I guess it should be L^(-3/2) just like the case of a particle in 3D box. Time dependence is already incorporated in time-dependent Schrodinger equation.
Joydev,
A wave-function, if it is not in an infinite well, has no 'edge'.
There will always be a non-zero component at an arbitrary distance from its centre.
If one wants, one can calculate the '1/e' 'size' of it, but by its nature you cannot bound it (without an infinite well - and they're not terribly physical)
I think it is L^(-4/2) for true 4D. Because the forth coordinate is -ict (or ict). Here c is the speed of light. The Schroedinger equation (as Vikash Pandey mentioned) in my opinion does not dictate the units of the wave function. In that 4D space it must be rewritten in a new way, I cannot say how it would look like, but I do not see how this equation could have any effect on the units. Wave function is normalized to 1 anyway, so I bet on L^-2 :)
To check if the dimensions are OK (dimensional analysis) is the first step to figure out if an equation is correct. Unfortunately it's not done often enough. The excellent question of Joydev, about the dimensions of a wave function, and the following differing answers, are prime examples of this shortcoming.
Let's take a steady state wave function - it's a square of wave function - probability of finding a particle - per unit volume, so it is L^-3. The problem arises if you try to go to wave function proper (which exists in complex space), The time dependence of a wave function is also taking place in the complex space. So should it be per unit of complex space and complex time?
One more point - Schrodinger equation is non-relativistic, we don't have to worry about spacetime.
We encountered a similar problem of other than ordinary 3D space in what units to use when the space is less than 3 (fractal,, say 2.7). Can you say "per L^2.7"? It concerned the dissipation of turbulence in fluids.
So I'm expanding on Joydev's question; what are the units in other than ordinary 3D coordinates?
First of all, the wavefunction is defined in phase space, not spacetime. So referring to spacetime is wrong.
Now the probability density is the square of the wavefunction and that multiplied by the phase space volume is dimensionless.
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