Physical Chemistry , 1st ed.

Solution

According to equation 11.53, the total angular momentum is dependent only

on and .For 4 and 5, the angular momenta of the electrons are

L(4)  4(4 1) (6.626 

10 ^34 J s)/2

L(5)  5(5 1) (6.626 

10 ^34 J s)/2

Evaluating the above expressions:

L(4) 4.716 

10 ^34 J s

L(5) 5.776 

10 ^34 J s

There is a third observable of interest. It is the zcomponent of the total an-

gular momentum,Lz. The relationships between angular momentum opera-

tors allow for the simultaneous knowledge of the total angular momentum

(through its square,L^2 ) and oneof its Cartesian components. By convention,

the zcomponent is chosen. This is due in part to the spherical polar coordi-

nate system, and the relatively simple definition of the zcomponent of the an-

gular momentum in terms of, as seen in the discussion of the 2-D rotating

system.

As before, the zcomponent of angular momentum is defined as

Lˆzi







 (11.54)

This is the same operator we used for 2-D rotation. Since the part of the

3-D rotational wavefunction is exactly the same as for the 2-D rotational wave-

function, it may not surprise you that the eigenvalue equation, and therefore

the value of the observable Lz, is exactly the same:

Lˆzm (11.55)

The zcomponent of the three-dimensional angular momentum, which has

components in the x,y, and zdirection, is quantized.Its quantized value de-

pends on the mquantum number.

Lzis only one component of the total angular momentum L. The other

components are Lxand Ly. However, the principles of quantum mechanics do

not allow us to know quantized values for these two components simultane-

ously with Lz. Therefore, only one of the three components of the total angu-

lar momentum can have a known eigenvalue simultaneously with the L^2 itself.

For convenience, we choose the zcomponent of the angular momentum,Lz,

to be the knowable observable.*

Graphically, the quantized total and z-component angular momenta are il-

lustrated in Figure 11.14. The length of each vector represents the total angu-

lar momentum and is the same for all five vectors where 2. However, the

zcomponents of the five vectors are different, each one indicating a different

value of the mquantum number (from 2 to 2). This figure also illustrates

that all of the momentum cannot be completely in the zdirection, since there

is no nonzero integer Wsuch that W ( W)(W 1).

348 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom

*Technically, we could choose the xor ycomponent of the total angular momentum to

be the knowable observable, but the zcomponent is typically chosen if one dimension is

unique compared to the other two.

Lz  0   Ltot   6 

Ltot   6  

Ltot   6  

Lz  1

Ltot   6  

Lz  2

Lz   2 Ltot   6

Lz   1

Figure 11.14 The same quantized value for L
can have different quantized values for Lz.For
2, there are 2 1 5 possible values ofm,
each having a different value ofLz.