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I'm interested to find a solution to the wave equation corresponding to

Gaussian initial conditions

[tex] \psi(0,x) = e^{-x^2/2} [/tex]

A solution which satisfies these initial conditions is (up to some constant factor)

[tex] \psi(t,x) = \int \frac{d^3k}{(2\pi)^3} e^{-k^2/2 + i(k \cdot x - \omega t)} [/tex]

where \omega = |k|. If we use spherical coordinates ( so that k.x = |k| r cos \theta )

then the angular dependence can be integrated out ... I get

[tex] \psi(t,x) = \int_{0}^{\inf} \frac{dk}{(2\pi)^2} \frac{2k \sin(kr)}{r} e^{-k^2/2 - \omega t} [/tex]

but can't get any further. Any ideas? Maybe this isn't the right direction to go anyway.

The reason I'm looking at this is because I'm interested to see the viability of

Hermite functions as a basis for state space in QM: the above function e^-x^2/2 is the

zeroth Hermite function. Hermite functions form a countable orthonormal basis, they

can be defined to have unit normalization, and all of their moments are finite. None

of these are true for the usual plane wave basis... so I feel the Hermite functions provide

a more concrete realisation of the Hilbert space of states. Of course, they won't be much

use unless I get them in closed form for t>0.

Thanks,

Dave

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# Closed-form solutions to the wave equation

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