- #1
confused_man
- 16
- 1
Hello everyone,
I'm trying to understand quantized vortices in rotating Bose gases. From what I understand, any system that is described by a quantum mechanical wavefunction [tex] \psi [\tex] will be irrotational (in other words, will only be able to get angular momentum through the nucleation of quantized vorticies).
-----feel free to skip this part-------------------------
This can be shown by using the continuity equation
[tex]
\frac{\partial n}{\partial t} + \nabla \cdot (n \mathbf{v}) = 0
[\tex]
where [tex] n = |\psi|^2 [\tex] and
[tex]
\mathbf{v} = \frac{-i\hbar}{2m}\frac{\psi^*\nabla\psi-\psi\nabla\psi^*}{|\psi|^2}
[\tex]
for the probability density [tex] n [\tex]. If generic wavefunction written in polar form [tex]\psi=\psi_0 e^{i\phi}[\tex] is inserted into the velocity equation, then it is found that the velocity is the gradient of a scalar function,
[tex]
\mathbf{v} = \frac{\hbar}{m}\nabla\phi
[\tex]
which means that the wavefunction is irrotational since the curl of the gradient of a scalar is zero ([tex] \nabla \times \mathbf{v} = 0 [\tex]. However, the wavefunction can have angular momentum if it possesses a singularity in the form of a vortex.
------ end of boring derivation --------------------
Now the problem that I am having is that I've only seen vortices discussed in the context of 4He or in Bose-Eintein condensates. What I was wondering is
- Is it possible to see these quantized vortices in Bose gases above the condensate transition temperature?
- What about fermi gases? There are experiments in the literature in which a fermi gas is Bose-condensed and then rotated, but I was wondering if the condensation part is necessary.
I would seem to me that as long as the system is quantum in nature that it should be irrotational and be able to produce quantized vortices when rotated. Am I missing something?
Any help or comments would be greatly appreciated. Thanks!
I'm trying to understand quantized vortices in rotating Bose gases. From what I understand, any system that is described by a quantum mechanical wavefunction [tex] \psi [\tex] will be irrotational (in other words, will only be able to get angular momentum through the nucleation of quantized vorticies).
-----feel free to skip this part-------------------------
This can be shown by using the continuity equation
[tex]
\frac{\partial n}{\partial t} + \nabla \cdot (n \mathbf{v}) = 0
[\tex]
where [tex] n = |\psi|^2 [\tex] and
[tex]
\mathbf{v} = \frac{-i\hbar}{2m}\frac{\psi^*\nabla\psi-\psi\nabla\psi^*}{|\psi|^2}
[\tex]
for the probability density [tex] n [\tex]. If generic wavefunction written in polar form [tex]\psi=\psi_0 e^{i\phi}[\tex] is inserted into the velocity equation, then it is found that the velocity is the gradient of a scalar function,
[tex]
\mathbf{v} = \frac{\hbar}{m}\nabla\phi
[\tex]
which means that the wavefunction is irrotational since the curl of the gradient of a scalar is zero ([tex] \nabla \times \mathbf{v} = 0 [\tex]. However, the wavefunction can have angular momentum if it possesses a singularity in the form of a vortex.
------ end of boring derivation --------------------
Now the problem that I am having is that I've only seen vortices discussed in the context of 4He or in Bose-Eintein condensates. What I was wondering is
- Is it possible to see these quantized vortices in Bose gases above the condensate transition temperature?
- What about fermi gases? There are experiments in the literature in which a fermi gas is Bose-condensed and then rotated, but I was wondering if the condensation part is necessary.
I would seem to me that as long as the system is quantum in nature that it should be irrotational and be able to produce quantized vortices when rotated. Am I missing something?
Any help or comments would be greatly appreciated. Thanks!
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