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roshan2004
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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?
Schrodinger's wave equation is a mathematical equation developed by Austrian physicist Erwin Schrodinger in 1926. It is a fundamental equation in quantum mechanics used to describe the behavior of particles in a quantum system.
Schrodinger's wave equation describes the wave function of a quantum system, which represents the probability of finding a particle in a certain location at a certain time. It also describes how this wave function changes over time.
Schrodinger's wave equation is used in quantum mechanics to predict the behavior of particles at the atomic and subatomic levels. It allows scientists to calculate the wave function of a system and determine the probabilities of different outcomes.
Schrodinger's wave equation revolutionized the field of quantum mechanics and provided a mathematical foundation for understanding the behavior of particles at the atomic level. It has been a crucial tool in making predictions and furthering our understanding of the microscopic world.
While Schrodinger's wave equation is a powerful tool in quantum mechanics, it does have limitations. It can only accurately describe non-relativistic systems, and it does not take into account the effects of gravity. Additionally, it cannot predict the exact position or momentum of a particle, but rather provides probabilities of finding the particle in certain locations or with certain momenta.