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In what conditions do we use time dependent and time independent Schrodinger's wave equations?
Gerenuk said:The time-dependent version is always the right choice.
However if your potential does not depend on time, but position only [itex]V(x)[/itex] (unlike [itex]V(x,t)[/itex]), then you can use the trial solution [itex]\psi(x,t)=\phi(x)e^{-i E t/\hbar}[/itex] and derive an equation for the special part [itex]\phi[/itex] of the wavefunction. You effectively get the time-independent version of the Schrödinger equation. From this time-indepedent special case you can first find [itex]\phi[/itex] and E and finally put it back into the full wavefunction [itex]\psi(x,t)[/itex]. Please try that above exercise with the trial solution.
Gerenuk said:It's not a matter of taste. What you want to use merely depends on whether your potential is [itex]V(x,t)[/itex] or [itex]V(x)[/tex]. What time-independent eigenstates do you want to find if you haven't even given a time-independent potential to deal with?
Maybe you are referring to some perturbation theory approximations?
I wanted to emphasize that there is only one equation that describes the physics. Everything else is a special case. This is to prevent misconceptions about "dualities" and "special cases".SpectraCat said:I understand that others will have a different take .. I was just providing a counter-example to your statement that, "The time-dependent version is always the right choice."
roshan2004 said:In what conditions do we use time dependent and time independent Schrodinger's wave equations?