Ciro Santilli
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# Time-independent Schrodinger equation

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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
• a space-only part that does not depend on time
$$Equation 21. Time-independent Schrodinger equation. [−2mℏ​∇2+V(x)]Ψ=EΨ(x) (21)$$
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It is interesting to note that the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation.
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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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