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
It is interesting to note that the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation.
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.
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".
- 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