양자 소용돌이: 두 판 사이의 차이

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소용돌이의 두께는 초유체의 회학 성분에 따라 다르며 액체 헬륨내에서 두께는 수 옹스트롬이다.
 
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A [[superfluid]] has the special property of having phase, given by the [[wavefunction]], and the velocity of the superfluid is proportional to the [[gradient]] of the phase. The [[circulation]] around any closed loop in the superfluid is zero, if the region enclosed is [[simply connected]]. The superfluid is deemed [[irrotational]]. However, if the enclosed region actually contains a smaller region that is an absence of superfluid, for example a rod through the superfluid or a vortex, then the circulation is,
 
:<math>\oint_{C} \mathbf{v}\cdot\,d\mathbf{l} = \frac{\hbar}{m}\oint_{C}\nabla\phi\cdot\,d\mathbf{l} = \frac{\hbar}{m}\Delta\phi,</math>
 
where <math>\hbar</math> is [[Planck's constant]] divided by <math>2\pi</math>, m is the mass of the superfluid particle, and <math>\Delta\phi</math> is the phase difference around the vortex. Because the wavefunction must return to its same value after an integral number of turns around the vortex (similar to what is described in the [[Bohr model]]), then <math>\Delta\phi = 2\pi n</math>, where n is an [[integer]]. Thus, we find that the circulation is quantized:
 
:<math>\oint_{C} \mathbf{v}\cdot\,d\mathbf{l} = \frac{2\pi\hbar}{m}n.</math>
 
==Vortex in a superconductor==
 
A principal property of [[superconductors]] is that they expel [[magnetic fields]]; this is called the [[Meissner effect]]. If the magnetic field becomes sufficiently strong, one scenerio is for the superconductive state to be "killed". However, in some cases, it may be energertically favorable for the superconductor to form a quantum vortex, which carries a quantized amount of magnetic flux through the superconductor. Meanwhile, the superconductive state prevails in the regions around the vortex. A superconductor that is capable of carrying a vortex is called a type-II superconductor.
 
Over some enclosed area S, the [[magnetic flux]] is
 
:<math>\Phi = \oint_S\mathbf{B}\cdot\mathbf{\hat{n}}\,d^2x = \oint_{\partial S}\mathbf{A}\cdot d\mathbf{l} </math>.
 
Substituting a result of London's second equation: <math>\mathbf{j}_s = -\frac{n_se_s^2}{m}\mathbf{A} - \frac{n_se_s\hbar}{m}\mathbf{\nabla}\phi</math>, we find
 
:<math>\Phi =-\frac{m}{n_s e^2}\oint_{\partial S}\mathbf{j}_s\cdot d\mathbf{l} +\frac{\hbar}{e_s}\oint_{\partial S}\mathbf{\nabla}\phi\cdot d\mathbf{l}</math>,
 
where n_s, m, and e_s are the number density, mass and charge of the [[Cooper pairs]].
 
If the region, S, is large enough so that <math>\mathbf{j}_s = 0</math> along <math>\partial S</math>, then
 
:<math>\Phi = \frac{\hbar}{e_s}\oint_{\partial S}\mathbf{\nabla}\phi\cdot d\mathbf{l} = \frac{\hbar}{e_s}\Delta\phi = \frac{2\pi\hbar}{e_s}n </math>.
 
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[[Category:소용돌이]]
[[Category:양자역학]]

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