Energy of many-particle quantum states*

Here, we derive a functional form for the energy of quantum many-particle systems from first principles. The trick is to understand the Hamiltonian as a generalized harmonic oscillator and use uncertainty relations to constrain allowed zero-point fluctuations. In particular, we write the expectation value of the many-particle Hamiltonian as the sum of classical and quantum contributions. Then, for a given charge density distribution, fluctuations of the Coulomb field and momentum can be related via the respective commutation of operators. When combined with the Lieb-Thirring bound on kinetic energy, we obtain the energy of interacting many-particle quantum states for non-uniform densities. The approach is applicable to bosonic as well as fermionic systems, and does not require exchange-correlation separation. In the case of uniform density, the functional form agrees with benchmark Quantum Monte Carlo data, including phase transitions. In addition to accuracy, our goal is to develop a deeper appreciation of the first principles at play in many-body quantum systems. (based on arXiv:2010.01656)