Conditions for Acceptable Wave Function

Conditions for Acceptable or Well Behaved Wave Function

Condition for a physically accepted, well behaved, realistic wave function are given below-
1. Ψ(x,t) should be finite, single-valued and continuous everywhere in space.
2. (dΨ/dx) should be continuous everywhere in space but dΨ/dx may be discontinuous in some cases as follows-
a. If the potential under which the particle is moving, has an infinite amount of discontinuity at some points
b. If the potential under which the particle is moving, is of dirac delta nature.
3. Ψ(x,t) should be square integrable i.e.

The probability of finding the particle at time t in an interval dx about the position x is proportional to
|ψ(x,t)|²dx

Wave Functions

In one dimension, wave functions are generally denoted by the symbol ψ(x,t). They are functions of the coordinate x and the time t. ψ(x,t) is not a real, but a complex function.
The wave function of a particle, at a particular time, contains all the information but the wave function itself has no physical interpretation. It is not measurable. However, the square of the absolute value of the wave function has a physical interpretation.
In one dimension, we interpret |ψ(x,t)|² as a probability density, a probability per unit length of finding the particle at a time t at position x.
This interpretation is possible because the square of the magnitude of a complex number is real. For the probability interpretation to make sense, the wave function must satisfy certain conditions discussed above.

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