A solution to time dependent SE but not Time independent SE?

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Discussion Overview

The discussion revolves around the relationship between solutions to the time-dependent Schrödinger equation (SE) and the time-independent Schrödinger equation. Participants explore the conditions under which a wave function can be a solution to one equation but not the other, particularly in the context of quantum mechanics and potential functions.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions how a wave function can be a solution to the time-dependent SE but not to the time-independent SE, suggesting that the time-dependent solutions consist of linear combinations of wave functions from another potential that spans Hilbert space.
  • Another participant asserts that if a solution is time-dependent, it inherently cannot be time-independent, emphasizing that all time-dependent solutions are distinct from time-independent solutions.
  • A participant raises the point that at t=0, one might consider whether the wave functions could be time-independent, but another counters that without the time-dependent factor, the function is not the same, indicating that time dependence does not simply factor out.
  • It is noted that the time-independent Schrödinger equation applies when separation of variables can be used, and while one could theoretically apply it at t=0, the resulting basis states would not be useful due to differing time evolution.
  • A later reply mentions that a teacher clarified that linear combinations of separable states as solutions to the SE do not apply to the time-independent SE, even at t=0.

Areas of Agreement / Disagreement

Participants express disagreement regarding the applicability of time-dependent solutions to the time-independent equation, with no consensus reached on the implications of time dependence in this context.

Contextual Notes

Limitations include the assumption that time-independent solutions can be derived from time-dependent solutions under certain conditions, which remains unresolved. The discussion also highlights the dependence on the ability to separate variables in the context of the time-independent Schrödinger equation.

faen
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A solution to time dependent SE but not Time independent SE??

How is it possible that a wave function is a solution to the time dependent schroedinger equation, but not to the time independent schroedinger equation (without time factors tacked on) with the same potential? I had this case on my quantum physics exam. I wrote that the time dependent set consisted of a linear combination of wavefunctions which were solutions from another potential which spanned hilbert space. But it still sounds contradicting that this function didnt fit into the time independent schroedinger equation when its basis functions span the space too.
 
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well... yeah. if it is time dependent then it is not time-independent... so all time dependent solutions are not time-independent solutions.
 
yea but without the time dependence factor (t=0), wouldn't they be time independent too?
 
faen said:
yea but without the time dependence factor (t=0), wouldn't they be time independent too?

yes, but without the time-dependent factor then it is not the same function. in general the time-dependence doesn't factor.
 
the time independent Schrödinger equation only applies when you can use separation of variables, while you could theoretically apply it at t=0 and get basis states for that particular instant, this would be of little use as the time evolution would be completely different.

so if you have a solution to the time independent equation this means the solution can be put in the form of e^(-iEt)y(x) whereas you can't do this for a solution to the time independent solution
 
Asked my teacher and found out that linear combination of separable states as solution to SE don't apply to the time independent SE but only to the time dependent SE. Not even at t=0.
 
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