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)e−iEt/ℏϕn(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.