# Solve Schroedinger Equation with Mathematica DSolve for given potential

1. Mar 11, 2010

### donquixote17

1. The problem statement, all variables and given/known data
I need to solve the Schroedinger equation (Using DSolve in Mathematica) for a potential that is infinite below z=0 and V=mgz for positive z.

2. Relevant equations
TISE:
$\psi \text{''}[z]+\frac{2 m}{\hbar }(\text{En}-V[z])\psi [z]==0$

3. The attempt at a solution
First I defined the potential as
$V[z]=\text{Piecewise}[\{\{\infty ,z<0\},\{m g z,z\geq 0\}\}];$

Then I told Mathematica to solve the Diffeq
$\text{DSolve}\left[\left\{\psi \text{''}[z]+\frac{2 m}{\hbar }(\text{En}-V[z])\psi [z]==0,\psi [0]==0,\psi [\infty ]==0\right\},\psi [z],z\right]$

The error mathematica gave me was
InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses. >>

I'm not really sure how to get mathematica to give me the solution. I know I need 2 conditions besides the diffeq in order to solve. The only boundary condition that I know to use is that the wavefunction must be zero at z=0 since the potential is infinite there.

Any ideas on what second boundary condition I should use or any ideas on where I'm going wrong? Thanks

Last edited: Mar 11, 2010
2. Mar 11, 2010

### vela

Staff Emeritus
It might be choking, in part, on the infinite potential. You know the wavefunction vanishes there, so try using V(z)=mgz and solve for just z>0.