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As shown at https://quantummechanics.ucsd.edu/ph130a/130_notes/node124.html from quantum physics by Jim Branson (2003), using separation of variables we can break up the general Schrodinger equation into:
  • a time-only part that does not depend on space and does not depend on the potential
  • a space-only part that does not depend on time:
    Equation 33. 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.
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It is interesting to note that the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation, since the value of the energy E can be any fixed value that can satisfy the boundary conditions conditions imposed.
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Then, just like in solving partial differential equations with the Fourier series of the heat equation, the boundary conditions may make it so that only certain discrete values of E are possible solutions.
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Then, the time part o the equation can be solved explicitly in the general case, as it does not depend on the potential , and it is just an exponential.
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Therefore, all we need to do to solve the general Schrodinger equation is to solve the time-independent version, and then decompose the initial condition in terms of it like as done in Section "Solving partial differential equations with the Fourier series".
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Furthermore:
  • we immediately see from the equation that the time-independent solutions are states of deterministic energy because the energy is an eigenvalue of the Hamiltonian operator
  • the probability density of such a state does not change with time because the exponential time part cancels out on the conjugate
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