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There is no one answer. The dimensions are inferred from the fact that $\left| \psi\right|^2$ represents a probability density. Perhaps the most straight-forward way is to consider the normalizatio...
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#2: Post edited
by
Anyon
·
2024-05-19T19:27:45Z (6 months ago)
Fixed duplicate word.
There is no one answer. The dimensions are inferred from the fact that $\left| \psi ight|^2$ represents a probability _density_. Perhaps the most straight-forward way is to consider the normalization condition, i.e. that the integral of the probability density over the state space equals a probability of one. As you know, probabilities are dimensionless by definition. If the wave function is represented in $d$ spatial dimensions, the dimension of the wave function thus becomes becomes $\left[ \psi ight]=L^{-d/2}$, where $L$ represents length (the SI unit for which is the metre). You can similarly work out the units for wave functions represented in momentum space.
- There is no one answer. The dimensions are inferred from the fact that $\left| \psi ight|^2$ represents a probability _density_. Perhaps the most straight-forward way is to consider the normalization condition, i.e. that the integral of the probability density over the state space equals a probability of one. As you know, probabilities are dimensionless by definition. If the wave function is represented in $d$ spatial dimensions, the dimension of the wave function thus becomes $\left[ \psi ight]=L^{-d/2}$, where $L$ represents length (the SI unit for which is the metre). You can similarly work out the units for wave functions represented in momentum space.
#1: Initial revision
by
Anyon
·
2023-06-10T21:23:10Z (over 1 year ago)
There is no one answer. The dimensions are inferred from the fact that $\left| \psi\right|^2$ represents a probability _density_. Perhaps the most straight-forward way is to consider the normalization condition, i.e. that the integral of the probability density over the state space equals a probability of one. As you know, probabilities are dimensionless by definition. If the wave function is represented in $d$ spatial dimensions, the dimension of the wave function thus becomes becomes $\left[ \psi \right]=L^{-d/2}$, where $L$ represents length (the SI unit for which is the metre). You can similarly work out the units for wave functions represented in momentum space.