金兹堡－朗道方程

理论

${\displaystyle f=f_{n0}+\alpha |\psi |^{2}+{\frac {\beta }{2}}|\psi |^{4}+{\frac {1}{2m^{*}}}\left|\left({\frac {\hbar }{i}}\nabla -{\frac {e^{*}}{c}}\mathbf {A} \right)\psi \right|^{2}+{\frac {H^{2}}{8\pi }}}$[6]

${\displaystyle \psi =0}$，则上式化为常态下的自由能 ${\displaystyle f_{n0}+{\frac {h^{2}}{8\pi }}}$${\displaystyle m^{*}}$表示有效质量${\displaystyle e^{*}}$表示有效电荷A磁矢势${\displaystyle H}$磁场强度。在后续的实验中，人们发现 ${\displaystyle e^{*}\approx 2{\mathit {e}}}$${\displaystyle {\mathit {e}}}$基本电荷）。

 ${\displaystyle \alpha \psi +\beta |\psi |^{2}\psi +{\frac {1}{2m^{*}}}\left({\frac {\hbar }{i}}\nabla -{\frac {e^{*}}{c}}\mathbf {A} \right)^{2}\psi =0}$

${\displaystyle \mathbf {J} ={\frac {c}{4\pi }}{\boldsymbol {\nabla }}\times \mathbf {H} }$，可推导出电流密度

${\displaystyle \mathbf {J} ={\frac {e^{*}}{m^{*}}}\left|\psi \right|^{2}\left(\hbar \nabla \varphi -{\frac {e^{*}}{c}}\mathbf {A} \right)}$[6]

分析

${\displaystyle f_{s}-f_{n}=\alpha |\psi |^{2}+{\frac {1}{2}}\beta |\psi |^{4}}$[6]

${\displaystyle |\psi |^{2}\propto 1-t^{4}\approx 4(1-t)}$
${\displaystyle \alpha \propto {\frac {1-t^{2}}{1+t^{2}}}\approx (1-t)}$
${\displaystyle \beta \propto {\frac {1}{(1+t^{2})^{2}}}\approx const}$

相干长度与穿透深度

${\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{2m|\alpha |}}}.}$

${\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{4m|\alpha |}}}.}$

${\displaystyle \lambda ={\sqrt {\frac {m}{4\mu _{0}e^{2}\psi _{0}^{2}}}},}$

解析解

${\displaystyle {\frac {\partial u}{\partial t}}-a{\frac {\partial ^{2}u}{\partial x^{2}}}-bu+c|u|^{2}u=0}$[7]

${\displaystyle sol[1]:=u=-(1/2)*b/{\sqrt {(}}c*b)+(1/2)*{\sqrt {(}}c*b)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*{\sqrt {(}}a*b)*x/a-(3/4)*b*t)/c}$
${\displaystyle sol[2]:=u=-(1/2)*b/{\sqrt {(}}c*b)+(1/2)*{\sqrt {(}}c*b)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*{\sqrt {(}}a*b)*x/a-(3/4)*b*t)/c}$
${\displaystyle sol[3]:=u=-(1/2*I)*b/{\sqrt {(}}-c*b)-(1/2)*{\sqrt {(}}-c*b)*tan(_{C}1+(1/4)*{\sqrt {(}}-2*a*b)*x/a-(3/4*I)*b*t)/c}$
${\displaystyle sol[4]:=u=-(1/2)*b/{\sqrt {(}}c*b)+(1/2)*{\sqrt {(}}c*b)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*{\sqrt {(}}a*b)*x/a-(3/4)*b*t)/c}$
${\displaystyle sol[5]:={\frac {-{\sqrt {(}}3)*exp(-1-(1/4)*{\sqrt {(}}3)*x+(9/4)*t)}{(exp(1+(1/4)*{\sqrt {(}}3)*x-(9/4)*t)+exp(-1-(1/4)*{\sqrt {(}}3)*x+(9/4)*t))}}}$

参考文献

1. ^ V. L. Ginzburg; L. D. Landau. On the Theory of Superconductivity. Zh. Eksp. Teor. Fiz. 1950, 20: 1064. doi:10.1016/B978-0-08-010586-4.50078-X.
2. ^ A.A. Abrikosov. On the Magnetic Properties of Superconductors of the Second Group. Zh.Eksp.Teor.Fiz. 1956-11, 32: 1442–1452.
3. ^ L.P. Gor'kov. Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity (PDF). Zh. Eksp. Teor. Fiz. 1959, 36: 1918–1923 [2018-01-18].
4. ^ A.A. Abrikosov; Beknazarov. Fundamentals of the theory of metals. Amsterdam: North-Holland. 1988: 589 [2018-01-19]. ISBN 0444870946.
5. ^ Neil W. Ashcroft; N. David Mermin. Solid state physics 27. repr. New York: Holt, Rinehart and Winston. 1977: 747. ISBN 0030839939.
6. Tinkham, Michael. Introduction to superconductivity 2nd ed. Mineola, NY: Dover Publications. 2004: 111. ISBN 0486435032.
7. ^ Inna Shingareva; Carlos Lizárraga-Celaya. Solving nonlinear partial differential equations with Maple and Mathematica. New York: Springer. 2011: 28 [2018-01-19]. ISBN 978-3-7091-0516-0.

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