Why do we treat particles as standing waves in QP?

Question

The Quantum Physics course I am taking starts with the Classical Wave Equation and a statement that we treat quantum particles as standing waves.

The explanation they give is that most of the time particles are bound by some kind of potential and that the solution for the Particle in a Box problem is a standing wave.

The way I understood it is that it comes naturally from the Classical Wave Equation, that it is the only solution with such boundary conditions. But then I looked at a guitar string in slow motion and it didn't behave like a standing wave! After some googling I found out that its behavior is called a pulse.

Now I am at a loss. Why do we only consider standing waves and discard other solutions? Are there some restrictions that should be applied that I am not aware of?

Answer 1

Your pulse is in fact a sum of standing waves, each with a different amplitude. Writing a pulse as a sum of standing waves can be done using Fourier transform and Fourier analysis.

It is possible to use sums of solutions because the Schrödinger equation is linear: the sum of solutions remains a solution.

Note that the solutions to the SE with definite energy will have a probability density independent of time, so you cannot get a propagating pulse as illustrated. On the other hand, when solutions with different energies are combined, it is possible to have time-dependent (i.e. time changing) probability densities.