Momentum Eigenstate: Meaning of <psi|p|psi>, etc.

[imagemania]

Homework Statement

I am trying to translate what is meant by:
<psi | p | psi>
<psi|p^2|psi>
<psi | x | psi>
In a mathematicaly context as shown by this link:

http://answers.yahoo.com/question/index?qid=20110521103632AASz9Hm


Can anyone specify what these mean?

Thanks!

 

[tiny-tim]

hi imagemania!

<| denotes a row vector

|| denotes a matrix

|> denotes a column vector ​

 

[imagemania]

Ok, but I am still not following how he got one for the first question:
<psi | p | psi> = 0

for the integral:
\psi = \int_{-\infty}^{\infty} {e}^{-\alpha {(k-{k}_{0})}^{2}}{e}^{ikx} dk

Thanks

 

[tiny-tim]

not following you …

ψ is as given, and p is the momentum operator ​

 

[imagemania]

Perhaps i'll ignore that post and go back to the fundamental question. From my understanding,
\bar{p}=\frac{hk}{2 \pi}. Knowing \psi is there a way to deduce a better answer to \bar{p} or is it just as I said here?

I am also unsure about the equation for mean value of x.

Thanks :)

 

[vela]

It's Dirac notation. For an operator A, you can write
\langle \psi | \hat{A} | \psi \rangle = \int \psi^*(x)\hat{A}\psi(x)\,dx
You've been given the wave function. What you need to do next is look up how to express the operators x, p, and p² appropriately.