Physical Chemistry , 1st ed.

Example 10.3

Which of the following operator/function combinations would yield eigen-

value equations? What are the eigenvalues of the eigenfunctions?

a.

d

d

x

2

(^2) cos (^4)
x

b.
d
d
x
(e^4 x)
c.^
d
d
x
(e^4 x
2
)
Solution
a.Since
d
d
x
2
(^2) cos (^4)
x
(^) 
1

1

6

cos

4

x

this is an eigenvalue equation with an eigenvalue of1/16.

b.Since

d

d

x

(e^4 x) 4(e^4 x)

this is an eigenvalue equation with an eigenvalue of4.

c.Since

d

d

x

(e^4 x

2

)  8 x(e^4 x

2

)

this is not an eigenvalue equation because although the original function is

reproduced, it is not multiplied by a constant. Instead, it is multiplied by an-

other function, 8 x.

Another postulate of quantum mechanics states that for every physical ob-

servable of interest, there is a corresponding operator. The onlyvalues of the

observable that will be obtained in a single measurement must be eigenvalues

of the eigenvalue equation constructed from the operator and the wavefunction

(as shown in equation 10.2). This, too, is a central idea in quantum mechanics.

Two basic observables are position (usually—and arbitrarily—in the xdi-

rection) and the corresponding linear momentum. In classical mechanics, they

are designated xand px. Many other observables are various combinations of

these two basic observables. In quantum mechanics, the position operator xˆis

defined by multiplying the function by the variable x:

xˆx (10.3)

and the momentum operator pˆx (in the xdirection) is defined in differential

form as

ˆpxi





x

(10.4)

where iis the square root of1 and is Planck’s constant divided by 2 ,h/2.

The constant is common in quantum mechanics. Note the definition of mo-

mentum as a derivative with respect to position, not with respect to time as

with the classical definition. Similar operators exist for the yand zdimensions.

278 CHAPTER 10 Introduction to Quantum Mechanics