What are the units, if any, of the particle in a box wavefunction. what does this mean?

Explanation:

We know that the square modulus of the wavefunction integrated over a volume gives us the probability of finding the particle in that volume. So the result of the integral

must be dimensionless, as represents a probability.

As the differentials has units of length

for the integral to be dimensionless, the units of the square modulus of the wavefunction has to be:

taking the square root this gives us :

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