130_notes.dvi

which becomes simple if the operator itself does not explicitly depend on time.

d

dt

〈ψ|A|ψ〉=

i

̄h

〈ψ|[H,A]|ψ〉

Expectation values of operators that commute with the Hamiltonianare constants of the motion.

We can apply this to verify that the expectation value ofxbehaves as we would expect for a classical
particle.

d〈x〉

dt

=

i

̄h

〈[H,x]〉=

i

̄h

〈[

p^2

2 m

,x

]〉

=

〈p

m

〉

This is a good result. This is called theEhrenfest Theorem.

For momentum,

d〈p〉

dt

=

i

̄h

〈[H,p]〉=

i

̄h

〈[

V(x),

̄h

i

d

dx

]〉

=−

〈

dV(x)

dx

〉

whichMr. Newtontold us a long time ago.

11.5 The Time Development Operator*.

We can actually make anoperator that does the time development of a wave function. We
just make the simple exponential solution to the Schr ̈odinger equation using operators.

i ̄h

∂ψ

∂t

=Hψ

ψ(t) =e−iHt/ ̄hψ(0)

whereHis the operator. We can expand this exponential to understand itsmeaning a bit.

e−iHt/ ̄h=

∑∞

n=0

(−iHt/ ̄h)n

n!

This is an infinite series containing all powers of the Hamiltonian. In some cases, it can be easily
computed.

e−iHt/ ̄his the time development operator. It takes a state from time 0 to timet.