# 格罗斯–皮塔耶夫斯基方程

${\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2},\dots ,\mathbf {r} _{N})=\psi (\mathbf {r} _{1})\psi (\mathbf {r} _{2})\dots \psi (\mathbf {r} _{N})}$

${\displaystyle H=\sum _{i=1}^{N}\left(-{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial \mathbf {r} _{i}^{2}}+V(\mathbf {r} _{i})\right)+\sum _{i

${\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}{\partial ^{2} \over \partial \mathbf {r} ^{2}}+V(\mathbf {r} )+{4\pi \hbar ^{2}a_{s} \over m}\vert \psi (\mathbf {r} )\vert ^{2}\right)\psi (\mathbf {r} )=\mu \psi (\mathbf {r} ),}$

## 方程形式

${\displaystyle g={\frac {4\pi \hbar ^{2}a_{s}}{m}}}$,

${\displaystyle {\mathcal {E}}={\frac {\hbar ^{2}}{2m}}\vert \nabla \Psi (\mathbf {r} )\vert ^{2}+V(\mathbf {r} )\vert \Psi (\mathbf {r} )\vert ^{2}+{\frac {1}{2}}g\vert \Psi (\mathbf {r} )\vert ^{4},}$

${\displaystyle \mu \Psi (\mathbf {r} )=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )+g\vert \Psi (\mathbf {r} )\vert ^{2}\right)\Psi (\mathbf {r} )}$

${\displaystyle N=\int \vert \Psi (\mathbf {r} )\vert ^{2}\,d^{3}r.}$

${\displaystyle i\hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )+g\vert \Psi (\mathbf {r} ,t)\vert ^{2}\right)\Psi (\mathbf {r} ,t).}$

## 方程解

### 精确解

#### 自由粒子

${\displaystyle \Psi (\mathbf {r} )={\sqrt {\frac {N}{V}}}e^{i\mathbf {k} \cdot \mathbf {r} }.}$

${\displaystyle E(\mathbf {k} )=N\left[{\frac {\hbar ^{2}k^{2}}{2m}}+g{\frac {N}{2V}}\right].}$

#### 孤子

${\displaystyle \psi (x)=\psi _{0}\tanh \left({\frac {x}{{\sqrt {2}}\xi }}\right)}$,

${\displaystyle \psi (x,t)=\psi (0)e^{-i\mu t/\hbar }{\frac {1}{\cosh \left[{\sqrt {2m\vert \mu \vert /\hbar ^{2}}}x\right]}},}$

### 托马斯-费米近似

${\displaystyle \psi (x,t)={\sqrt {\frac {\mu -V(x)}{NU_{0}}}}}$

### 玻戈留玻夫近似

${\displaystyle \psi =\psi _{0}+\delta \psi }$

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\delta \psi +V\delta \psi +g(2|\psi _{0}|^{2}\delta \psi +\psi ^{2}\delta \psi ^{*})=i\hbar {\frac {\partial \delta \psi }{\partial t}}}$
${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\delta \psi ^{*}+V\delta \psi ^{*}+g(2|\psi _{0}|^{2}\delta \psi ^{*}+\psi ^{2}\delta \psi )=i\hbar {\frac {\partial \delta \psi ^{*}}{\partial t}}}$

${\displaystyle \delta \psi =e^{-i{\frac {\mu }{\hbar }}t}(u({\boldsymbol {r}})e^{-i\omega t}-v^{*}({\boldsymbol {r}})e^{i\omega t})}$

${\displaystyle (-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V+2gn-\mu -\hbar \omega )u-gnv=0}$
${\displaystyle (-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V+2gn-\mu +\hbar \omega )v-gnu=0}$

${\displaystyle \hbar \omega =\epsilon _{\boldsymbol {q}}={\sqrt {{\frac {\hbar ^{2}{\boldsymbol {q}}^{2}}{2m}}({\frac {\hbar ^{2}{\boldsymbol {q}}^{2}}{2m}}+2gn)}}}$

${\displaystyle {\boldsymbol {q}}}$很大时，色散关系呈现为${\displaystyle {\boldsymbol {q}}}$的平方，正如所料类似于非相互作用的激子。当${\displaystyle {\boldsymbol {q}}}$很小，色散关系为线性，

${\displaystyle \epsilon _{\boldsymbol {q}}=s\hbar q}$

## 参考文献

1. ^ Gross, E.P. Structure of a quantized vortex in boson systems. Il Nuovo Cimento. May 1961, 20 (3): 454–457. doi:10.1007/BF02731494.
2. ^ Vortex Lines in an Imperfect Bose Gas. Soviet Physics JETP. 1961, 13 (2): 451–454 [2011-03-31]. （原始内容存档于2012-03-20）.
3. ^ Hugenholtz, N. M.; Pines, D. Ground-state energy and excitation spectrum of a system of interacting bosons. Physical Review. 1959, 116 (3): 489–506. Bibcode:1959PhRv..116..489H. doi:10.1103/PhysRev.116.489.
4. ^ Evidence for a Critical Velocity in a Bose–Einstein Condensed Gas C. Raman, M. Köhl, R. Onofrio, D. S. Durfee, C. E. Kuklewicz, Z. Hadzibabic, and W. Ketterle

Theory of Bose_Einstein condensation in trapped gases Franco Dalfovo and Stafano Giorgini Reviews Modern Physics

## 更多阅读

• Pethick, C. J. & Smith, H. Bose–Einstein Condensation in Dilute Gases. Cambridge: Cambridge University Press. 2002. ISBN 0521665809..
• Pitaevskii, L. P. & Stringari, S. Bose–Einstein Condensation. Oxford: Clarendon Press. 2003. ISBN 0198507194..