130_notes.dvi

and
p(op)=p.

The (op) notation used above is usually dropped. If we see the variablep, use of the operator is
implied (except in state written in terms ofplikeφ(p)).

Gasiorowicz Chapter 3

Griffiths doesn’t cover this.

Cohen-Tannoudji et al. Chapter

6.3 Expectation Values

Operators allow us to compute the expectation value of some physics quantity given the wavefunc-
tion. If a particle is in the stateψ(x,t), the normal way tocompute the expectation valueof
f(x) is

〈f(x)〉ψ=

∫∞

−∞

P(x)f(x)dx=

∫∞

−∞

ψ∗(x)ψ(x)f(x)dx.

We can move thef(x) between just beforeψanticipating the use of linear operators.

〈f(x)〉ψ=

∫∞

−∞

ψ∗(x)f(x)ψ(x)dx

If the variable we wish to compute the expectation value of (likep) is not a simple function ofx, let
its operator act onψ(x). Theexpectation value ofpin the stateψis

〈p〉ψ=〈ψ|p|ψ〉=

∫∞

−∞

ψ∗(x)p(op)ψ(x)dx

The Dirac Bra-ket notation (See section 6.4) shown above is a convenient way to represent the
expectation value of a variable given some state.

• See Example 6.7.1:A particle is in the stateψ(x) =

( 1

2 πα

) 1 / 4

eik^0 xe−

x 4 α^2

. What is the expectation
value ofp?*

For any physical quantityv, the expectation value ofvin an arbitrary stateψis

〈ψ|v|ψ〉=

∫∞

−∞

ψ∗(x)v(op)ψ(x)dx