Question

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

Answer 1

• 
  `[\psi]= [Length^(-3/2)]`
• This means that the integral of the square modulus over the space is dimensionless.

Step-by-step 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

`\int\limits^(x_f)_(x_0) \int\limits^(yf)_(y_0) \int\limits^(z_f)_(z_0) |\psi|^2 \, dz \,  dy \,  dx`

must be dimensionless, as represents a probability.

As the differentials has units of length

`[dx]=[dy]=[dz]=[Length]`

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

`[\psi]^2 = [Length^(-3)]`

taking the square root this gives us :

`[\psi] = [Length^(-3/2)]`