There’s what looks like an important article in PhysOrg on this. I’ll let you read it rather than try to summarize since I don’t think I’m getting it. But they seem to bury the lead as this is at the very end.
I think the idea is that even though quantum phenomena are in a very real sense just probability smears, those smears still have a trajectory and wake which have effects of their own. Is this sort of close and if so, what sorts of interactions do they have in mind?
Any system can be described by some set of parameters. Quantum phase space is just the space of parameters needed to describe some quantum-mechanical system. In classical mechanics, phase spaces usually have a relatively small number of dimensions: For instance, the phase space of a single classical point particle is six dimensional, with three parameters for its position at some time, and three more for its momentum (or velocity, if you prefer) at that time. Quantum mechanical phase spaces, though, are usually infinite-dimensional, except in some highly-constrained systems. Usually, you express them in terms of the eigenfunctions of some observable such as energy: If the system can have a discrete set of possible measurable energies, then you can break down any state as a sum of states corresponding to those energies. The coordinates of the phase space would then be the coefficients of the different energy eigenstates in the sum.
ok, that sort of makes sense I guess if I think of it in terms of vectors or something. Would have to look up eigenvalues although I remember the term from matrix algebra.
But what I’m getting from the article is that previously, because of the stochastic nature of the quantum wave function, you couldn’t get neatly defined visualizations of quantum interactions. Now they seem to be saying that via the Wigner function, they’re able to pin down what these interactions “look like” in some abstract sense and that is leading to new discoveries. That’s what I’m not getting. What exactly are they “seeing?”
Let’s talk about the phase space of everyone’s favorite simple quantum system, the harmonic oscillator. As in the classical case, there are two dimensions to this phase space, position and momentum. In the classical case, the oscillator traces a one dimensional curve (trajectory) in this phase space. In appropriately-scaled coordinates, this curve is a perfect circle for a lossless oscillator, or a spiral converging on zero for a lossy oscillator.
In the quantum case, there is no one-dimensional trajectory, since at any instant of time the state of the oscillator cannot be reduced to a single point in phase space. If it were a single point, that means we would know both the position and momentum accurately at that instant in time, which is impossible due to the uncertainty principle. We can however represent the state at any time with a wave function in position. If we fourier transform that wave function, we get a momentum wave function. There are many possible quantum states.
A particular case would be a wave function that is shaped like a bell curve, displaced from the origin. If the uncertainty represented by the width of the bell curve is the same as that in the “ground state”, i.e. the lowest energy state, then the wave function will simply oscillate back and forth, just like a classical oscillator, without any changes in the spread of the wave function. In momentum space, it will look the same, oscillating 90 degrees out of phase. The trajectory in phase space can now be mapped out. You have a fuzzy ball of uncertainty moving in a circle around the origin, without changing shape. This is just like the classical trajectory, simply replacing the discrete point at every instant of time with a ball of uncertainty.
Now that was a lot of hand waving. The Wigner dsitribution is just a precise mathematical definition of that fuzzy ball of uncertainty in phase space. Heisenberg’s equation tells us how a specified Wigner function evolves in time. The ball of uncertainty has an area whose units are the product of momentum and position, the same units as Planck’s constant. A “minimum uncertainty wave packet” will have a ball of uncertainty whose area is exactly the minimum allowed by the uncertainty principle. It might be round or eilliptical. If you squeeze the uncertainty in position, the momentum uncertainty has to get bigger, preserving the area of the ellipse. The ellipse will rotate in phase space at the same time that its center is revolving around the origin, all the while conserving area while the uncertainty oscillates in magnitude between the position axis and the momentum axis.
A few pictures would make this a lot clearer, but the fundamental notion is that thinking about quantum systems in phase space, using the Wigner function to precisely quantify the state is often helpful. Energy eignestates are grossly over emphasized in my view. It is hopeless to try to explain classical systems in terms of energy eigenstates, which are completely stationary, with no time evolution. The Wigner distribution picture goes easily from describing energy eigenstates to explaining highly excited dynamical states, which can represent classical behavior in the limit.
Perhaps I should post this in MPSIMS, but I actually sort of understood that. Neat for you and your use of imagery.
I think the main point of quantum mechanics on phase space is that you can essentially conceive of it as a deformation (about what exactly that means, later) of classical (statistical) mechanics. JWT Kottekoe has done a good job at explaining the relevant notions, I’ll just try and add some more things from this perspective.
First, if you do classical mechanics on phase space, your whole system is represented by a point, whose coordinates give you the positions and momenta of all the constituents of your system. So, in general, if you have a system of N point particles, this means 6N coordinates in a three-dimensional position space: three coordinates for the position of each of the N particles, and again three for the momenta. The evolution of your system then is a trajectory through phase space—in JWT Kottekoe’s example, for instance, a circle (as depicted on this wiki page).
Now, the phase space area for the harmonic oscillator has dimensions of [positionmomentum]=[energytime]=Js, which is not coincidentally the dimension of Planck’s constant ℏ. As you may know, the uncertainty principle constrains the accuracy with which one can know both position and momentum of a quantum system simultaneously. Roughly, the product of the uncertainty in each quantity, ΔxΔp, must always be at least ℏ or greater. But this means that your system can’t be represented by a point in phase space anymore, but rather, must be constrained to a spot of area at least ℏ.
There is another situation in physics in which a system does not occupy a definite spot in phase space: statistical mechanics. There, the precise state of the system is simply not known; rather, what is known is a probability distribution over possible states. In general, and in principle, one can always ‘sharpen’ this distribution to a point, by measuring all parts of the system. This is not possible in quantum mechanics, due to the uncertainty principle, as we have seen. This emphasizes the irreducible nature of probability in quantum mechanics: it doesn’t just quantify our ignorance (as it does in the statistical mechanics case), but rather, is fundamental.
But still, it is natural to suppose that, if ℏ were vanishingly small, quantum mechanics on phase space will transition to ordinary statistical mechanics. This is indeed the case, and is the reason why I have called quantum mechanics a deformation of phase space statistical mechanics. In the mathematical sense, a deformation of some object is a related object that goes over into the original object upon the vanishing of some deformation parameter. For instance, an ellipse is a deformation of a circle in this sense: if the eccentricity goes to zero, an ellipse simply becomes a circle.
The same is now the case with phase space quantum mechanics: here, the deformation parameter is simply ℏ, and if that tends to zero, we get classical statistical mechanics back out. In particular, the Wigner function is simply a deformed version of the probability distribution on phase space (known as the Liouville distribution); it gives you exactly the same information as, for instance, the quantum wave function or density matrix (to the latter of which it is related by the Weyl-Wigner transform).
Doing quantum mechanics in this way has some advantages, for instance, in quantum optics; additionally, it is sometimes useful to be able to do quantum mechanics in the same arena as classical mechanics (rather than more abstract ones such as Hilbert spaces). But I think the greatest gain is conceptual: while nothing in the discussion I gave should be mistaken for rigorous, the intuition it builds through being able to conceptualize quantum systems, I think, can be very useful (the same point, incidentally, as is being raised by the article linked in the OP).
It should be noted here that hbar is a fundamental constant, and so, strictly speaking, it doesn’t make sense to speak of the limit as it approaches zero. Strictly speaking, what physicists mean when we say things like that is that any other relevant quantities in the problem with the same units are much larger than hbar.
Thanks for the explanations. I guess I’m just not seeing how this is really going to contribute to facilitating new discoveries. I mean anyone doing basic research understands that sure, for certain purposes you’ll treat an electron as a point particle, for others a wave and for still other as a probability distribution, so I guess it’s utility is lost on me.
It’s quite often the case, especially in quantum mechanics, that someone will come up with a new mathematical way of describing something, and prove that the new method is exactly equivalent to the old, established methods. But even though the methods are exactly equivalent, often, one method or the other makes it easier to think about certain problems, and provides useful guidance on how to arrange the equations and such to find a solution.
The basic description of an electron is as a point particle, however that point particle has a very abstract, very math-heavy thing called a quantum state (at least when it’s isolated) which is described by a wavefunction.
The wavefunction is not a physical wave nor does it (depending on your pov) directly describe a physical property of the electron, instead the result of a measurement of any of the different physical properties an electron(whilst still being an electron) can have are determined by the wavefunction, usually probabilistically. Sometimes wave-like behavior emerges from this description.
A lot of the practical issues in QM center around describing the quantum state as in practice the kind of quantum systems that you want to describe are much more complicated than a single isolated electron and writing down the quantum state can be challenging. Because the actual description of the quantum state is totally abstract and mathematical it allows you to re-frame it in different ways, without changing anything important e.g. the Schrodinger picture vs the Heisenberg picture or Hilbert space vs phase space as talked about above.
Using alternative descriptions of the quantum state (whilst remaining in mainstream quantum mechanics) has zero implications practically (i.e. as in what the theory predicts) and most of the time doesn’t even have philosophical implications. However it can have big implications for people doing the calculations in terms of finding shortcuts to long calculations, finding approximations (approximation is very important in physics as exact calculations 99.9% of the time are either impractical or impossible) and offering insights.