Physical Chemistry Third Edition

16.2 The Third Postulate. Mathematical Operators and Mechanical Variables 687

The following facts about commutation are useful:

1. Every operator commutes with itself.


2. Multiplication operators commute with each other.


3. A constant multiplication operator commutes with all other operators of the types
   that we will consider.


4. Operators that act on different independent variables commute with each other.


5. A derivative operator almost never commutes with a multiplication operator con-
   taining the same independent variable.

EXAMPLE16.2

Find the operator̂C(K̂+V̂)^2 ifK̂andV̂are two operators that do not commute.

Solution

Ĉ(K̂+V̂)^2 (K̂+V̂)(K̂+V̂)K̂^2 +K̂V̂+V̂K̂+V̂^2

The termsK̂V̂andV̂K̂are different from each other if the two operators do not commute.

Exercise 16.2

a.Find the operator (K̂+V̂)^3 ifK̂andV̂do not commute.

b.Find the operator (Â+̂B)^3 if̂AandB̂commute.

PROBLEMS

Section 16.2: The Third Postulate. Mathematical
Operators and Mechanical Variables

16.1 Find an expression for the commutator

[

x,

d^2

dx^2

]

.

16.2 Find an expression for the commutator

[

x

d

dx

,x

]

.

16.3 Find a simplified expression for the operator

[

1

x

+

d

dx

] 2

.

16.4Afunction of an operatoris defined through the Taylor
series that represents the function. For example, the
functionf(x)exis represented by the Taylor
series

ex 1 +x+

1

2!

x^2 +

1

3!

x^3 + ···

1

n!

xn+ ···

wheren! stands fornfactorial, the product of all of

the integers beginning withnand ranging down to 1.

The exponential of an operator is defined as the

series

e

Â

 1 +Â+

1

2!

Â^2 +^1

3!

Â^3 + ··· +^1

n!

̂An+ ···

where the operator products are defined in the usual way,

as successive operations of the operator.

a.Write the formula for the result whenêAoperates on

an eigenfunction ofÂ.

b.Write the expression for the first three terms ofe(Â+̂B),

whereÂand̂Bare two operators that do not

necessarily commute.

16.5 a.Find the expression for sin(Â).

b.Find the expression for cos(̂A).

c.Find the expression for ln(Â).

d.Write the expression for the first two terms of

sin(Â+B̂), wherêAandB̂are two operators that do

not necessarily commute.