Why is a wave function time dependent? And what does it physically mean when the probability density density function is varying with time. I'm not getting how to physically interpret that. Will taking example of a particle in 1d box with superposition state two energy levels help? Because it's simplest system(I can think of) whose probability density is time dependent.
Edit: I'm not looking for a proof or a mathematical explanation of how it depends on time but I'm questioning why should the wavefunction depend on time. I am getting answers derived from time dependent Schrodinger's equation but then we are already assuming that wave function time dependent. So to be more clear my intent was question "where is the time dependence of Schrodinger's equation coming from? Where did the time dependence come from?"
Edit: I actually started questioning about why the probability density of superposition of states is time dependent, on second look I realised it came from the time dependence of wave function and that again was because of time dependence term in Schrodinger's equation.
Probability density<--wave function<---schrodingers equation<--- [???????]
Is it the end or any more layers are there? Please go down as many layers as possible.