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mr Goodwill [35]
3 years ago
7

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

Physics
1 answer:
SIZIF [17.4K]3 years ago
5 0

Answer:

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

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}]

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