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Physics LibreTexts

10.3: Completeness and time-dependence

( \newcommand{\kernel}{\mathrm{null}\,}\)

In the discussion on formal aspects of quantum mechanics I have shown that the eigenfunctions to the Hamiltonian are complete, i.e., for any ψ(x,t)

ψ(x,t)=n=1cn(t)ϕn(x)

where

ˆHϕn(x)=Enϕn(x).


We know, from the superposition principle, that

ψ(x,t)=n=1cn(0)eiEt/ϕn(x),

so that the time dependence is completely fixed by knowing c(0) at time t=0 only! In other words if we know how the wave function at time t=0 can be written as a sum over eigenfunctions of the Hamiltonian, we can then determibe the wave function for all times.


This page titled 10.3: Completeness and time-dependence is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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