Gross-Pitaevskii equation

The Gross-Pitaevskii equation (after Eugene P. Gross and Lew Petrowitsch Pitajewski ) describes the temporal development of the condensate of a quantum mechanical many-body system in an external potential : ${\displaystyle \psi ({\vec {r}},t)}$ ${\displaystyle V({\vec {r}},t)}$

${\displaystyle \mathrm {i} \hbar {\frac {\partial \psi }{\partial t}}=\left[-{\frac {\hbar ^{2}}{2m}}{\vec {\nabla }}^{2}+V({\vec {r}},t)+g|\psi |^{2}\right]\psi }$

The function is the order parameter of the phase transition. The parameter describes whether the interaction is attractive ( ) or repulsive ( ). ${\displaystyle \psi ({\vec {r}},t)}$ ${\displaystyle g\,}$ ${\displaystyle g<0\,}$ ${\displaystyle g>0\,}$

The Gross-Pitaevskii equation plays an important role in the theoretical treatment of bosonic quantum fluids such as Bose-Einstein condensates (BEC), superconductors, and superfluids . It includes solitary solutions (non-linear waves) and vortices (quantized eddies). It corresponds to a molecular field approximation with the interaction with the mean field of the other bosons in the nonlinear term.

Taking into account also electrically charged particles ( charge , vector potential ), so must the momentum operator replace: . In this case, the Gross-Pitaevskii equation becomes the Ginzburg-Landau equation , which is used for the phenomenological description of superconductors. ${\displaystyle q\,}$ ${\displaystyle {\vec {A}}}$ ${\displaystyle -\mathrm {i} \hbar {\vec {\nabla }}\rightarrow -\mathrm {i} \hbar {\vec {\nabla }}+q{\vec {A}}}$

interpretation

The degree of freedom of the Gross-Pitaevskii equation, a classic complex-valued field, can be interpreted as the mean value of a field operator . The approximation of the field operator by the mean value is permissible if many particles are in the same quantum mechanical single-particle state, which is only possible with bosons. In the context of quantum mechanics, the Gross-Pitaevskii equation corresponds in this sense to the Maxwell equations. ${\displaystyle \psi ({\vec {r}},t)}$ ${\displaystyle {\hat {\psi }}({\vec {r}},t)}$

In this case , the non-linearity does not apply and there is formal agreement with the 1-particle Schrödinger equation . However, the degrees of freedom of the Schrödinger equation are the particle coordinates. A derivation of the Gross-Pitaevskii equation from the Schrödinger equation is possible with the help of the formalism of the second quantization . ${\displaystyle g=0}$

Energy and dispersion

The energy density of a system described by the Gross-Pitaevskii equation is given by:

${\displaystyle \epsilon \left[\psi \right]={\frac {\hbar ^{2}}{2m}}|{\vec {\nabla }}\psi |^{2}+V|\psi |^{2}+{\frac {1}{2}}g|\psi |^{4}}$

The dispersion relation is:

${\displaystyle E=\hbar \omega ={\sqrt {{\frac {\hbar ^{2}k^{2}}{2m}}\left({\frac {\hbar ^{2}k^{2}}{2m}}+2g|\psi |^{2}\right)}}}$

literature

• Anthony James Leggett : Bose-Einstein Condensation in the Alkali Gases: Some Fundamental Concepts , Reviews of Modern Physics, Vol. 73, 2001, pp. 307-356
• Original works:
  • EP Gross, Structure of a quantized vortex in boson systems , Il Nuovo Cimento, Vol. 20, 1961, pp. 454-457, Hydrodynamics of a superfluid condensate , J. Math. Phys., Vol. 4, 1963, p. 195 -207
  • LP Pitaevskii: Vortex Lines in an Imperfect Bose Gas , Soviet Physics JETP, Vol. 13, 1961, pp. 451-454.