Letters in Mathematical Physics

, Volume 97, Issue 2, pp 151–164 | Cite as

A Simple Derivation of Mean Field Limits for Quantum Systems

Article

Abstract

We shall present a new strategy for handling mean field limits of quantum mechanical systems. The new method is simple and effective. It is simple, because it translates the idea behind the mean-field description of a many particle quantum system directly into a mathematical algorithm. It is effective because, with less effort, the strategy yields better results than previously achieved. As an instructional example we treat a simple model for the time-dependent Hartree equation which we derive under more general conditions than what has been considered so far. Other mean-field scalings leading, e.g. to the Gross-Pitaevskii equation can also be treated (Pickl in Derivation of the time dependent Gross Pitaevskii equation with external fields, preprint; Pickl in Derivation of the time dependent Gross Pitaevskii equation without positivity condition on the interaction, preprint).

Mathematics Subject Classification (2000)

35Q40 35Q55 81V70 

Notations

\({\frac{\partial} {\partial t}}\)

Partial time derivative

\({\frac{{\rm d}} {{\rm d} t}}\)

Total time derivative

\({\Delta_j, \nabla_j}\)

Laplacian and gradient in the coordinate xj

\({\Psi_N^t}\)

Solution of the Schrödinger equation (1)

\({\varphi^t}\)

Solution of the Hartree equation (9)

\({\langle\cdot,\cdot\rangle}\)

Scalar product on \({L^2(\mathbb{R}^3)}\)

\({\langle\langle\cdot,\cdot\rangle\rangle}\)

Scalar product on \({L^2(\mathbb{R}^{3N})}\)

\({|\varphi\rangle\langle\chi|}\)

Dirac notation for the operator on \({L^2(\mathbb{R}^3)}\) given by \({|\varphi\rangle\langle\chi|\xi=\langle\chi,\xi\rangle\space\varphi}\)

\({|\varphi(x_j)\rangle\langle\chi(x_j)|}\)

Dirac notation for the operator on \({L^2(\mathbb{R}^{3N})}\) given by \({|\varphi(x_j)\rangle\langle\chi(x_j)|\Psi=\varphi(x_j)\int \chi^*(x_j) \Psi(x_1,\ldots,x_N) {\rm d}^3 x_j}\)

\(\mathcal{O}_N(1)\)

Landau’s symbol. Used for functions which tend to zero as \({N \to \infty}\) .

\({p_j^\varphi}\)

Operator on \({L^2(\mathbb{R}^{3N})}\) given by \({p_j^\varphi=|\varphi(x_j)\rangle\langle\varphi(x_j)|}\)

\({q_j^\varphi}\)

Operator on \({L^2(\mathbb{R}^{3N})}\) given by \({q_j^\varphi=1-p_j^\varphi}\)

\({\mathcal{A}_k}\)

Set given by \({\mathcal{A}_k=\{(a_1,a_2,\ldots,a_N): a_j\in\{0,1\}\;;\;\sum_{j=1}^N a_j=k\}}\)

\({P_{N,k}^\varphi}\)

Operator on \({L^2(\mathbb{R}^{3N})}\) given by \({P_{N,k}^\varphi=\sum_{a\in\mathcal{A}_k}\prod_{j=1}^N(p_{j}^{\varphi})^{1-a_j}(q_{j}^{\varphi})^{a_j}}\)

\({\widehat{n}^{\varphi}}\)

Operator on \({L^2(\mathbb{R}^{3N})}\) given by \({\widehat{n}^{\varphi}=\sum_{k=0}^Nn(k)P_{N,k}^{\varphi}}\) . (For the example in Section 3 we choose n(k) = k/N)

\({\alpha_N(\Psi_N,\varphi)}\)

Functional \({L^2(\mathbb{R}^{3N}) \times L^2(\mathbb{R}^{3}) \to \mathbb{R}^+}\) given by \({\alpha_N(\Psi_N,\varphi)=\langle\langle\Psi_N,\widehat{n}^{\varphi}\Psi_N\rangle\rangle}\)

Keywords

mean field limits Hartree equation many body quantum mechanics. 

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Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of MunichMunichGermany

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