Physical Chemistry Third Edition

16.2 The Third Postulate. Mathematical Operators and Mechanical Variables 685

can take on. The third and fourth postulates must be used to obtain the possible values

of other mechanical variables such as the momentum and the angular momentum. The

third postulate is:

Postulate 3. There is a linear hermitian mathematical operator in one-to-one correspon-

dence with every mechanical variable.

This postulate states that for each operator there is one and only one variable, and

for each variable there is one and only one mathematical operator. Amathematical

operatoris a symbol that stands for performing one or more mathematical operations.

When the symbol for an operator is written to the left of the symbol for a function, the

operation is to be applied to that function. For example,d/d xis a derivative operator,

standing for differentiation of the function with respect tox;cis a multiplication

operator, standing for multiplication of the function by the constantc;h(x) is also a

multiplication operator, standing for multiplication by the functionh(x). We usually

denote an operator by a letter with a caret (̂) over it.

The result of operating on a function with an operator is another function. IfÂis an

operator andfis a function,

Af̂ (q)g(q) (16.2-1)

wheregis another function. The symbolqis an abbreviation for whatever independent

variablesfandgdepend on. Figure 16.1 shows an example of a function,f(x)ln(x),

and the functiong(x) 1 /xthat results when the operatord/d xis applied to ln(x).

0

Value of function^0

1

2

21

123

x

df/dx 5 g(x) 5 1/x

f(x) 5 In(x)

4

Figure 16.1 A Function and Its

Derivative.

Operator Algebra

There is anoperator algebraaccording to which we can symbolically manipulate

operator symbols without specifying the functions on which the operators act. An

operator can be set equal to another operator in anoperator equation. For example, if

Af̂ (q)Bf̂ (q) (16.2-2)

is valid for any well-behaved functionf(q), we can write the operator equation

ÂB̂ (16.2-3)

An operator equation means that the operators on the two sides of the equation always

produce the same result when applied to any well-behaved function.

The operator that always produces the same function as the one on which it operates

is called theidentity operatorand is denoted byÊ. The symbolÊcomes from the

German word “Einheit,” meaning “unity.” It is equivalent to multiplying by unity:

Ef̂ (q)f(q) (16.2-4)

Equation (16.2-4) can be written as the operator equation:

Ê 1 (16.2-5)

where 1 is the operator for multiplication by unity.

Thesum of two operatorsis defined by

(Â+B̂)f(q)Af̂ (q)+Bf̂ (q) (16.2-6)