Particle-in-a-potential

A "classical" problem in quantum mechanics which every student faces. Here we find out the energy values of a particle trapped in a potential. Only the mathematica code part is here which helps us know the problem analytically.

The specfic problem we are dealing with here is

The potential is symmetric under y axis as shown in the graph below obtained by running the mathematica code:

So we will look for solutions that are parity eigenstates. Solving Schrodinger wave equation (SWE) for x>=x0 we get a combination of parabolic cylindrical functions (Shown in mathematica code in traditional form for textbook like looks!).

Matching appropriate boundary conditions and a bit of simplification we get some useful expressions in terms of Hermite functions. (Shown in code) The odd (solid) and even parity hermite polynomials are

Finally, we can plot the wave functions after finding appropriate coefficients of the solution wave functions/eigen functions of SWE. the selected eigen functions (1st and 3rd) are plotted as:

TO DO

• calculate energies in tabular form

License

• GNU GPL v3.0 or later on your opinion.