Calculating Expectation Values for x, x^2 in 1D Box

[thenewbosco]

Homework Statement

Calculate the expectation values of x, $$x^2$$ for a particle in a one dimensional box in state $$\Psi_n$$

Homework Equations

$$\Psi_n = \sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a})$$

The Attempt at a Solution

i formed the integral
$$\int_{-\infty}^{+\infty}\Psi ^2 x dx$$ as the expectation value of x. (Psi squared simply because this psi is not complex)
this gives $$<x>=\frac{2}{a} \int_{-\infty}^{+\infty}x sin^2(\frac{n\pi x}{a})dx$$. The problem is i do not know a way to simplify this integral, similarly i have the problem when there is an x^2 in the integral.
any help would be appreciated

 

[G01]

First, the limits of integration are the ends of the box, not infinity...(since psi is zero outside the box anyway)

To solve, try integration by parts. (BTW I just did the exact same problem!)

 

[gardman007]

I am working on this same problem for x^2. Since x is just a multiplicative operator, shouldn't you just be able to put in x^2 in front of the sin term? It doesn't seem to give me the correct result.
Thanks

 

[G01]

I am working on this same problem for x^2. Since x is just a multiplicative operator, shouldn't you just be able to put in x^2 in front of the sin term? It doesn't seem to give me the correct result.
Thanks

Can I see your work? You may be making a computational error. You should be able to do that integral by integration by parts as well, just like the previous one. It just takes one more step.

 

[gardman007]

I found your solution on another page G01, I'm still not sure what I was doing wrong though. It seems like my method should work. Any ideas?

 

[G01]

I found your solution on another page G01, I'm still not sure what I was doing wrong though. It seems like my method should work. Any ideas?

What was your method? If you mean putting the x^2 in front of the sine term that should be fine. How did the solution you find solve it?

 

[gardman007]

I am just calculating the actual values, so I'm letting my calculator due the integration. Could you explain the process to get to the correct integration?

 

[gardman007]

So, for <x^2>, I should just be able to replace x with x^2 in front of the sin term in the last equation of the first post?