Abstract
We present a simple proof of the Hartree equation with quantitative estimates, using the method of coherent states. Along the way, we introduce second quantization for bosonic many-body systems.
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Notes
- 1.
To evaluate \(W(f)\varOmega \) we made use of the Baker-Campbell-Hausdorff formula for operators A, B with the property that \([[A,B],A] = [[A,B],B]=0\):
$$\begin{aligned} e^{A+B} = e^{-\frac{1}{2}[A,B]} e^A e^B. \end{aligned}$$Since \(a(f)\varOmega =0\) one then simply has to expand the exponential of \(a^*(f)\).
- 2.
Systematically, time derivatives of the form \((\partial _t e^{-A(t)})e^{A(t)}\) (with A(t) a sufficiently regular family of operators) can be calculated as follows. Start by writing
$$\begin{aligned}&(\partial _t e^{-A(t)})e^{A(t)} = \lim _{h \rightarrow 0} \frac{1}{h} \int _0^1d\lambda \frac{d}{d \lambda }\left( e^{-A(t+h)\lambda }e^{A(t)\lambda }\right) = - \int _0^1d\lambda \,e^{-A(t)\lambda }\dot{A}(t) e^{A(t)\lambda }. \end{aligned} $$Now one uses the Baker-Campbell-Hausdorff formula \(e^{-A}Be^A = B - \int _0^1 d \rho \, e^{-\rho A}[A,B]e^{\rho A}\) for operators A, B and iterates. In the application here, the iteration breaks off immediately because the commutator is just a complex number.
- 3.
We speak of constants since these are not operators, but rather just complex numbers. Of course they depend on time and on N.
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Benedikter, N., Porta, M., Schlein, B. (2016). Coherent States Approach. 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_3
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DOI: https://doi.org/10.1007/978-3-319-24898-1_3
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