Physical Chemistry , 1st ed.

Example 10.2

For each of the following combinations of operator and function, write the

complete mathematical operation and evaluate the expression.

Oˆ

d

d

x

Bˆ

d

d

x

2

2 Sˆexp ( ) [raising 2.7182818... to some power]

 1  2 x 4  2  3  3 sin 4x

a.Sˆ 2

b.Oˆ 1

c.Bˆ 3

Solution

a.Sˆ 2 exp(3) e^3 0.04978...

b.Oˆ 1 

d

d

x

(2x4)  2

c.Bˆ 3 

d

d

x

2

2 (sin 4x)  (^) d

d
x
(4 cos 4x) 16 sin 4x
In the examples above, the combination of operator and function yield an
expression that could be mathematically evaluated. However, suppose the def-

initions are Lˆln ( ) and 10. The expression Lˆcannot be evaluated

because logarithms of negative numbers do not exist. Not all operator/function
combinations are mathematically possible, or yield meaningful results. Most
operator/function combinations of interest to quantum mechanics willhave
meaningful results.
When an operator acts on a function, some other function is usually gen-
erated. There is a special type of operator/function combination that, when
evaluated, produces some constant or group of constants times the original
function.For instance, in Example 10.2c, the operator d^2 /dx^2 is applied to the
function sin 4x, and when the operator is evaluated, a constant times sin 4xis
produced:

d

d

x

2

2 (sin 4x)  (^) d

d
x
(4 cos 4x) 16 sin 4x
If we want to use the more concise symbolism for the operator and the func-
tion, the above expression can be represented as

BˆK (10.2)

where Kis a constant (in this case,16). When an operator acts on a function
and produces the original function multiplied by any constant (which may be
1 or sometimes 0), equation 10.2 is referred to as an eigenvalue equationand
the constant Kis called the eigenvalue.The function is called an eigenfunction
of the operator. Not all functions are eigenfunctions of all operators. It is a rare
occurrence for any random operator/function combination to yield an eigen-
value equation. In the example above, the eigenvalue equation is

d

d

x

2

    2 (sin 4x) 16(sin 4x)

where the parentheses are used to isolate the original function. The eigenfunc-
tion of the operator is sin 4xand the eigenvalue is 16.

10.3 Observables and Operators 277