How useful are quantum-classical hybrid theories?

[andresB]

I have seen several attempts at building a consistent quantum-classical theory, i.e., a theory where a quantum (sub) system interacts with a classical one somehow, and both affect each other (do note that this definition rules out things like just plugging an external classical EM field into the Scrhödinger equation).

Examples: (1) https://iopscience.iop.org/article/10.1088/1742-6596/442/1/012006.
(2) https://arxiv.org/abs/1802.04787

However, they tend to focus more on theoretical aspects (consistency, existence of generalized brackets and/or joint Hamiltonian/unitary evolution) and less on application to real systems. I guess, hybrid theories are supposed to give a good approximation in mesoscopic systems, where some things are big enough to be considered classical but not that big that all quantum effects can be ignored.

So, is there an actual physical systems (not toy models) where quantum-classical theories give good results for practical calculations?

 

[Haborix]

I'm afraid I can't answer your question directly. But I thought this paper might be of interest (and perhaps its future research lineage). Generally speaking, I would expect to find more real-world applications coming out of physical chemistry and related fields.

 

[hilbert2]

So, is there an actual physical systems (not toy models) where quantum-classical theories give good results for practical calculations?

For instance, if you have a system where a fast-moving proton or alpha particle passes a hydrogen atom at close distance, you can approximate the heavy charged particle as moving on a linear classical trajectory and causing a time-dependent additional electric potential in the Hamiltonian.

 

[A. Neumaier]

Quantum-classical systems are very useful in practice. Indeed, most practical quantum reasoning is a quantum-classical hybrid.

For example in Stern-Gerlach experiments or Bell inequality experiments one treats the paths of the particles as classical and only the internal degrees of freedom by quantum mechanics.

In most of quantum chemistry (except for very small molecules) one treats the nuclei as classical and the electrons as quantum.

There is an extended literature on quantum-classical systems -- see, e.g., the discussion and references in
Sections 7.8-7.8 of my book Coherent Quantum Physics (which are based on Sections 4.6 and 4.7 of Part III of my papers on the thermal interpretation).

 

[hilbert2]

Would it be possible to calculate the problem of hydrogen atom excitation by a fast proton wave packet fly-by completely quantum mechanically with modern computers? The two protons and an electron would mean 9 position coordinates in $\Psi (x_i ,t)$, but some can be ignored in a CMS system. I guess solving the TDSE in high resolution with implicit finite difference would still be difficult because of the large array of points. Of course you would at least have to ignore any QED corrections.

 

[andresB]

For instance, if you have a system where a fast-moving proton or alpha particle passes a hydrogen atom at close distance, you can approximate the heavy charged particle as moving on a linear classical trajectory and causing a time-dependent additional electric potential in the Hamiltonian.

but by considering the external proton's trajectory to be fixed you are forfeiting any quantum backreaction on the classical subsystem (in this case, the external proton). In the end, the combined system reduces entirely to the quantum mechanical problem of the hydrogen atom plus a time-varying electric field.

I have seen several attempts at building a consistent quantum-classical theory, i.e., a theory where a quantum (sub) system interacts with a classical one somehow, and both affect each other (do note that this definition rules out things like just plugging an external classical EM field into the Scrhödinger equation).