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).
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Abbreviations
- \({\frac{\partial} {\partial t}}\) :
-
Partial time derivative
- \({\frac{{\rm d}} {{\rm d} t}}\) :
-
Total time derivative
- \({\Delta_j, \nabla_j}\) :
-
Laplacian and gradient in the coordinate x j
- \({\Psi_N^t}\) :
-
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 \({\alpha_N(\Psi_N,\varphi)=\langle\langle\Psi_N,\widehat{n}^{\varphi}\Psi_N\rangle\rangle}\)
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Pickl, P. A Simple Derivation of Mean Field Limits for Quantum Systems. Lett Math Phys 97, 151–164 (2011). https://doi.org/10.1007/s11005-011-0470-4
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DOI: https://doi.org/10.1007/s11005-011-0470-4