Dimensional analysis on equation including scattering length

[KFC]

Hi all,
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 Schrödinger 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 Schrödinger 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?

 

[fzero]

A normalized wavefunction with have units of $(\text{volume})^{-1/2}$, so in 3d SI units, this would be $\text{m}^{-3/2}$.

 

[KFC]

A normalized wavefunction with have units of $(\text{volume})^{-1/2}$, so in 3d SI units, this would be $\text{m}^{-3/2}$.

Sorry I didn't mention that before, I am considering the 1 dimensional case

 

[fzero]

Sorry I didn't mention that before, I am considering the 1 dimensional case

In that case, $a_s$ would no longer have units of length, but $(\text{length})^{-1}$. In 1d, the Coulomb interaction is linear, so the details of the S-wave scattering are different than in 3d.

 

[KFC]

In that case, $a_s$ would no longer have units of length, but $(\text{length})^{-1}$. In 1d, the Coulomb interaction is linear, so the details of the S-wave scattering are different than in 3d.

Thanks a lot. Following that comment, I have the consistent units on both side now. However, I still don't quite understand why $a_s$ is in the inverse of length. If that's the case, why we still call it scattering 'length'? You mention the S-wave scattering in 3D, so what's the unit should be for 2D case?

In the literature, I found that the s-wave scattering length for sodium 23 is about 2.6nm, so does it mean it is only true for 3D case? If you make the system 1 dimensional, should the scattering length becomes 1/2.6 [$\text{nm}^{-1}$] or something else?

Thanks a lot for your pointing out the mistake I made.

 

[fzero]

Thanks a lot. Following that comment, I have the consistent units on both side now. However, I still don't quite understand why $a_s$ is in the inverse of length. If that's the case, why we still call it scattering 'length'? You mention the S-wave scattering in 3D, so what's the unit should be for 2D case?

Let's go back to that wikipedia page and look at the 2nd equation, which is the model Hamiltonian with the contact potential term
$$ V_{\text{cont.}}( \mathbf{r}_i - \mathbf{r}_j) = \sum_{i<j} \frac{4\pi \hbar^2 a_s}{m} \delta( \mathbf{r}_i - \mathbf{r}_j).$$
On dimensional grounds, the parameter $a_s$ has units of length, and if we were to go through the analysis, we'd find that the cross-section for scattering from this delta function potential is $\sigma \sim 4\pi a_s^2$ at low-energies. Therefore, $a_s$ can be identified with the scattering length.

In lower dimensions, the dimensionality of the delta function changes. In 2d, the parameter $a_s$ would be dimensionless, while in 1d it would have units of inverse length. A detailed analysis of scattering in these dimensions appears in http://arxiv.org/abs/1009.1918. In the 1d result there, what we're calling $a_s$ is genuinely the inverse of the scattering length, see eq (62). In 2d, the analysis is more subtle and the parameter $a_s$ doesn't appear to be physically significant. The scattering length actually appears in a logarithmic term, see eq (74). Actually this reference obtains the contact potential as a limit of the square well. It is possible that some differences will occur if one starts with the delta function. But in 2d there will probably be logarithmic divergences anyway, leading to similar results.

Now, in practice, we are in 3d and a system can be effectively 2d or 1d due to some phenomenon. In http://massey.dur.ac.uk/resources/ngparker/chapter2.pdf, section 2.3, a quasi-1d system is examined, corresponding to an atom trap where harmonic oscillator potentials are applied in two directions, trapping atoms in a 1d channel. In terms of the GP equation, the term $g_{3d} |\psi|^2$ is replaced by $g_{1d} = g_{3d} /(2\pi l_r)^2$, see eq. (2.21). Here $l_r$ is the length scale associated with the harmonic oscillator potential.

In the literature, I found that the s-wave scattering length for sodium 23 is about 2.6nm, so does it mean it is only true for 3D case? If you make the system 1 dimensional, should the scattering length becomes 1/2.6 [$\text{nm}^{-1}$] or something else?

The physical scattering length there is derived from 3d experiments. If you used the atom trap from that 2nd reference I mention above, the effective 1d scattering length is probably $(2\pi l_r)^2 /a_{3d}$.

 

[KFC]

Let's go back to that wikipedia page and look at the 2nd equation, which is the model Hamiltonian with the contact potential term
$$ V_{\text{cont.}}( \mathbf{r}_i - \mathbf{r}_j) = \sum_{i<j} \frac{4\pi \hbar^2 a_s}{m} \delta( \mathbf{r}_i - \mathbf{r}_j).$$
On dimensional grounds, the parameter $a_s$ has units of length, and if we were to go through the analysis, we'd find that the cross-section for scattering from this delta function potential is $\sigma \sim 4\pi a_s^2$ at low-energies. Therefore, $a_s$ can be identified with the scattering length.

In lower dimensions, the dimensionality of the delta function changes. In 2d, the parameter $a_s$ would be dimensionless, while in 1d it would have units of inverse length. A detailed analysis of scattering in these dimensions appears in http://arxiv.org/abs/1009.1918. In the 1d result there, what we're calling $a_s$ is genuinely the inverse of the scattering length, see eq (62). In 2d, the analysis is more subtle and the parameter $a_s$ doesn't appear to be physically significant. The scattering length actually appears in a logarithmic term, see eq (74). Actually this reference obtains the contact potential as a limit of the square well. It is possible that some differences will occur if one starts with the delta function. But in 2d there will probably be logarithmic divergences anyway, leading to similar results.

Now, in practice, we are in 3d and a system can be effectively 2d or 1d due to some phenomenon. In http://massey.dur.ac.uk/resources/ngparker/chapter2.pdf, section 2.3, a quasi-1d system is examined, corresponding to an atom trap where harmonic oscillator potentials are applied in two directions, trapping atoms in a 1d channel. In terms of the GP equation, the term $g_{3d} |\psi|^2$ is replaced by $g_{1d} = g_{3d} /(2\pi l_r)^2$, see eq. (2.21). Here $l_r$ is the length scale associated with the harmonic oscillator potential.
The physical scattering length there is derived from 3d experiments. If you used the atom trap from that 2nd reference I mention above, the effective 1d scattering length is probably $(2\pi l_r)^2 /a_{3d}$.

wow, that's absolutely a great explanation. I appreciate the detail information and references.