Quantum homework - Average Expectation Values?

[jeebs]

Quantum homework - Average Expectation Values??

Hi people,
I'm struggling with my quantum mechanics homework - I've included links to photographs of my attempts at solutions, but i know they are wrong because I am given what the answers are supposed to be. Can somebody help me spot where I am going wrong? I don't have a keyboard key for psi(x) so i'll use Y(x) for wavefunctions. Here is the question:

My particle, described by the wavefunction Y(x) = Ax(L-x) is confined to a region 0<x<L.
A is a constant.

a) Normalize the wavefunction to unity.
b) Compute the average value of position <x> , <x^2> , <p> and <p^2>.

Here is my solution to part a).
Have i done what is being asked of me, ie. square the wavefunction, and set the integral of that, within the limits of x, equal to 1??

http://i52.photobucket.com/albums/g33/long_john_cider/normalization.jpg

Here are my solutions to part B.
The thing about these ones is, i don't think there is a problem with my actual calculations since I have done them a few times and got the same result. I think its with the initial equations for <x>, <x^2>, <p> and <p^2> that I start out with.

<x>: http://i52.photobucket.com/albums/g33/long_john_cider/xavexpv.jpg

<x^2>: http://i52.photobucket.com/albums/g33/long_john_cider/x2avexpv.jpg

I am supposed to use these answers to get this result, but my answer is nowhere near:
Δx = sqrt(<x^2> - <x>^2) = L / sqrt(28)


<p>: http://i52.photobucket.com/albums/g33/long_john_cider/p2avexpv.jpg

<p^2>: http://i52.photobucket.com/albums/g33/long_john_cider/p2avexpv.jpg

I am supposed to use these answers to get this result, but again my answer is nowhere near:
Δp = sqrt(<p^2> - <p>^2) = H(sqrt(10) / L

where H = h/2pi , h bar in other words.


Can anybody spot what I am doing wrong? I greatly appreciate any help because I have followed my notes perfectly but I am still getting the wrong answers.
Thanks.

 

[CompuChip]

Well, first of all, you almost did a) correctly: you calculated the integral of psi^2 (actually, psi* psi) and set it to one. But then somehow you missed the point of the exercise: L is the given width of the "box" your particle is in, while A is the normalisation factor. So the idea is to solve the expression you have for A.

Then you will get, for example, for <x> a very nice answer. It's instructive to try and guess what it should be in advance, based on your physical intuition (sketch the wavefunction, where do you think will be the most probable place for the particle?)
I'm not sure about your <x^2>, it looks unfamiliar. Maybe if you express it just in terms of L (by rewriting A, using a)) you will find the correct answer? I think you should have something withan L^5, not L^2, there.

 

[jeebs]

aah thank you CompuChip, my answer for <x> turns out to be L/2 (which seems to make sense), since my normalization constant is A^2 = 30/(L^5).

thats my problem with quantum, I've only been doing it a few weeks and i often have no idea how to attack a problem. If you hadn't suggested it there was no way I would ever have thought that A was the normalization constant, or that I was even supposed to be looking for one. My lecturer is awful, he never explains anything.

 

[CompuChip]

Very well, so you can take Y(x) = sqrt[30]/L^(5/2) x (L - x)
and calculate <x^2>.

And I know your problem, I had a QM1 lecturer like that. He raced through the theory and then spent his time showing nice Mathematica plots which seem to have been very interesting. He refused to explain Fourier transformations, as only a small part of the class had taken that optional course, and told us to read up on them ourselves.