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From quantum mechanics, I learnt:

  1. Completeness of energy eigenfunction: The energy eigenfunctions of the Schrodinger equation span the space, i.e. any state can be expanded as linear combination of energy eigenstates.\ That is $\Psi(x)=\sum_{n=0}^{\infty}b_{n}\psi_{n}(x)$.
    And $\Psi(x,t)=\sum_{n=0}^{\infty}b_{n}e^{-\frac{itE_{n}}{\hbar}}\psi_{n}(x)$

  2. Linearity of Schrodinger equation: Linear combination of solutions of Schrodinger equation can also solve the Schrodinger equation.

Therefore, can we say that any wavefunction can solve an arbitrary Schrodinger equation? But that sounds weird.

The question actually comes from the MIT8.06 lecture (1:02:46), where the professor stated that the trial wave function is not the solution of Schrodinger equation, but can be expanded as energy eigenstates, so I'm confused.

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The energy eigenstates are the solutions to the time independent Schrodinger equation, however linear combinations of the energy eigenfunctions are not time independent, so they are not solutions to the time independent SE - they are solutions to the time dependent SE.

You are quite correct that we can take any function of position only (not time) and subject to a few sensible constraints like normalisation we can write it as a superposition of the energy eigenfunctions. It will then be a solution to the time dependent SE, and the time dependent SE will tell us how this function evolves with time.

What you cannot do is take any arbitrary function of position and time and make it a solution to the time dependent SE by writing it as a superposition of the energy eigenstates. That's because the coefficients in the sum of energy eigenstates would be functions of time and would in general change with time in a way that did not agree with the time dependent SE.

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  • $\begingroup$ For the last paragraph, is it mean that $\Psi(x,t)=\sum_{n=0}^{\infty}b_{n}e^{-\frac{itE_{n}}{\hbar}}\psi_{n}(x)$ is not a solution of time dependent SE, becuase it is not a linear superposition? $\endgroup$
    – Lucas
    Jul 23, 2022 at 8:14
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    $\begingroup$ @Lucas what you've written is the evolution described by the TDSE so your equation is a solution of the TDSE. What I'm saying is that for some randomly chosen function of time and space, f(t,x), when you expanded this as a sum of the energy eigenfunctions the time dependence of the coefficients would be different from your equation and hence your random function f(t,x) would not be a solution of the TDSE. $\endgroup$ Jul 23, 2022 at 11:23
  • $\begingroup$ Thanks! Can you be more specfic? In which way are the coefficients different? $\endgroup$
    – Lucas
    Jul 23, 2022 at 17:44

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