Mathematical Methods for Physics and Engineering : A Comprehensive Guide

QUANTUM OPERATORS

later algebraic convenience:

[

A, eλB

]

=

[

A,

∑∞

n=0

(λB)n

n!

]

=

∑∞

n=0

λn

n!

[A, Bn]

=

∑∞

n=0

λn

n!

nBn−^1 [A, B], using the earlier result,

=

∑∞

n=1

λn

n!

nBn−^1 [A, B]

=λ

∑∞

m=0

λmBm

m!

[A, B],writingm=n−1,

=λeλB[A, B].

Now consider the derivative with respect toλof the function

f(λ)=eλAeλBe−λ(A+B).

In the following calculation we use the fact that the derivative ofeλCisCeλC; this is the
same aseλCC, since any two functions of the same operator commute. Differentiating the
three-factor product gives

df

dλ

=eλAAeλBe−λ(A+B)+eλAeλBBe−λ(A+B)+eλAeλB(−A−B)e−λ(A+B)

=eλA(eλBA+λeλB[A, B])e−λ(A+B)+eλAeλBBe−λ(A+B)

−eλAeλBAe−λ(A+B)−eλAeλBBe−λ(A+B)

=eλAλeλB[A, B]e−λ(A+B)

=λ[A, B]f(λ).

In the second line we have used the result obtained above to replaceAeλB,andinthelast
line have used the fact that[A, B]commutes with each ofAandB, and hence with any
function of them.
Integrating this scalar differential equation with respect toλand noting thatf(0) = 1,
we obtain

lnf=^12 λ^2 [A, B] ⇒ eλAeλBe−λ(A+B)=f(λ)=e

1

2 λ^2 [A,B].

Finally, post-multiplying both sides of the equation byeλ(A+B)and settingλ=1yields

eAeB=e

(^12) [A,B]+A+B
.
19.2 Physical examples of operators
We now turn to considering some of the specific linear operators that play a
part in the description of physical systems. In particular, we will examine the
properties of some of those that appear in the quantum-mechanical description
of the physical world.
As stated earlier, the operators corresponding to physical observables are re-
stricted to Hermitian operators (which have real eigenvalues) as this ensures the
reality of predicted values for experimentally measured quantities. The two basic