Free Electron passing over Potential Well

[jegues]

Homework Statement

See figure attached.

Homework Equations

The Attempt at a Solution

What condition must be satisfied in order for the electron to pass over the well?

Thanks again!

 

[ehild]

The solutions of the time dependent Schrödinger equation are waves both in free space and over the potential well. How are the parameters of the wave related to the energy of the electron and the depth of the well?

ehild

 

[jegues]

The solutions of the time dependent Schrödinger equation are waves both in free space and over the potential well. How are the parameters of the wave related to the energy of the electron and the depth of the well?

ehild

When E < Ub, because it is a matter wave, the electron has a finite probability of tunneling through the barrier and materializing on the other side, moving rightward with energy E as though nothing had happened in the region of 0 ≤ x ≤ L.

But I'm still lost on how to solve this one or what equation(s) I should be looking at.

EDIT: $$\frac{d^{2}\psi}{dx^{2}} + \frac{8\pi^{2}m}{h^{2}}[E-U(x)]\psi = 0$$

This relates the mechanical energy of the particle to the potential of the well.

 

[jegues]

Giving this another shot.

$$K = \frac{1}{2}mv^{2}$$

$$p = mv$$

So,

$$E = \frac{p^{2}}{2m}$$

but,

$$p=\frac{h}{\lambda}$$

So,

$$E = \frac{h^{2}}{2m \lambda^{2}}$$

then,

$$\lambda = \frac{h}{\sqrt{2mE}} = 430pm$$

 

[ehild]

That is all right, but E in your equation is E-Ub=8 eV really, is not it?

You could have approached the problem also by solving the Schrödinger equation.
The electron is free as it has positive energy. Its wave function is of the form A exp(ikx). Substituting into the Schrödinger equation,

k=sqrt(2m(E-Ub))/(h/(2pi)).

But you know that k=2pi/lambda.

ehild

 

[jegues]

That is all right, but E in your equation is E-Ub=8 eV really, is not it?

You could have approached the problem also by solving the Schrödinger equation.
The electron is free as it has positive energy. Its wave function is of the form A exp(ikx). Substituting into the Schrödinger equation,

k=sqrt(2m(E-Ub))/(h/(2pi)).

But you know that k=2pi/lambda.

ehild

Well it states the electron is passing over the well so it's energy would be (5+3)eV=8eV

 

[ideasrule]

No, energy is conserved, so it would still be 3 eV. 8 eV is the energy above the bottom of the well, which is the quantity you want in this case.