I Physical meaning of two independent, non-interacting parts

[vibhuav]

I keep coming across this descriptor, "two (or three) independent, non-interacting parts," in many books on QM (for example, Penrose's Shadows of the Mind). It is usually followed by a mathematical description (for example, state vector |A>|B>). I can wrap my mind around the quantum paradox of superposition, w|A> + z|B>, from a physical POV, but not this AND concept. Can someone give a non-mathematical, physical example of such a pair (or triads)? What does it mean physically?

There is also a sentence which says we can "simply adjoin" (say) a photon with another one on (say) the moon to get the "two-independent, non-interacting pair." But how would one "simply adjoin" them physically and practically? (Even a thought experiment would be good enough for me.)

And finally, are entangled particles considered as non-interacting or interacting?

 

[hilbert2]

For instance, suppose you have two hydrogen atoms in vacuum at a large enough distance from each other to not interact, and the state vectors of those at time $t_0$ are $\left|\psi_1 (t_0 )\right.\rangle$ and $\left|\psi_2 (t_0 ) \right.\rangle$. You can consider those as two different systems, but it's also possible to see them as parts of one system with the combined state being $\left|\psi (t_0) \right.\rangle = \left|\psi_1 (t_0 )\right.\rangle\left|\psi_2 (t_0) \right.\rangle$

That the systems are non-interacting means that the state at a later time $t_0 + \Delta t$ is just

$\left|\psi (t_0 + \Delta t) \right.\rangle = \left|\psi_1 (t_0 +\Delta t)\right.\rangle\left|\psi_2 (t_0 +\Delta t) \right.\rangle$.

So the time evolution of each subsystem can first be calculated separately and then the time-propagated vectors combined to the product state $\left|\psi_1 (t_0 +\Delta t)\right.\rangle\left|\psi_2 (t_0 +\Delta t) \right.\rangle$. This also holds for any value of $\Delta t$.

The normal modes of a coupled harmonic oscillator system can also be considered as non-interacting oscillators, despite being parts of the same interacting system. This is just a curious property of systems with Hooke's law interactions.

Particles can be entangled despite not interacting at all, if you deliberately construct a system where this happens.

 

[vanhees71]

Well, that's a somewhat delicate issue. It's dependent on whether you deal with identical or non-identical particles, where one refers to the intrinsic quantum numbers (spin, various charges).

For two non-identical particles the state ket of two non-interacting uncorrelated (non-entangled) particles is given as a product state,
$$|\Psi \rangle = |\psi_1 \rangle \otimes |\psi_2 \rangle.$$
If the particles are identical bosons or fermions such a state must be symmetric or antisymmetric under exchange of the two particles, and the state of two non-interacting uncorrelated (non-entangled) particles is given as a correspondingly symmetrized or anti-symmetrized product state
$$|\Psi \rangle = \frac{1}{\sqrt{2}} (|\psi_1 \rangle \otimes |\psi_2 \rangle \pm |\psi_2 \rangle \otimes |\psi_1 \rangle).$$
Another important point is that such uncorrelated states, when prepared at an initial time, stays such an uncorrelated state, if the particles are non-interacting, i.e., the Hamiltonian has the form
$$\hat{H}=\hat{H}_1 \otimes \hat{1} + \hat{1} \otimes \hat{H}_2,$$
where $\hat{H}_1$ and $\hat{H}_2$ are single-particle operators.

 

[Ken G]

Perhaps it could be said that the physical meaning of the AND concept is that you want to be able to talk about two systems in the same sentence, while at the same time treat them completely separately. That's a kind of semantic, rather than physical, merging of two systems, the default mode for all non-interacting distinguishable systems in both classical and quantum mechanics. But as pointed out above, this is not correct if the systems are indistinguishable, because even if they don't interact in the sense of inducing forces on each other, they still produce correlations in their behaviors (of the fermionic or bosonic types described mathematically above). So I would say the answer to your question is that the AND way of connecting two systems is not the one to be asking about, because it's the way we always thought one could connect two systems that don't exert any forces on each other. The issue is why the simplistic "semantic" AND approach doesn't always work, even when there are no direct forces!

The key hint that such a problem existed, prior to quantum mechanics, was the Gibbs paradox in classical statistical mechanics (https://en.wikipedia.org/wiki/Gibbs_paradox). This paradox said that if you have two identical boxes of indistinguishable gas particles, put the boxes next to each other and open a door between them, the resulting system cannot change at all (in the thermodynamic limit) when the door is opened, so must have the same entropy as before. But every particle now has access to twice the volume, so how is that not increasing the number of states they can be in? The resolution is we do not count states for particles, we count states for the system as a whole, and the indistinguishability of the particles means that any permutation of the particle labels must not count as a different state (since particle labels cannot be real for a indistinguishable particles). In quantum mechanics, that means combined wavefunctions can only change by multiplication by a physically irrelevant global phase factor when particles labels are interchanged, and the two ways to do that are given above.

So thinking in terms of wavefunctions and global phase factors is a quantum mechanical thing to do, but even in classical physics we already saw that descriptions of indistinguishable systems had to find some way to "erase" any physical meaning for separately labeling those noninteracting systems when combining them in the AND mode. You do that by simply not counting permutations of system labels as separate states in classical thermodynamics, and then you find that opening the door between the boxes does not change the entropy. So even before quantum mechanics, we already knew there was something subtle about the AND operation applied to indistinguishable systems, whereas that combination continues to be completely mundane for noninteracting distinguishable systems. All quantum mechanics does is give us an elegant way to account for the impossibility of indistinguishable system labels-- you make sure that using the labels can only introduce a + or - sign to the whole wavefunction when you interchange the (physically unreal) labels.