Commutator Formulas
Shervin Fatehi
September 20, 2006
1 Introduction
A commutator is defined as1
[ˆ
A, ˆ
B] = ˆ
Aˆ
B−ˆ
Bˆ
A(1)
where ˆ
Aand ˆ
Bare operators and the entire thing is implicitly acting on
some arbitrary function. We’ve seen these here and there since the course
began, most recently in the Heisenberg equation of motion for an expectation
value’s time dependence:2
d
dthAit=1
i¯hh[ˆ
A, ˆ
H]it
It’s worth recalling that any operator ˆ
Afor which [ ˆ
A, ˆ
H] = 0 is a constant
of the motion – that is, its expectation value hAit=hAiwill be independent
of time altogether.
Commutators can be a little tricky (Problem Set 2 should’ve proved that
to you!), so let’s have a look at the different formulas we can use to make
manipulating them a little easier.
1Most quantum mechanics books will discuss commutators in some detail. You can
also check out the Wikipedia page, http://en.wikipedia.org/wiki/Commutator, for more
information.
2The notation I’m using here is shorthand. Wherever the h...itappears, you should
remember that it means we’re calculating
hΨ(t)|ˆ...|Ψ(t)i=Z∞
−∞
Ψ∗(x, t)( ˆ...)Ψ(x, t) dx
1
