10.6.3 The expectation value ofxin the state√^12 (u 0 +u 1 ).
1
2
〈u 0 +u 1 |x|u 0 +u 1 〉 =1
2
√
̄h
2 mω〈u 0 +u 1 |A+A†|u 0 +u 1 〉=
√
̄h
8 mω〈u 0 +u 1 |Au 0 +Au 1 +A†u 0 +A†u 1 〉=
√
̄h
8 mω〈u 0 +u 1 |0 +√
1 u 0 +√
1 u 1 +√
2 u 2 〉=
√
̄h
8 mω(
√
1 〈u 0 |u 0 〉+√
1 〈u 0 |u 1 〉+√
2 〈u 0 |u 2 〉+√
1 〈u 1 |u 0 〉
+√
1 〈u 1 |u 1 〉+√
2 〈u 1 |u 2 〉)=√
̄h
8 mω(1 + 1)
=
√
̄h
2 mω10.6.4 The expectation value of^12 mω^2 x^2 in eigenstate
The expectation ofx^2 will have some nonzero terms.
〈un|x^2 |un〉 =
̄h
2 mω〈un|AA+AA†+A†A+A†A†|un〉=
̄h
2 mω〈un|AA†+A†A|un〉We could drop theAAterm and theA†A†term since they will produce 0 when the dot product is
taken.
〈un|x^2 |un〉 =
̄h
2 mω(〈un|√
n+ 1Aun+1〉+〈un|√
nA†un− 1 〉)=
̄h
2 mω(〈un|√
n+ 1√
n+ 1un〉+〈un|√
n√
nun〉)=
̄h
2 mω((n+ 1) +n) =(
n+1
2
)
̄h
mωWith this we can compute the expected value of the potential energy.
〈un|1
2
mω^2 x^2 |un〉=1
2
mω^2(
n+1
2
)
̄h
mω=
1
2
(
n+1
2
)
̄hω=1
2
En