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10.3 Completeness and time-dependence

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

$\displaystyle \psi$(x, t) = $\displaystyle \sum_{n=1}^{\infty}$cn(t)$\displaystyle \phi_{n}^{}$(x), (10.5)

where

$\displaystyle \hat{H}$$\displaystyle \phi_{n}^{}$(x) = En$\displaystyle \phi_{n}^{}$(x). (10.6)

We know, from the superposition principle, that

$\displaystyle \psi$(x, t) = $\displaystyle \sum_{n=1}^{\infty}$cn(0)e-iEt/$\scriptstyle \hbar$$\displaystyle \phi_{n}^{}$(x), (10.7)

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.



© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk