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I post a question about the dimension of wave function in position space before and people help me to identify that the unit of wave function should be in the unit of ##\text{m}^{-1/2}##. I am verifying that by doing the dimension analysis on Schrodinger equation

##i\hbar\frac{\partial \Psi}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V\right)\Psi##

Checking the left side and term about potential on the right side, note that the unit for ##\hbar## is ##\text{J}\cdot\text{s}##, unit of ##\Psi## is ##\text{m}^{-1/2}##, unit of ##V## is ##\text{J}##, so the left and right side end up with unit of ##\text{J}\cdot\text{m}^{-1/2}##

Today I am reading something about s-wave scattering online and I saw this nonlinear Schrodinger equation

https://en.wikipedia.org/wiki/Gross–Pitaevskii_equation

##\left(-\frac{\hbar^2}{2m}\nabla^2 + V + \frac{4\pi\hbar^2 a_s}{m}|\Psi|^2\right)\Psi = \mu \Psi##

The first two terms on the left side give the same unit as I got above, i.e. ##\text{J}\cdot\text{m}^{-1/2}##, but for the 3rd term about scattering length, I got something strange. Here is what I did

I think the unit for ##a_s## is meter so the unit for the third term is ##\frac{\text{J}^2\cdot\text{s}^2\text{m}}{\text{kg}}\frac{1}{\text{m}}\text{m}^{-1/2}## but this is not the same as ##\text{J}\cdot\text{m}^{-1/2}##, what mistake I made?

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# Dimensional analysis on equation including scattering length

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