Can the Free Particle Wave Equation Be Solved for Periodic Boundary Conditions?

[sniffer]

for a free particle, the wave equation is a superposition of plane waves,

$$\psi(x,0)= \int_{-\infty}^{\infty}g(k)\exp(ikx)dk$$
and
$$g(k)= \int_{-\infty}^{\infty}\psi(0,0)\exp(-ikx)dx$$

one is the Fourier transform of the other. some cases to solve this is when we assume a small delta k, so g(k) behaves like a pulse "delta" function.

is there any more general case we can solve this?

i have been thinking hard if we have definite periodic x, say from 0 up to $2\pi L$, is it solvable?

what would be the (periodic) eigen energy function (if it is)?

 

[vanesch]

for a free particle, the wave equation is a superposition of plane waves,

$$\psi(x,0)= \int_{-\infty}^{\infty}g(k)\exp(ikx)dk$$
and
$$g(k)= \int_{-\infty}^{\infty}\psi(0,0)\exp(-ikx)dx$$

one is the Fourier transform of the other. some cases to solve this is when we assume a small delta k, so g(k) behaves like a pulse "delta" function.

The above pair is, as you say, a Fourier transform pair. Any "nice" function can be written that way, so the above is not a "wave equation" or something, it is a general way of writing a function.
The quantum state of a single scalar particle is described by just such a nice function, called the wave function. At any time, it can be (almost) any function. However, what the wave equation (not written here) gives you, is how this wavefunction AT A CERTAIN TIME t0 will change into the wavefunction at another time t1. This equation will be different according to the situation at hand (free particle, particle in a potential...).

cheers,
Patrick.