Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR VECTOR SPACES
In general A
~
B
~
6ˆ B
~
A
~
. The di€erence A
~
B
~
B
~
A
~
is called the commutator of A
~
and B
~
and is denoted by the symbol ‰A
~
; B
~
]:
‰A
~
; B
~
ŠA
~
B
~
B
~
A
~
: …5:18†
An operator whose commutator vanishes is called a commuting operator.
The operator equation
B
~
ˆ A
~
ˆ A
~
is equivalent to the vector equation
B
~
jiˆ A
~
ji
for any ji:
And the vector equation
A
~
jiˆ ji
is equivalent to the operator equation
A
~
ˆ E
~
where E
~
is the identity (or unit) operator:
E
~
jiˆ ji
for any ji:
It is obvious that the equation A
~
ˆ is meaningless.
Example 5.11
To illustrate the non-commuting nature of operators, let A
~
ˆ x; B
~
ˆ d=dx. Then
A
~
B
~
f …x† ˆx d dx f …x†;
and
Thus,
B A f …x† ˆ d
xf …x† ˆ dx
f ‡ x df
~ ~ dx dx dx ˆ f ‡ x df
dx ˆ…E ‡A B † f :
~ ~ ~
…A
~
B
~
B
~
A
~
† f …x† ˆE
~
f …x†
or
d
x;
dx
ˆ x d dx d dx x ˆE :
~
Having de®ned the product of two operators, we can also de®ne an operator
raised to a certain power. For example
A m jiˆ A A A ji:
~ ~ |‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚}
~ ~
m factor
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