Expectation values and operators.

  • #1
i'm just not sure on this little detail, and its keeping me from finishing this problem.

take the arbitrary operator [tex]\tilde{p}^{n}\tilde{y}^{m}[/tex] where p is the momentum operator , and x is the x position operator

the expectation value is then <[tex]\tilde{p}^{n}\tilde{y}^{m}[/tex] >

is this the same as <[tex]\tilde{p}^{n}[/tex]> <[tex]\tilde{y}^{m}[/tex]>?

if not, how would i go about calculating <[tex]\tilde{p}^{n}\tilde{y}^{m}[/tex] >?
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  • #2
In general, they are not the same. The expectation value of an operator [itex]\hat A[/itex] is
[tex]\int \Psi^*(x) \hat A(x) \Psi(x) \, \mathrm{d}x[/tex]
where [itex]\Psi(x)[/itex] is your wavefunction (assuming you are talking QM here).
In this case,
[tex]\int \Psi^*(x) \hat p^n \hat y^m \Psi(x) \, \mathrm{d}x
\left( \int \Psi^*(x) \hat p^n \Psi(x) \, \mathrm{d}x \right)
\left( \int \Psi^*(x) \hat y^m \Psi(x) \, \mathrm{d}x \right).
You could write out [itex]\hat p[/itex] in the position basis and work out what [itex]\hat p^n(y^m \Psi)[/itex] looks like.

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