How to determine ##\psi(x,0)## in general situation?

[Kim hyeon su]

TL;DR

I want to know how can we determine $\psi(x,0)$ in a general situation.

First of all, please understand that I am not good at English so I used the translator.

(EDITED by mentor: small edits + added latex...)

After I studied free particles, I wanted to apply what I learned to other situations.

But as you know, to determine the solution of the free particle, you need to have $psi(x,0)$. Usually, $psi(x,0)$ is given by the textbook problem (like: $psi(x) = A e^{-x²}$).

However, in other situations (not in the book), there no information is given about psi(x,0).

So I want to know how can we determine $psi(x,0)$ in a general situation.
1) if $\psi(x,0)$ is determined by experiment, we have to measure $-\infty$ to $+\infty$? How?

2) What kind of $\psi(x,0)$ can a free particle have? And what makes them different?

 

[jambaugh]

The value of the wave-function $\psi(x,0)$ is not something that is measured, that is except for the case where the particle being described has been observed at a single point in which case the wave-function is a Dirac delta function centered at that point.

Typically one instead, solves the differential equation (Schrödinger's equation) for the wave function satisfying certain conditions. One typically must know the Hamiltonian (defining the system energy and time evolution via Schrödinger's eqn) and assume, for example the particle has a well defined energy.

Typical examples are things like a particle in an infinite square well potential (sinusoidal wave function), or a particle in a parabolic potential well (harmonic oscillator which has the $Ae^{-x^2}$ shaped wave function for its lowest energy "state") or particle in a -k/r potential (e.g. electron in a hydrogen atom).