How to Solve the Schrödinger Equation for a Finite Potential Well?

[Arbitrationer]

Homework Statement

Hi everyone. This is my first time on her so I hope I make what I'm looking for clear!

The question in the book says,

If V(x) = ∞, x<0 ; -Vo, 0 < x < a ; 0, x > a

Solve the Schrödinger equation for E < 0 inside and outside the well. Apply the boundary conditions at x = 0 and x = a to obtain and equation that determines the allowed values of E.
[/B]

Homework Equations

We just finished going over Finite Potential Wells. Inside the book and in class, we went over the simple case where:

V(x) = 0, -a/2 < x < a/2
V(x) = Vo, elsewhere

I think I get this. We came to the conclusion that, for one case,

ψ(x) = Ce^κx for the region to the left (the, what we called in class, "classically forbidden region" (CF)), 2Acos(kx) for the center region (classically allowed (CA)) and Ce^-κx for the region to the right (CF)

and for the other case:

ψ(x) = Ce^κx (CF), 2iAsin(kx) (CA), -Ce^-κx (CF)

Where A and C are just the constants obtained from solving the differential equation once V(x) is plugged into the Schrödinger Equation, and k is different from κ.
[/B]

The Attempt at a Solution

I wish I could say that I made an attempt. I am really confused on where to begin. Any and all help is greatly appreciated!

 

[vela]

Homework Statement

Hi everyone. This is my first time on her so I hope I make what I'm looking for clear!

The question in the book says,

If V(x) = ∞, x<0 ; -Vo, 0 < x < a ; 0, x > a

Solve the Schrödinger equation for E < 0 inside and outside the well. Apply the boundary conditions at x = 0 and x = a to obtain and equation that determines the allowed values of E.
[/B]

Homework Equations

We just finished going over Finite Potential Wells. Inside the book and in class, we went over the simple case where:

V(x) = 0, -a/2 < x < a/2
V(x) = Vo, elsewhere

I think I get this. We came to the conclusion that, for one case,

ψ(x) = Ce^κx for the region to the left (the, what we called in class, "classically forbidden region" (CF)), 2Acos(kx) for the center region (classically allowed (CA)) and Ce^-κx for the region to the right (CF)

and for the other case:

ψ(x) = Ce^κx (CF), 2iAsin(kx) (CA), -Ce^-κx (CF)

Where A and C are just the constants obtained from solving the differential equation once V(x) is plugged into the Schrödinger Equation, and k is different from κ.
[/B]

The Attempt at a Solution

I wish I could say that I made an attempt. I am really confused on where to begin. Any and all help is greatly appreciated!

You can start by understanding how the form of the wave function was derived for each region. You're going to take the same basic approach for the current problem.

 

[ehild]

Find k and K by replacing the solutions back into the Schroedinger equation.
The general solution in the potential well is a linear combination of sin(kx) and cos (kx). Apply the boundary conditions to find the relations among the constants. What are the boundary conditions?