- #1
schieghoven
- 85
- 1
Hi all,
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
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