# What exactly is a 'quantum mechanical physical system'?

I was reading the Wikipedia article about the Zero-point energy in which it states

Zero-point energy...is the lowest possible energy that a quantum mechanical physical system may have...

And I just wondered what it was referring to as a 'quantum mechanical physical system'? I understand if it was, for example, the particle in a box which is confined, but what abut something that has no boundary. I'm guessing that when talking about the zero-point energy, it refers to everything: the vacuum (no matter how small a piece of the vacuum you take), a particle, a particle and a box... It seems to be the smallest possible unit of energy.

I think I've got that right, however I'm not sure what to regard as a 'quantum mechanical physical system' in general? Is a single particle a system? Or is it the particle and it's surroundings?

• All physical systems obey quantum mechanics. Wikipedia is trying to say something more along the lines of "the lowest possible energy that a physical system can have, when we model it using quantum mechanics". Sometimes we don't because the quantum mechanics is too complicated or unnecessary. – knzhou Sep 5 '16 at 20:24
• I mostly agree with @knzhou except that you could argue that we don't quite understand how to explain classical behavior starting from quantum mechanics because we have to insert a really weird rule about the outcomes of measurements. Of course, physics theories are perfectly allowed to be weird. Weird does not mean invalid. – DanielSank Sep 6 '16 at 1:25

A physical system consists of a notion of states, observables and a dynamical law. In case of quantum mechanics, (pure) states are elements $\psi$ of a Hilbert space $\mathcal{H}$ normalized to 1 (which you can also consider as rays in a Hilbert space or rank-1 orthogonal projections). Observables are hermitian operators on this Hilbert space, and in the Schrödinger picture the dynamical law is the Schrödinger equation $\mathrm{i} \partial_t \psi(t) = H \psi(t)$ where $H$ is the energy observable, the Hamiltonian. There are equivalent formulations (e. g. the Heisenberg picture) and generalizations (e. g. the notion of mixed states).

You have to choose $H$ and $\mathcal{H}$ according to the problem at hand, e. g. whether your model is a continuum or a discrete model and whether your particle has (pseudo)spin. It can be a single particle system or a many-particle system. For energies well below the pair creation threshold you typically don't have to take into account that particles can be created or annihilated. Each model comes with its own range of validity, so the predictive value is only within that range of validity.

The question of zero point energy has to be asked with a specific model in mind, because the Dirac equation admits no such zero point (more specifically, the “zero point” is $-\infty$). There are models like the hydrogen atom which admit a lowest energy state (such as the hydrogen atom) and others that do not (e. g. a free, non-relativistic spinless particle moving in $\mathbb{R}^d$ has no ground state even though it has a lowest energy). Note that the existence of a ground state and finiteness of energy minimum are two separate questions, the former being far harder than the latter.

It's certainly a rough term, which really doesn't mean much more than a system where you get a very wrong answer if you treat it with classical physics. Thus, if it's close to its "zero point energy", it's redundant to call it a quantum mechanical system, as that goes without saying! But one must certainly avoid equating such systems with systems with only one particle, or even a small number of particles. There are plenty of situations in which a single particle acts classically, those are basically situations where a quantity called the "action", which is a bit like classical angular momentum, is much greater than Planck's constant. Also, we have white dwarfs and neutron stars, which contain a ghastly number of particles, but are still "quantum mechanical systems", in the sense that we cannot treat them classically.

I think I've got that right, however I'm not sure what to regard as a 'quantum mechanical physical system' in general? Is a single particle a system? Or is it the particle and it's surroundings?

Here is where the Heisenberg Uncertainty principle (HUP) is important. When modeling the physics of any system the model should be quantum mechanical if the HUP is important, i.e. for the variables under study h_bar cannot be assumed to be zero.

It is the HUP that constrains variables to follow quantum mechanical equations, and this can happen in even macroscopic dimensions, as with superconductivity and superfluidity. If h_bar can be assumed zero, the system is in classical dimensions.