16.2 The Third Postulate. Mathematical Operators and Mechanical Variables
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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 containing the same independent variable.
E X A M P L E 16.2
Find the operator
C (
K +
V ) 2 if
K and
V are two operators that do not commute.
Solution
C (
K +
V ) 2 (
K +
V )(
K +
V )
K
2 +
K
V +
V
K +
V
2
The terms
K
V and
V
K are different from each other if the two operators do not commute.
Exercise 16.2
a. Find the operator (
K +
V ) 3 if
K and
V do not commute.
b. Find the operator (
A +
B ) 3 if
A and
B commute.
P R O B L E M S
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
.
16.4 A function of an operator is defined through the Taylor
series that represents the function. For example, the
function f (x) e x is represented by the Taylor
series
e
x
1 + x +
1
2!
x
2
+
1
3!
x
3
+ · · ·
1
n!
x
n
+ · · ·
where n! stands for n factorial, the product of all of
the integers beginning with n and ranging down to 1.
The exponential of an operator is defined as the
series
e
A
1 +
A +
1
2!
A
2 +
1
3!
A
3 + · · · +
1
n!
A
n + · · ·
where the operator products are defined in the usual way,
as successive operations of the operator.
a. Write the formula for the result when e
A operates on
an eigenfunction of
A.
b. Write the expression for the first three terms of e (
A+
B) ,
where
A and
B are two operators that do not
necessarily commute.
16.5 a. Find the expression for sin(
A).
b. Find the expression for cos(
A).
c. Find the expression for ln(
A).
d. Write the expression for the first two terms of
sin(
A +
B), where
A and
B are two operators that do
not necessarily commute.