- #1

doggydan42

- 170

- 18

## Homework Statement

A particle of mass m in one dimension has a potential:

$$V(x) =

\begin{cases}

V_0 & x > 0 \\

0 & x \leq 0

\end{cases}

$$

Find ##\psi(x)## for energies ##0 < E < V_0##, with parameters

$$k^2 = \frac{2mE}{\hbar^2}$$

and

$$\kappa^2 = \frac{2m(V_0 - E)}{\hbar^2}$$

Use coefficients such that ##\psi(0) = -\frac{k}{\kappa}##

## Homework Equations

The time-independent schrodinger equation:

$$\hat H \psi(x) = E \psi(x)$$

## The Attempt at a Solution

I started with the potential being 0, which gave.

$$\psi''(x) = -k^2\psi(x)$$

This would give solutions in the form

$$\psi(x \leq 0) = Asin(kx) + Bcos(kx)$$

For the potential ##V_0##, I got

$$\psi''(x) = -\kappa^2\psi(x)$$

Similarly, $$\psi(x > 0) = Asin(\kappa x) + Bcos(\kappa x)$$

I was not sure how to choose coefficients.

I was thinking that for ##x \leq 0##, I could find ##B = -\frac{k}{\kappa}##. Though I could not figure out what to do for A, and for ##x > 0##.

Thank you in advance