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I'm doing an undergraduate introductory course in quantum mechanics, and I'm having a hard time understanding the interpretation. I've recently learned about the famous "Particle in a box" problem, in which we solve the time-independent schrodinger wave equation to get the energy eigenvalues.

After solving we find that the particle can only have definite and discrete energy levels. (via an expression in terms of the quantum number n). Now my question is, how do we know which energy level the particle will occupy? Does it depend on initial conditions, if yes then how? or is it that it exists in a superposition of the various states, if yes then how do we calculate the probability factors of each state?

I have understood how the general form of the wave function for a particular situation describes a given system, but am not able to understand how to incorporate the initial conditions of the system.

For example, in classical mechanics, we solve the differential equation of motion, which gives us the general form of all solutions, and then we substitute the initial conditions (like initial position, initial velocity etc.) to get the right solution. In the same way how do we do this in QM. How do we take into consideration things like initial position and velocity of the particle. How do we interpret initial conditions.

I would love to gain a further insight into this concept.