How Does an Electron Behave in a 1D Crystal Potential Well?

[wavingerwin]

Homework Statement

An electron in a one dimensional crystal is bound by:

$$U(x) = \frac{-\overline{h}^{2}x^{2}}{mL^{2}\left(L^{2}-x^{2}\right)}$$
for
$$\left|x\right| < L$$

and
$$x = infinity$$
for
$$\left|x\right| \geq L$$

Show that a stationary state for the electron in the potential well
$$\psi(x) = A\left(1-\frac{x^{2}}{L^{2}}\right)$$
satisfies the Schrödinger's Equation

and find E

Homework Equations

$$\frac{-\overline{h}^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+U(x)\psi = E\psi$$

The Attempt at a Solution

from Schrödinger's:

$$\frac{d^{2}\psi}{dx^{2}} = \frac{-2m}{\overline{h}^{2}}\left(\frac{\overline{h}^{2}x^{2}}{mL^{2}\left(L^{2}-x^{2}\right)}+E\right)\psi$$

and from the guess solution:

$$\frac{d^{2}\psi}{dx^{2}} = \frac{-2A}{L^{2}} = \frac{-2}{\left(L^{2}-x^{2}\right)}\psi$$

and so equating $$\frac{d^{2}\psi}{dx^{2}}$$,
I deduced that it satisfies the Schrödinger's equation but only when E = 0 and x = L.

Am I right?

I am also concerned because the potential, U, is negative 'inside' the well...

 

[nickjer]

It can't be a stationary state if it only satisfies the Schrödinger eqn at a single 'x' point. I think you need to double check your work on this problem, since you made an incorrect assumption.

I suggest simplifying the two terms you list in the parenthesis of the first eqn in your attempt at a solution. Convert it to one giant fraction. Then try to find an energy that will remove the 'x'-dependence in the numerator of that fraction. You will want to keep the x-dependence in the denominator since it needs to equal your 2nd equation. You want this equality to be applied to any value of 'x' and not just one specific value like before.

 

[wavingerwin]

Hi nickjer

After putting it as a single fraction, I got:

$$\frac{d^{2}\psi}{dx^{2}} = \left(\frac{-2}{\left(L^{2}-x^{2}\right)}\frac{x^{2}\left(\overline{h}^{2}-mE\right)+mL^{4}E}{L^{2}\overline{h}^{2}}\right)\psi$$

I just cannot figure out how to remove the x dependencies in the numerator.

If I equate this with
$$\frac{d^{2}\psi}{dx^{2}} = \frac{-2}{\left(L^{2}-x^{2}\right)}\psi$$

I can get a value for E, however it depends on x, which i think is wrong...

Maybe, just maybe, is the question wrong and
$$\psi(x) = A\left(1-\frac{x^{2}}{L^{2}}\right)$$
is not a possible solution to the situation?

 

[nickjer]

Double check your math for the numerator. I believe you are missing something. Also, once you correct the numerator, then you see the x^2 has a coefficient next to it in parenthesis. Very similar to what you have now (although yours is slightly wrong). Since you are solving for the energy that makes this stationary state satisfy Schrödinger's eqn, then you are free to choose an E that will make this coefficient in parenthesis go to 0. Then you will have 0*x^2 = 0. So you canceled out the x-dependence.

 

[wavingerwin]

If I'm right this time, I was missing L²

And so I equate 0 = hbar² - L²mE

and also, after eliminating x², I equate to the guess solution,

and both of them give me a consistent answer:

$$E = \frac{\overline{h}^{2}}{mL^{2}}$$

Now, I think this is right!

If it is, thanks very much