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In what conditions do we use time dependent and time independent Schrodinger's wave equations?
I don't agree. It makes sense to use the time-dependent Schrodinger equation (TDSE) when solving a problem for a time-dependent potential. However quite often what one is most interested in is the energy spectrum and/or eigenstates of a time-independent potential. In such a case, it is preferable to solve the time-independent Schrodinger equation (TISE), either analytically (this is only rarely possible for meaningful real-world problems), or numerically (i.e. using the variational method and diagonalizing a large matrix).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.
Read my first sentence again .. you said it is *always* best to use the TDSE. I said it is best to use the TISE when solving a time independent potential.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 refering 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".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."
If you are interested in a) the possible results of an energy measurement and b) the probability distribution of those results, then you must solve the energy eigenvalue equation where the energy operator is the Hamiltonian [tex]\hat H = - {{\hbar ^2 } \over {2m}}{{\partial ^2 } \over {\partial x^2 }} + V(x)[/tex] in one dimension for a time independent potential. This energy eigenvalue equation is what we call the time independent Schrodinger equation.In what conditions do we use time dependent and time independent Schrodinger's wave equations?