Exponential Probability Density of Electron in 1D Potential Well

[lozzyjay]

Homework Statement

An electron is in a 1 dimensional potential well. If the energy of the electron is < V0, show that the probability density P(x) for the electron falls exponentially:

P(x) = Aexp(-x/L)


I honestly have no clue, I've been trying all day to do this!

 

[berkeman]

You need to show us what you have been doing "all day", so that we can help to guide you in the right direction. What is the basic equation that you should be using to approach this problem? It has a catchy name...

 

[lozzyjay]

$$\frac{-h bar^{2}}{2m}$$ $$\frac{d^2\psi}{dx^2}$$ = (E - V$$_{0}$$)$$\psi$$

So that's the Schrödinger equation for when the energy is less than V0 I think...

Then to show the probability density...

P(x) = $$\int$$$$\psi$$(x)* $$\psi$$(x) = 1

I have no idea what to do next...

 

[Dick]

Ok, so the Schrödinger equation is psi''=k*psi, where k is a positive constant. What do the solutions to that equation look like?

 

[lozzyjay]

The solutions are of the form

$$\psi$$ = Ae$$^{ikx}$$ + Be$$^{-ikx}$$

 

[Dick]

Noooo. The k in psi''=k*psi is real and positive. How about A*exp(sqrt(k)*x)+B*exp(-sqrt(k)*x)?