Variational principle and mean-field approximation

Let us consider a N-body quantum system with Hamiltonian $\hat{H}$. The exact time-dependent Schrödinger equation can be obtained by imposing the quantum last action principle on the Dirac action

$$\int \psi \hbar {\partial \over \partial t} \hat{H} \psi$$ (1)

where $\psi$ is the many-body wavefunction of the system. Looking for stationary points of S with respect to variation of the conjugate wavefunction $\psi^{*}_{}$ gives

$$\hbar {\partial \over \partial t} \psi \hat{H} \psi$$ (2)

As is well known, it is usually impossible to obtain the exact solution of the many-body Schrödinger equation and some approximation must be used.

In the mean-field approximation the total wavefunction is assumed to be composed of independent particles, i.e. it can be written as a product of single-particle wavefunctions $\phi_{j}^{}$. In the case of identical fermions, $\psi$ must be antisymmetrized [6]. By looking for stationary action with respect to variation of a particular single-particle conjugate wavefunction $\phi_{j}^{*}$ one finds a time-dependent Hartree-Fock equation for each $\phi_{j}^{}$:

$$\hbar {\partial \over \partial t} \phi_{j}^{} {\delta \over \delta \phi_j^*} \psi \hat{H} \psi \hat{h} \phi_{j}^{}$$ (3)

where $\hat{h}$ is a one-body operator. The main point is that, in general, the one-body operator $\hat{h}$ is nonlinear. Thus the Hartree-Fock equations are non-linear (integro-)differential equations. These equations can give rise, in some cases, to chaotic behaviour (dynamical chaos) of the mean-field wavefunction.