vector.

There is nothing unique about thezdirection. One could choosêLxor̂Lyas a

member of a set of commuting observables instead of̂Lz. In that event, theΦfunctions

would be different, and would correspond to cones in Figure 17.4 that would be oriented

around either thexaxis or theyaxis. We choose to emphasizeLzsince its operator is

simpler in spherical polar coordinates than those of the other components.