- #1

RJLiberator

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## Homework Statement

A beam of particles of mass m and energy E is incident from the right unto a square well potential given by ##V(x)=-V_0## for ##-a<x<0##, and ##V(x)=0## otherwise.

Solve the Schrodinger equation to determine the wave function which describes this situation. Determine the transmission and reflection coefficients, and show R+T=1.

## Homework Equations

The Schrodinger Equation

[tex] \frac{-ħ}{2m} \frac{ \partial^2 \psi}{\partial x^2} + V(x) \psi(x) = E \psi(x)[/tex]

## The Attempt at a Solution

This is a pretty standard question apparently as there are a plethora of examples available from my notes/textbook/INTERNET.

My problem with this question is that the instructor has switched it such that the energy is incident FROM the right.

I'm working with the example from my book, so I will skip some of the calculations, but present you this:

I split the regions into 3 obvious regions. Region 1 will be the beginging region from the right. Region 2 will be the finite well, and region 3 will be the ending region.

Now, solving the Schrodinger equation for each region has left me with the following:

[tex] \psi (x) = Ae^{ikx}+Be^{-ikx}, Region 1 [/tex]

[tex] \psi (x) = I \sin{- \alpha x}+J \cos{ \alpha x}, Region 2 [/tex]

[tex] \psi (x) = Ce^{ikx}+De^{-ikx}, Region 3 [/tex]

Now my questions that I need help on and the reason why I am posting here is:

1) I need to understand which of these parts I can use.

For example, in Region 1, in class we had the incident wave being the Ae part. However, in this question, I feel like A = 0 since it is incident from the right so that means we must use the Be part.

Similarly, in region 3, we would then use the De part as C = 0.

**Is my thinking correct here?**

2) For Region 2, my book and notes differ. My book uses the sin and cosine representation as it is a 'even potential problem' so we can then just use the cosine part.

However, my notes uses the exponentials for a similiar problem.

Am I correct in using the cosine/sine representation? Does it really matter since e^(ix) can be represented as a sin cosine wave anyway?

Thank you for helping me with these 2 questions.