Distinguishability and tagging of particles

Question

I was reading Sakurai's book and here is an extract:

In classical physics it is possible to keep track of individual particles even though they may look alike. When we have particle 1 and particle 2 considered as a system, we can, in principle, follow the trajectory of 1 and that of 2 separately at each instant of time. For bookkeeping purposes, you may color one of them blue and the other red and then examine how the red particle moves and how the blue particle moves as time passes. In quantum mechanics, however, identical particles are truly indistinguishable. This is because we cannot specify more than a complete set of commuting observables for each of the particles; in particular, we cannot label the particle by coloring it blue. Nor can we follow the trajectory because that would entail a position measurement at each instant of time, which necessarily disturbs the system; in particular, the two situations (a) and (b) shown in Figure 7.1 cannot be distinguished-not even in principle.

From this I understand two points (maybe not entirely independent of each other):

1. Since classically you can track the trajectory, in order to distinguish a particle from another you can just "follow" that particle. For instance, a particle formed at LHC can be followed and hence at any instant of time can be distinguished. This cannot be done quantum mechanically, so bye-bye to trying to distinguish by following its trajectory.

2. It is mentioned that a particle cannot be tagged "blue" or "red". What does this tagging exactly mean? Four quantum numbers specify an electron, does this tagging mean somehow introducing new such degrees?

To put my question in perspective, I think of something in these lines. Take two identical electrons, put them in separate 1D boxes. Pour "blue" stuff in the whole of 1 box. Pour "red" stuff in the other. Now take them out of these boxes. Will we not able to distinguish between them now? Of course, this would only make sense if we know what we mean by "coloring". So to sum up my question:

What would tagging of particles even mean? and if it has a sensible meaning what's wrong with my example? (This is probably related to "we cannot specify more than a complete set of commuting observables for each of the particles" but I do not understand this.)

Answer 1

You're pretty much spot on by raising the question of what "coloring" means. Let's say our classical intuition about particles comes from thinking about rocks that you and your friends throw around. If you paint some color on the rocks and keep throwing them the way you used to, you don't think that the coloring process changes anything fundamental about the rocks. It's the same rocks, but maybe now you can find out which of your friends just hit you with a rock.

This also holds true mathematically. If you trace the movement of the earth around the sun, a good simplification is to think about two points with different masses interacting by gravitation. You can label one of them as, say, "A" for the sun and the other as the "B" for the earth and distinguish them by their masses. But even if they had the same masses, and thus were physically identical in this picture, you are at liberty of naming one "A" and the other "B" and you can track the motion of "A" and "B" in your mathematical model of this system, which means that if you calculate the state of the system at a later time, you will find two point masses at different locations in space, but there is still a unique way of telling which point mass was "A" and which was "B" at the original time. The names "A" and "B" are strapped on the particle without any influence on the physics, much like paint brushed on rocks.

In quantum mechanics, this intuition fails. First of all, because the process of painting without changing physics doesn't have a counterpart in QM. There is no property you can add to one particle to make it more traceable. You can do this for particles as small as large molecules and consider them as classical particles (molecular dyes are for example used in soft matter physics). But when you talk about fundamentally quantum particles like electrons, this is bound to fail, because their fundamental interactions are fixed and introducing a dye means to introduce a new interaction into the system. Also, on a very literal level, single atoms and anything smaller doesn't show color (https://www.fnal.gov/pub/science/inquiring/questions/colorofatoms.html).

This is, I think, what Sakurai means when he says that one cannot specify more than a complete set of commuting observables for a particle: there is no further (commuting) property of the particle that could be strapped to it and would serve as some kind of label while changing nothing about the physical process. Indeed, it is hard to imagine what it should mean to color an electron. It's a point particle, so there is no form of irregularity one can stick a coloring dye to.

And this is also rooted in the mathematical description of quantum systems. When talking about wave functions in the Schrödinger picture of QM, one must use the spin statistics theorem (which is, despite the name, a postulate in classical quantum mechanics and can only be derived as an actual theorem when using quantum field theory as an underlying Lorentz covariant theory). The statement is that a wave function $\Psi_0(x_1,x_2)$ that represents an initial state of a system of two identical particles at time $t=t_i$ is (anti-)symmetric with respect to interchanging the two particles, $$\Psi_{t_i}(x_1,x_2) = \pm \Psi_{t_i}(x_2,x_1).$$ Now, if, as before, we want to look at a state at a later time $t = t_f > t_i$, which is represented by the wave function $\Psi_{t_f}(x_1,x_2)$ satisfying $$\Psi_{t_f}(x_1,x_2) = \pm \Psi_{t_f}(x_2,x_1).$$ While there is the probability interpretation of the wave function $\rho_t(x_1,x_2)\textrm{d}x_1\textrm{d}x_2 = |\Psi_t(x_1,x_2)|^2\textrm{d}x_1\textrm{d}x_2$, which can be read as "the probability of finding particle 1 at $x_1$ and the probability of finding particle 2 at $x_2$" (more exactly, in small volumes around $x_1$ and $x_2$), there really isn't a way of distinguishing "the probability of finding particle 1 at $x_1$ and the probability of finding particle 2 at $x_2$" from "the probability of finding particle 1 at $x_2$ and the probability of finding particle 2 at $x_1$". Even if one had in some way found a way to fix this meaning at the initial time $t_i$, there is no way of knowing from the state at $t_f$ which particle should be assigned to $x_1$ and $x_2$ because of the symmetry.

Answer 2

1. indeed, having distinct continuous trajectories means we can distinguish particles. In quantum theory there are no trajectories so this method is not available.

2. tagging means the particles have and carry some property by which we can distinguish them: for example, we can distinguish electron and proton by their mass. Then we operate with a model of the system where "particle 1 is at $\mathbf x_1$ and particle 2 is at $\mathbf x_2$" is a meaningful statement: we can check validity of this statement in principle by looking at the tag.

This is how Schroedinger's equation for many electron atoms is introduced - electrons are tagged by index $k$, and psi function depends on their coordinates $\mathbf r_k$. Then it is customary to argue that because we can't tag and check which electron has which tag (masses are the same, charges are the same...), this indistinguishability should be preserved also by the state vector (psi function) in the calculations.

Let there be an asymmetric function $f(\mathbf x,\mathbf y)$. Then psi function $\psi(\mathbf r_1,\mathbf r_2) = f(\mathbf r_1,\mathbf r_2)$ implies different predictions than $\psi(\mathbf r_1,\mathbf r_2) = f(\mathbf r_2,\mathbf r_1)$ does. So $f$ and both psi functions should be rejected. Otherwise we could distinguish the particles in the calculations based on the $\psi$ function.