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Mean-Field Regime for Fermionic Systems

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Effective Evolution Equations from Quantum Dynamics

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 7))

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Abstract

Fermionic systems behave very differently from bosonic systems due to their high kinetic energy. In particular, the mean-field limit is naturally coupled to a semi-classical limit. We explain this fact in detail and then introduce Hartree-Fock theory, with some comments on the relation to Thomas-Fermi theory and the Vlasov equation. We introduce particle-hole transformations and derive the Hartree-Fock equation by a method analogous to the bosonic coherent state method.

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Notes

  1. 1.

    However \(\widetilde{\omega }_N\) is only approximately an projection.

  2. 2.

    See [4] for the proof that for bounded potentials the exchange term \(X_t\) can be neglected with a very small error, see (6.46).

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Niels Benedikter

  2. Institute of Mathematics, University of Zurich, Zürich, Switzerland

    Marcello Porta & Benjamin Schlein

Authors
  1. Niels Benedikter
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  2. Marcello Porta
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  3. Benjamin Schlein
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Correspondence to Niels Benedikter .

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Benedikter, N., Porta, M., Schlein, B. (2016). Mean-Field Regime for Fermionic Systems. In: Effective Evolution Equations from Quantum Dynamics. SpringerBriefs in Mathematical Physics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-24898-1_6

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