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Completeness and time

# Completeness and time - eigenfunctions(10.9(The functions...

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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 (10.5) where (10.6) We know, from the superposition principle, that (10.7) so that the time dependence is completely fixed by knowing at time only! In other words if we know how the wave function at time can be written as a sum over eigenfunctions of the Hamiltonian, we can then determibe the wave function for all times. Simple example The best way to clarify this abstract discussion is to consider the quantum mechanics of the Harmonic oscillator of mass and frequency , (10.8) If we assume that the wave function at time is a linear superposition of the first two

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Unformatted text preview: eigenfunctions, (10.9) (The functions and are the normalised first and second states of the harmonic oscillator, with energies and .) Thus we now kow the wave function for all time: (10.10) In figure 10.1 we plot this quantity for a few times. Figure 10.1: The wave function ( 10.10 ) for a few values of the time . The solid line is the real part, and the dashed line the imaginary part. The best way to visualize what is happening is to look at the probability density, (10.11) This clearly oscillates with frequency . how that . Another way to look at that is to calculate the expectation value of : (10.12) This once again exhibits oscillatory behaviour!...
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