- #1

timn

- 19

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## Homework Statement

Solve the time-independent Schrödinger equation in one dimension for a potential step, i.e. V(x) = 0 for x<0 and V(x) = V_0 for x>0.

## Homework Equations

[tex] - \frac{\hbar^2}{2m} \frac{d^2u(x)}{dx^2} + V(x)u(x) = Eu(x) [/tex]

## The Attempt at a Solution

Rewrite as (4-3)

[tex]

\frac{d^2u(x)}{dx^2} + \frac{2m}{\hbar^2}( E - V(x) )u(x) =

\frac{d^2u(x)}{dx^2} + k^2( 1 - V(x)/E )u(x) = 0

[/tex]

Looking at x<0, i.e. V(x)=0:

[tex]

\frac{d^2u(x)}{dx^2} + k^2u(x) = 0

\Leftrightarrow u(x) = Ae^{ikx} + Be^{ikx}

[/tex]

The derivation in my textbook claims, at this step:

Is it implicit that all multiples of the suggested u(x) are solutions (for x<0)? Furthermore, from the paragraph on the probability flux, it looks like |R(x)| < 1. Why?

It seems like I'm missing some assumption. Why would the x<0 case be different from the case of a free particle?

**Edit:**I'm having a similar problem with x>0.

Gasiorowicz talks about the waves having a direction. What is meant by this? Why is the term e^-iqx dismissed?

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