1.15 More Fun with Operators
We find the time development operator (See section 11.5) by solving the equationi ̄h∂ψ∂t=Hψ.
ψ(t) =e−iHt/ ̄hψ(t= 0)This implies thate−iHt/h ̄is the time development operator. In some cases we can calculate the
actual operator from the power series for the exponential.
e−iHt/ ̄h=∑∞
n=0(−iHt/ ̄h)n
n!We have been working in what is called the Schr ̈odinger picture in whichthe wavefunctions (or
states) develop with time. There is the alternate Heisenberg picture (See section 11.6) in which the
operators develop with time while the states do not change. For example, if we wish to compute the
expectation value of the operatorBas a function of time in the usual Schr ̈odinger picture, we get
〈ψ(t)|B|ψ(t)〉=〈e−iHt/ ̄hψ(0)|B|e−iHt/ ̄hψ(0)〉=〈ψ(0)|eiHt/h ̄Be−iHt/ ̄h|ψ(0)〉.In the Heisenberg picture the operatorB(t) =eiHt/h ̄Be−iHt/h ̄.
We use operator methods to compute the uncertainty relationshipbetween non-commuting variables
(See section 11.3)
(∆A)(∆B)≥i
2〈[A,B]〉
which gives the result we deduced from wave packets forpandx.
Again we use operator methods to calculate the time derivative of anexpectation value (See section
11.4).
d
dt〈ψ|A|ψ〉=
i
̄h〈ψ|[H,A]|ψ〉+〈
ψ∣
∣
∣
∣
∂A
∂t∣
∣
∣
∣ψ〉
ψ(Most operators we use don’t have explicit time dependence so the second term is usually zero.)
This again shows the importance of the Hamiltonian operator for timedevelopment. We can use
this to show that in Quantum mechanics the expectation values forpandxbehave as we would
expect from Newtonian mechanics(Ehrenfest Theorem).
d〈x〉
dt=
i
̄h〈[H,x]〉=i
̄h〈[
p^2
2 m,x]〉=〈p
m〉
d〈p〉
dt=
i
̄h〈[H,p]〉=
i
̄h〈
[V(x),
̄h
id
dx]
〉
=−
〈
dV(x)
dx〉
Any operatorAthat commutes with the Hamiltonian has atime independentexpectation value.
The energy eigenfunctions can also be (simultaneous) eigenfunctions of the commuting operatorA.
It is usually a symmetry of theHthat leads to a commuting operator and hence an additional
constant of the motion.