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The cool thing about the Time-independent Schrödinger equation is that we can always reduce solving the full Schrödinger equation to solving this slightly simpler time-independent version.
This is actually the approach that we will always take when solving the schrodinger equation, see e.g. quantum harmonic oscillator.
Before reading any further, you must understand heat equation solution with Fourier series, which uses separation of variables.
Once that example is clear, we see that the exact same separation of variables can be done to the Schrödinger equation. If we name the constant of the separation of variables for energy, we get::
  • a time-only part that does not depend on space and does not depend on the Hamiltonian at all. The solution for this part is therefore always the same exponentials for any problem, and this part is therefore "boring":
  • a space-only part that does not depend on time, bud does depend on the hamiltonian:
    Equation 9. Time-independent Schrodinger equation
    Since this is the only non-trivial part, unlike the time part which is trivial, this spacial part is just called "the time-independent Schrodinger equation".
    Note that the here is not the same as the in the time-dependent Schrodinger equation of course, as that psi is the result of the multiplication of the time and space parts. This is a bit of imprecise terminology, but hey, physics.
Because the time part of the equation is always the same and always trivial to solve, all we have to do to actually solve the Schrodinger equation is to solve the time independent one, and then we can construct the full solution trivially.
Once we've solve the time-independent part, call it for each discrete energy as , we proceed exactly as we did in heat equation solution with Fourier series: we make a weighted sum over all possible to match the initial condition, which is analogous to the Fourier series in the case of the heat equation to reach a final full solution:
Equation 10. Solution of the Schrodinger equation in terms of the time-independent and time dependent parts
and this is a solution by selecting such that at time we match the intial condition:
The fact that this approximation of the initial condition is always possible from is mathematically proven by some version of the spectral theorem bsaed on the fact that The Schrodinger equation Hamiltonian has to be Hermitian and therefore behaves nicely.
It is interesting to note that solving the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation where:
The only difference from usual matrix eigenvectors is that we are now dealing with an infinite dimensional vector space.
Furthermore:

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