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Quantum Harmonic Oscillator
[edit]It follows from momentum space definition of position operator as , that by using Schrodinger's equation and taking , we get:
Since momentum basis Schrodinger equation is same as that of position basis Schrodinger equation, the real-valued momentum wavefunction solution is same as momentum wavefunction corresponding to spatial wavefunction up-to an arbitrary phase which can be found to be for energy eigenfunction. Hence the change , , and in any equation expressed in terms of those variables gives another valid equation.
Configuration space
[edit]Finding from given curve in configuration space, .