Homework Help: Calculating the expectation value for a particles energy in a 1_D well

1. Aug 14, 2012

solas99

expectation value for a particle in a 1-D well

how do i calculate the expectation value for the particles energy in a 1-D well.

i have attached a word file, with my working out, just not quite sure if im on the right track...

i appreciate any help...thanks a mill

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2. Aug 14, 2012

VortexLattice

Re: expectation value for a particle in a 1-D well

Erm, unless it means something else, L_z is an operator, not something you'd find in a wave function. So that wave function doesn't really make sense.

3. Aug 14, 2012

Clever-Name

I can't really read that document at all, the equations are extremely blurry and I can't make out the symbols. However, recall that the expectation value for a given operator is the following:

$$\langle Q_{op}\rangle = \int_{-\infty}^{\infty} \Psi^{*} Q_{op} \Psi dx$$

4. Aug 14, 2012

solas99

Re: expectation value for a particle in a 1-D well

can you explain to me what you did not understand from the question or my working out...so i can clear it up..

Lz is not an operator, its just part of the function to describe the the particle in a 1D well.

5. Aug 14, 2012

solas99

Re: expectation value for a particle in a 1-D well

have a hand written version attached..

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6. Aug 15, 2012

vela

Staff Emeritus
Re: expectation value for a particle in a 1-D well

Your calculation of $\hat{H}\psi$ isn't correct in both of your attempts. I'm not sure how you're getting your result. Do you know how to calculate a derivative?

7. Aug 15, 2012

solas99

i know how to calculate the derivative, can you please guide me.

thanks a mill

8. Aug 15, 2012

vela

Staff Emeritus
If you know how to differentiate, can you explain how you got your results?

9. Aug 15, 2012

solas99

Im not sure if im doing it right,
what are the steps of solving this problem,

1st. find a eigen value of hamiltonian
2. find the time dependance wave function for stationary states
am i right..

can you please guide me through it

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10. Aug 15, 2012

vela

Staff Emeritus
As Clever-Name implied above, you want to calculate $\langle \psi \lvert \hat{H} \rvert \psi \rangle$, so calculating $\hat{H}\psi(z)$ is a good first step. You have
$$\hat{H}\psi(z) = -\frac{\hbar^2}{2m}\frac{d^2}{dz^2} \sqrt{\frac{10}{L_z}}\left(1-\frac{z^2}{L_z^2}\right).$$ So far so good. The problem is, the rest of what you wrote doesn't make sense. First, you claimed
$$-\frac{\hbar^2}{2m}\frac{d^2}{dz^2} \sqrt{\frac{10}{L_z}}\left(1-\frac{z^2}{L_z^2}\right) \rightarrow -\frac{\hbar^2}{2m}\frac{n^2\pi^2}{L_z}\left(1-\frac{z^2}{L_z^2}\right)$$ which is simply wrong. This is basic differentiation. I don't understand how you got your result if you know how to calculate a derivative. Next, you then somehow claim that your answer is of the form
$$E\sqrt{\frac{10}{L_z}}\left(-\frac{z^2}{L_z^2}\right)$$ which is obviously inconsistent with what you wrote earlier (what happened to the 1 inside the parentheses?). In fact, the relationship $\hat{H}\psi = E\psi$ only holds if $\psi$ is an eigenstate of the Hamiltonian, and the given wave function isn't an eigenstate.

You never said what the actual potential here is. I'm assuming you have an infinite square well as opposed to a finite square well.

11. Aug 21, 2012

solas99

i have attached a file with integration of wavefunction.

so i integrated the wavefunction, im just not sure of the steps.

for example> if i want to work out the expectation value of energy. what exact steps do i follow..

im just confused with the terminology of what im asked.

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