# Fourier Series, QM 1D Potential well

Cuthbert Nyack
In Quantum mechanics "particles" have wave properties( wavelength and frequency) as well as momentum and energy.
Consider a potential well with V(x) = 0 for 0 £ x £ a and V(x) = ∞ elsewhere.
Solution of Schroedinger's equation shows that the particle can be in any one of a discrete set of states with wave function y(x)n and energy En given by:-. A particle can also be in a combination state y(x) given by:- In such a state, the particle has a probability Pn of being in state n. This probality is derived from the Fourier Series of y(x). with wavefunction y(x), the particle has energy E. In this case the particle has an effective quantum number of ~2.205. In the applet below y(x) is shown in red, the component functions are in green and the corresponding probabilities are in orange.
Eg parameters (21, 1, 0.0, 2.8), this shows y(x) summed to 21 terms and the component at n = 1. Changing Time shows the evolution of both functions in time.
Eg parameters (41, 27, 0.0, 209.0), this shows y(x) summed to 41 terms and the component at n = 27. Changing Time shows the evolution of both functions in time.

When activated the following gif image show how the applet should appear. 