130_notes.dvi

10.6.5 The expectation value of p

2

2 min eigenstate

The expectation value of p

2

2 mis

〈un|

p^2

2 m

|un〉 =

− 1

2 m

m ̄hω

2

〈un|−AA†−A†A|un〉

=

̄hω

4

〈un|AA†+A†A|un〉

=

̄hω

4

((n+ 1) +n) =

1

2

En

(See the previous section for a more detailed computation of the same kind.)

10.6.6 Time Development Example

Start off in the state att= 0.

ψ(t= 0) =

1

√

2

(u 1 +u 2 )

Now put in the simple time dependence of the energy eigenstates,e−iEt/h ̄.

ψ(t) =

1

√

2

(u 1 e−i

(^32) ωt
+u 2 e−i
(^52) ωt
) =

1

√

2

e−i

(^32) ωt
(u 1 +e−iωtu 2 )
We can compute the expectation value ofp.
〈ψ|p|ψ〉 = −i

√

m ̄hω

2

1

2

〈u 1 +e−iωtu 2 |A−A†|u 1 +e−iωtu 2 〉

=

√

m ̄hω

2

1

2 i

(

〈u 1 |A|u 2 〉e−iωt−〈u 2 |A†|u 1 〉eiωt

)

=

√

m ̄hω

2

1

2 i

(√

2 e−iωt−

√

2 eiωt

)

= −

√

m ̄hωsin(ωt)

10.7 Sample Test Problems

1. A 1D harmonic oscillator is in a linear combination of the energy eigenstates

ψ=

√

2

3

u 0 −i

√

1

3

u 1

Find the expected value ofp.

Answer