Struggling to understand Identical particles I'm struggling with the concept of identical particles in QM. Say I have two electrons, with one trapped on my left $\left|\uparrow\right\rangle_1$ and one trapped on my right $\left|\downarrow\right\rangle_2$. The 2-particle wavefunction is then clearly $\left|\uparrow\right\rangle_1 \otimes \left|\downarrow\right\rangle_2$. If an exchange happens, clearly the wavefunction would be $\left|\downarrow\right\rangle_1 \otimes \left|\uparrow\right\rangle_2$, which is totally different and I can detect this exchange by measuring in the appropriate basis. The 2-particle system does not seem to violate any law of QM to me, and I can do all kind of operations on them including entanglement.
Yet, why is it that when I open QM textbook like Griffiths and Sakurai etc. in the Identical Particles chapters, they always say that this is impossible, that in order to construct a many-particle wavefunction, one must know that

God doesn't know which is which, because there is no such thing as
"this" electron, or "that" electron

and

we (can't) follow the trajectory because that would entail a position
measurement at each instant of time, which necessarily disturbs the
system

I mean, am I tracking the electrons if I trap them on 2 sides of my table? Even if I am tracking the electron, what is the bad thing about it that makes my many-particle state $\left|\uparrow\right\rangle_1 \otimes \left|\downarrow\right\rangle_2$ invalid to QM? What am I misunderstanding?
 A: Tracking the two electrons you're talking about would requiere knowledge of both position and momentum, which contradicts Heisenberg uncertainty principle. Since you cannot track them you don't know which electron is (spin) up $|\uparrow\rangle$ and which one is down $|\downarrow\rangle$. As a consequence, your state for the system must be a combination of both situation, i.e.:
$$
|{\rm state\ of\ 2\ electrons}\rangle = C_1|\uparrow\rangle_1\otimes |\downarrow\rangle_2 + C_2 |\downarrow\rangle_1\otimes |\uparrow\rangle_2
$$
Because of normalization and extracting a global phase (that according to QM is irrelevant) we get
$$
|{\rm state\ of\ 2\ electrons}\rangle = \frac{1}{\sqrt{2}}(|\uparrow\rangle_1\otimes |\downarrow\rangle_2 + e^{i\theta} |\downarrow\rangle_1\otimes |\uparrow\rangle_2)
$$
And now, as an empirical fact, nature tells us that $e^{i\theta} = -1$ for fermions (odd half-integer spin: electrons and similar particles) and $+1$ for bosons (integer spin particles).
From the theory point of view, this empiral fact is proven in quantum field theory and it's related to commutation and anticommutation rules of the fields describing such particles.
A: Your mistake is that you assume that you may put a number according to the positions. But actually the identical electron state would look like,
\begin{equation}
\frac{1}{2}\Big(|\uparrow, \text{left}\rangle_1\otimes|\downarrow, \text{right}\rangle_2-|\downarrow, \text{right}\rangle_1\otimes|\uparrow, \text{left}\rangle_2\Big) 
\end{equation}
I.e. exchange operation exchanges all properties between two particles. Or equivalently it simply exchanges the number you associate with particles between them.
As to where this principle comes from, what prohibits the symmetric states for fermions and anti symmetric for bosons, you won't get the answer in QM. It may be understood only from the spin-statistic theorem in QFT.
A: 
I'm struggling with the concept of identical particles in QM. Say I have two electrons... What am I misunderstanding?

Not every solution to the Schrodinger equation is a "physical state." (Not every solution describes the natural world as it actually is.)
Consider, for example, a solution to the two-particle Schrodinger equation:
$$
\Psi(\vec r_1, s_1;\vec r_2, s_2)\;.
$$
If the above function is a solution, then so too is
$$
\Psi(\vec r_2, s_2;\vec r_1, s_1)
$$
a solution of the Schrodinger equation having the same energy, since the Hamiltonian is symmetric with respect to particle index exchange. If the states are different then this degeneracy is called "exchange degeneracy."
It is a fact of nature that the wave functions that actually describe two physical particles always obey either:
$$
\Psi(\vec r_2, s_2;\vec r_1, s_1)
=\Psi(\vec r_1, s_1;\vec r_2, s_2)\tag{bosons}
$$
or
$$
\Psi(\vec r_2, s_2;\vec r_1, s_1)
=-\Psi(\vec r_1, s_1;\vec r_2, s_2)\tag{fermions}\;.
$$
Therefore a state with only spin dependence such as
$$
|\Psi\rangle \sim |\uparrow; \downarrow\rangle
$$
is not a physical fermion state.
But a state like
$$
|\Psi\rangle \sim |\uparrow; \downarrow\rangle
-
|\downarrow; \uparrow\rangle
$$
is a physical fermion state, since exchanging the indices returns the same state (multiplied by negative one).

Here is a quote from Hans Bethe to make you feel better:

"Now it is a fact of nature, established by many observations, that all actual wave functions in physics obey $P_{ij}\Psi = \pm\Psi$, either with the plus or the minus sign. In other words, from the great multitude of mathematical solutions of $H\Psi = E\Psi$, nature has selected only the nondegenerate ones... Which symmetry applies depends on the type of identical particles involved; experiment shows that $\Psi$ is antisymmetric for electrons, protons, neutrons, mu-mesons..."

A: 
I mean, am I tracking the electrons if I trap them on 2 sides of my table?

Not really. Your trap isn't perfect. You can't literally have an infinite square well to ensure the particle stays trapped. Hence, you can't really know if the particles tunneled and exchanged positions, for example. The only way for you to know whether the particle actually stayed in the same trap the whole time is by measuring its position continuously. If they do switch places, you won't know, because there is (by definition) no way to distinguish.
The main point is that if you stop measuring the position of a particle, you have no way of knowing whether that is still the same particle. Traps aren't perfect, so there is a slight probability your particle tunneled out of it and was replaced by a different one. There are no observational differences, but it is not the same particle anymore.
Hence, in QM, it doesn't make sense to speak of a particular particle when you have another identical particle. You are unable to distinguish them, so you can only describe the pair, or perhaps the left particle. But there exist no tags on the particles telling you which is $1$ and which is $2$. Unless you are willing to continuously measure their positions and hence affect the experiment.
A: The state of the electrons in the situation you describe is actually not $|\uparrow, {\rm left}\rangle_1 \otimes |\downarrow, \rm{right}\rangle_2$. It is instead
\begin{equation}
|\Psi\rangle = \frac{1}{\sqrt{2}}\Big(|\uparrow, {\rm left}\rangle_1 \otimes |\downarrow, {\rm right} \rangle_2 - |\downarrow, {\rm right} \rangle_1 \otimes |\uparrow, {\rm left} \rangle_2 \Big)
\end{equation}
where the label "left" or "right" refers to the spatial location of the particle, and the indices $1$ and $2$ refer to the "identity" of the particle.
In particular, you cannot say whether electron 1 or electron 2 is the one that is trapped at your left, with spin up.
This may feel surprising and counterintuitive. However, if you think through the experimental setup carefully, you will find there is no way for you to say whether electron 1 or 2 is "really" the one in the left trap. There is no way for you to "mark" either of the two electrons.
If it helps, you can view indistinguishable particles as a mathematical possibility that quantum mechanics allows for. Maybe you don't intuitively believe electrons must be indistinguishable, but you have to accept that quantum mechanics logically allows indistinguishable particles. Indistinguishable particles live on a subset of Hilbert space where the state is fully symmetric or fully antisymmetric under exchange of particles.
It then becomes an experimental question about whether real-world electrons are described by the quantum mechanics distinguishable or indistinguishable particles. Electrons exhibit Fermi-Dirac statistics, a fact that forms the basis of modern condensed-matter theory, and therefore in some sense is experimentally tested every time you use a conductor or insulator, or every time a material scientist designs electronic properties of a material. So as an experimental question, it is settled that electrons behave as indistinguishable fermions.
The fact that electrons are indistinguishable fermions is also understood theoretically using relativistic quantum field theory.
A: Exchange symmetry doesn't apply to measurable properties like position and spin. It only applies to unphysical labels.
Your wave function is
$$\left|\uparrow\right\rangle_1 \otimes \left|\downarrow\right\rangle_2$$
Each particle in this wave function has three properties: a spin ($\uparrow$ or $\downarrow$), a subscript ($1$ or $2$), and a position in the tensor product (left or right). But your particles have only two physical properties, spin and position. Obviously spin is spin, but it's not clear what you mean by the other two properties in the wave function. The answer to your question depends on what they mean.
Other answers have assumed that they are both unphysical labels, redundantly encoding the same which-particle information. If they are, then the correct antisymmetrized wave function would be $\left|\uparrow\right\rangle_1 \otimes \left|\downarrow\right\rangle_2 - \left|\downarrow\right\rangle_1 \otimes \left|\uparrow\right\rangle_2$ (up to a scalar factor), where I obtained the second term from the first by reversing the product and swapping $1$ and $2$. But this doesn't really make sense, because physical position, which is crucial to your thought-experiment, is not represented in your wave function at all, so you can't model the swapping of the particles.
If only the tensor-product order is meaningless, and the subscripts represent physical position, then the antisymmetrized wave function would be $\left|\uparrow\right\rangle_1 \otimes \left|\downarrow\right\rangle_2 - \left|\downarrow\right\rangle_2 \otimes \left|\uparrow\right\rangle_1$, where I obtained the second term from the first by reversing the product only. If you physically swap the particles, then the wave function becomes $\left|\uparrow\right\rangle_2 \otimes \left|\downarrow\right\rangle_1 - \left|\downarrow\right\rangle_1 \otimes \left|\uparrow\right\rangle_2$, which I obtained from the previous wave function by swapping $1$ and $2$.
A: You seem to be talking about two particles in the same quantum well or in the same atom. If you had two particles trapped in two adjacent quantum wells, they wouldn't necessarily be bound by the Pauli exclusion principle to have opposite up and down spins. I don't know if quantum entanglement changes that situation.
