How Do You Calculate the Expectation Value for a Particle in an Infinite Box?

[shlomo127]

Homework Statement

A particle in an infinite box is in the first excited state (n=2).
Obtain the expectation value (1/2)*<x*p+p*x>

Homework Equations

The Attempt at a Solution

I'm completely baffled by this problem.
Can anyone just please point me in the right direction and then ill respond with what i get (because i know I am going to get stuck again)

Thanks!

 

[Coto]

Hey shlomo. As a hint, the general formula for an expectation value is is given by:

&lt;Q(x,p)&gt; = \int_{-\infty}^{+\infty} \psi^* Q(x,p) \psi dx

where x, p are the position and momentum (quantum mechanical) operators.

This should hopefully give you a push in the right direction. As a first suggestion, you will have to find the wave equation \psi for your specific situation.

 

[shlomo127]

ok, I am going to work on that,
are there any websites you can recommend to help me learn this stuff?
my Prof doesn't explain anything well...
thanks!

 

[Coto]

Well I suppose it's dependent on the situation you're in. What's the outline of the course?

Personally, I find it to be overwhelming if given a resource that goes into far more detail than the course requires.

 

[shlomo127]

i believe the wave equation is:
\psi(x) = Asin(kx) + Bcos(kx)
and k=sqrt((2mE)/(h-bar)^2)

so then the normalized wave function would be:
\psi n(x)=sqrt(2/L)*sin((n*pi*x)/L)

and the problem said that n=2.

And that's where I get stuck. Am i doing it right thus far? and how do i go further?

 

[soul]

As far as I know so far you did right. My suggestion for the further is that you can calculate <xp> and <px> seperately ans sum them up. To do so, apply the operators with their usual orders and take their integral. I know, it seems a bit tidious but this is the only way that I think.

 

[shlomo127]

ok, sounds good, can u set up on of the integrals so i can see how to do it?
then ill try to evaluate both integrals and ill post what i get.

Thanks!

 

[soul]

For the p*x take the derivative of the x*wave function's conjugate and multiply it with the wave function itself and take the integral.

 

[EnershyMethod]

Have you figured it out yet? I don't understand 'soul's last comment/step. I understand up to normalizing the wave function. I don't know how to derive px and xp.

Help!

 

[EnershyMethod]

BUMP,
We need it for tomorrow.