Hi, I'm trying to get my head around Schrödinger's equation and quantum wave theory. I'll try to shortly state how I understand it so you may see where I'm wrong and better answer the question. In classical mechanics if you solve a linear differential equation, the sum of the solutions is also a solution. This happens in quantum mechanics too obviously, but there's a difference: in classical mechanics you must choose one solution based on initial conditions, i.e. the speed and shape of a string, or the speed of a particle inside a gravitational or electromagnetic field when t=0. Then you get a single solution. In quantum mechanics you solve the equation, but you can't possibly know the initial conditions because of the uncertainty principle. So instead we add up all the solutions to form a localized wave packet, which represents both the uncertainty in position and momentum, through the Fourier transform. I understand the maths behind this process, what I don't understand is the physics. How do you determine the weigh of each possible solution? For a free particle, for instance, why would you build a gaussian packet instead of another one? On a similar note, with bound states (particle in a box for example) how do you weigh each possible state and in which way do these things model and reflect a particular problem and initial conditions? Is there even such a thing as initial conditions in QM? Those are the main questions, but I'll sneak a last one here. With bound states, you get a wave function which is the weighed sum of all the solutions of the SE. Each of these has a determined energy so we call them energy states. We say that the particle exists in a superposition of all these states until it's observed, where it must collapse to one and only one of the states: it can only have one energy value and it must be that of one of the possible energy states. Does this imply that each state is linearly independent from every other state? I can verify it in the particle in a box experiment, but does it always happen? Can it be proved from the SE? I could elaborate on the math to say why I think they should be LI but I'm away from home now and I need my notes for this. If it is unclear, I'll try and explain it better or post another question when I get back home. Thanks a lot in advance!