- #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