Schroedinger’s first interpretation of the wavefunction was that Ψ would describe how a particle dissipates. Where the wavefunction Ψ was the highest, then that was where more of the particle was present. Max Born disagreed saying that a particle would not dissintegrates, choosing another direction to move. Max Born proposed that the wavefunction would actually describe the probability of a particle inhabiting a space. Both Schroedinger and Einstein were initially opposed to the idea of a probabilistic interpretation of the Schroedinger equation. The probabilistic interpretation of Max Born however later became the consensus view of quantum mechanics.
The wavefunction Ψ therefore describes the probability of finding a particle at position x at time t, not the amount of the particle that exists there.
Since the Schroedinger equation is both a function of position and time, it can only be solved for one variable at a time. Solving for position is preferable due to the fact that if the wavefunction is known for all x, this can provide information for how the wavefunction is at a later time.
Of the limits regarding the wavefunction, it is also said that the wavefunction must be convergent. The wavefunction therefore does not approach a finite constant as x approaches infinity.
We also recall that a wavefunction may also be multiplied by a number. It would appear that doing so would violate the above expression. The answer regarding this conjecture is that the above formula represents a normalized wavefunction. Yet it turns out that not all wavefunctions are normalizable. The case of multiplying the wavefunction with a magnitude in fact would still be normalizable, however. A wavefunction can be normalized if the integral is a finite number less than infinity using the following method: