The Schrodinger Wave Equation (SWE) is a differential equation that governs the behavior of all non-relativistic matter on a quantum mechanical level. Psi Spectrum solves the one dimensional time independent SWE in the following form:
For the purposes of this simulation, the mass has been defined such that the coefficient of the second derivative is unity.
Solving the SWE requires finding the specific energies (E) at which the boundary conditions are satisfied. We always require the solution to be continuous for all x and to approach zero as x goes to infinity or negative infinity. All values of E for which this occurs are allowed energy states for the particle being modeled; E values that do not meet these conditions are not possible energy states. Psi Spectrum finds the E values at which the SWE has a solution by trial and error. The Shooting Method is used to narrow a range of energy levels until that range is narrow enough to be considered a single energy value.
Solutions to the SWE can be found for a variety of potential functions. Once these solutions are found, Psi Spectrum can also calculate the energy transitions possible between two potential functions. This is done by calculating <A|x|B> where
Squaring the value of this integral gives the probability of the given transition occuring. Psi Spectrum represents this data in two ways. First, a transition matrix can be calculated giving the value of the above integral for each of the transitions between each energy level in the two potentials. Second, a spectrum can be calculated which shows the probability of such a transition taking place and the wavelength of electromagnetic radiation which would be released by such a transition.