I am a beginner to quantum mechanics and am trying to make sense of Schrodinger's Equation. I am attempting to find probabilities in the case of a free particle in the general case.

It is my understanding that the solution to Schrodinger's Equation in the general case of a free particle is as follows:

$$\psi(X,T) = e^{i/\hslash ( px - Et)}$$

The modulus square of this is 1, which means the probability density function is uniform.

Two questions:
1. Over what values of x is this pdf defined? Can we eliminate all values of x > ct?
2. Am I correct to interpret x as the distance from the (known) starting position of the particle at t = 0?

Thanks.
 Notice that that wave-function is not normalizable. The integral of the modulus square of that wave-function over all space is infinite. A free particle wave function cannot actually be what you gave, but must be a wave-packet.

 Quote by Matterwave Notice that that wave-function is not normalizable. The integral of the modulus square of that wave-function over all space is infinite. A free particle wave function cannot actually be what you gave, but must be a wave-packet.
So are you saying that the solution I mentioned does not describe a free particle wave? I'm not sure how that could be, I have read from multiple sources that it is.

Is there a different approach that needs to be taken to achieve a normalizable function?