[Fixed typo in Schrodinger’s equation below—AW]

Previously I talked about interference, the chief weird thing about QM that makes it different from Classical Mechanics.  You have to think about complex-valued “amplitudes”, from which you derive (real-valued) probabilities.  From this you can also derive the notion of a Hilbert Space of states.  We discussed the space of states for the polarization of a photon (a 2 state system), and how there are many different choices of “basis”, representing different ways of identifying a mutually exclusive set of two possibilities.

Now let’s consider a more complicated system: a single particle moving around in empty space.  There are infinitely many states, because space is a continuum.  Hence we need to use an infinite dimensional Hilbert space.  This is harder to visualize than a two-dimensional one, but it will still be true that, in any given basis, each state can be regarded as a quantum superposition of a bunch of possibilities.

There are many possible choices of basis, but two of them are particularly nice.  You can choose to either express the system as superposition of position states, or as a superposition of momentum states, but you can’t specify both at the same time, because they are two different bases of the Hilbert Space!  This is the origin of the Heisenberg Uncertainty Principle.

Of course, since the position and momentum are continuous variables, the probability of having any particular exact value of position or momentum is always 0.  So we have to generalize the framework slightly and talk about amplitude densities.  (However, there are other choices of basis where you don’t have to do that).

An amplitude density is an amplitude per unit square-root-of-volume.  I know these units sound a bit strange, but that way when you square it, you get a probability per unit volume, which is as things should be for purposes of doing measurements.  The amplitude density is more commonly called the wavefunction of the particle.  So the wavefunction can be written as a function of position: \(\Psi(x,\,y,\,z)\), or as a function of momentum: \(\Psi(p_x,\,p_y,\,p_z)\), but not both at the same time.  However, if you know one of them, you can calculate the other one by a Fourier transform (should you be lucky enough to know what that is).

If you have multiple particles, you shouldn’t think that each particle has a separate wavefunction.  Instead, you use a single wavefunction which depends on the positions (or momenta) of all of the particles.  For example, if there are two different particles which we’ll call #1 and #2, then you’d write:$$\Psi(\vec{r}_1, \vec{r}_2),$$ where I’m now using vector notation as a shorthand; but each vector still has x, y, and z components.  Hence the wavefunction for 2 particles actually is a function living in a 6 dimensional space!  (More generally, the wavefunction of \(N\) particles will live in \(3N\)-dimensions, assuming that space is 3 dimensional.)

(Given these two particles, it might be that the wavefunction factorizes, so that $$\Psi(\vec{r}_1, \vec{r}_2) = \Psi_1(\vec{r}_1)\Psi_2(\vec{r}_2).$$That’s what would happen if you independently prepare each particle in a state, and don’t let them interact with each other.  But in general, there’s lots of wavefunctions you could write down which do not factorize in this way.  This allows the particles to be correlated in strange ways not allowed by classical physics.  We call this phenomenon entanglement.)

Now the two particles might be either different type, or the same type.  One of the principles of particle physics is that apart from a limited number of attributes such as position/momentum, “spin”, and a few other things, all particles of a given type are identical.  (E.g. all electrons have identical properties, and all photons also have identical properties.)

If the two particles are identical, then it shouldn’t make any difference which particle we choose to label as “1” and which we choose to label as “2”.  So there should be a symmetry of the wavefunction if we switch the two particles.  (Remember, in QM we have interference whenever two histories end up in the same place, so to get things right we have to obsess about exactly when two situations count as exactly the same, and when they don’t.)  There are two different ways to implement this symmetry.  The obvious thing to do is to say that:$$\Psi(\vec{r}_1, \vec{r}_2) = \Psi(\vec{r}_2, \vec{r}_1),$$so that the amplitude is the same in both cases.  This sensible approach is taken by identical bosons, which includes particles such as photons, gluons, gravitons, mesons, He-4 nuclei, and so on.

Another, more perverse way to implement the symmetry is to insert a minus sign:$$\Psi(\vec{r}_1, \vec{r}_2) = -\Psi(\vec{r}_2, \vec{r}_1).$$This bizarre form of identicalness is used by fermions such as electrons, quarks, neutrinos, protons, neutrons, and He-3 nuclei.  (In general, something made out of an odd number of fermions is also a fermion, since if you switch two copies, you’ll get an odd number of minus signs.)

So photons are strictly identical, while electrons are almost identical, but you get a minus sign if you switch them.  But remember, the overall phase of a QM system doesn’t matter.  So you won’t actually notice anything weird if you definitely switch two fermions.  The minus sign only matters in situations where they might-or-might-not have gotten switched, because then the interference between the two histories will be different.

A somewhat more straightforward implication is that no two identical fermions are ever in exactly the same position, because then the weird antisymmetry tells us that \(\Psi(x_1, x_1) = – \Psi(x_1, x_1) = 0\).  This is a special case of the Pauli Exclusion Principle, which is the reason for the Periodic Table.  Since electrons can be either “spin up” or “spin down”, you can only put 2 distinct electrons in each energy level of an atom.  Then the energy levels get full, and you have to put the electrons into higher energy shells.

Bosons, on the other hand, are gregarious and love to be in the same place.  Or rather, to speak less anthropomorphically, their probability to be in the same place is greater than you would expect from classical probability theory.  This is what makes lasers (a bunch of photons all in the same state) practically possible.

I’ve mentioned “spin” several times, but I haven’t actually said what it is.  In QM, some particles also have an intrinsic angular momentum or polarization, which gives them a certain sort of directionality in space (even though they are point particles).  Unlike a lot of the cute terms used in particle physics such as “color” or “charm”, the term “spin” really does refer to actual literal angular momentum.  But it works in a weird way.  The angular momentum along any axis is quantized, meaning it has to be either an integer or an integer + 1/2 (times the Planck constant \(\hbar\)).  The maximum possible angular momentum along any axis is called the “spin” of the particle.

In Nature, there is a rule called spin-statistics which says that particles with integer spin are always bosons, and particles with half-integer spin are always fermions.  (You can prove this rule mathematically in QFT, but it requires Special Relativity and some additional physical assumptions.)

Every known fundamental fermion is spin 1/2, which means that along any given axis it is either spinning clockwise (a.k.a. “up” or +1/2) or counterclockwise (a.k.a. “down” or -1/2).  You only get to specify the spin along one axis, say the vertical one.  This is not to say that an electron can’t spin “left”, “right”, “in”, or “out”, but these states are quantum superpositions of the “up” and “down” states.  By rotational symmetry, we could pick a different basis (e.g. right / left) and instead think of up and down as superpositions of right and left.  The (2 complex dimensions = 4 real dimensions) space of possible electron spins is called a spinor.

A spinor needs to be rotated by 720º (2 full circles) to get back to its original state.  Yes, you read that right.  If you only rotate it by 360º (1 full circle) then it comes back to itself with an extra minus sign in the amplitude.  Just like when you switch two electrons.  They’re just perverse that way.

Most of the fundamental bosons are spin 1, so their polarization is given by a vector, as in the case of the photon which we discussed last time.  Vectors get a minus sign when you rotate them by 180º, and return back to the way they were after 360º, just like you were taught in school.

However, the Higgs boson (which gives mass to most of the other fundamental particles) is a spin-0 or scalar field.  That means it doesn’t change at all when you rotate it. On the other hand, the graviton is a spin-2 particle, which means it gets a minus sign when you rotate it 90º, and goes to itself under 180º.  Its polarization is described by a matrix, but let’s not get into that here.

The bottom line is that for anything more complicated than a scalar field, in addition to the position or momentum variables you also need to include the spin degrees of freedom.  So if we have one electron and one photon, the wavefunction will look like e.g. $$\Psi(\vec{p}_e, s_e,\vec{p}_\gamma, s_\gamma),$$where \(s\) represents the spin of the electron or photon along e.g. the z-axis.  This is still a 6 dimensional space since \(s_e\) can only take the values \((+1/2, -1/2)\), and \(s_\gamma\) can only take the values \((+1,0,-1)\).  (Incidentally, since the photon is massless, its spin is always required to be perpendicular to its momentum, so there are really only 2 polarization states, not 3.  But the explanation of this involves relativity and gauge symmetry and a bunch of other things from QFT…)

There is also a third kind of basis, distinct in general from both the position and the momentum basis, in which time evolution is particularly simple.  This is the basis where the energy of the system takes on a definite value.  In this basis, the only thing that changes is the phase of each energy state.  The phase changes with time at a speed proportional to the energy.

So one way to specify the dynamics of a QM system is simply to say what the formula for the energy \(H\) is, as a function of all the positions and momenta of all the particles in the problem.  (You think of this an operator, a gadget which acts on the wavefunction to get another wavefunction.  So if you are in the position basis, the “momentum operator” is given by \(\vec{p} \Psi = (i / \hbar) \vec{\nabla} \Psi\), which is equivalent to switching from the position to the momentum basis, multiplying by p, and then switching back.)  Then you can figure out how the wavefunction changes with time by using the Schrodinger equation:$$\frac{d\Psi}{dt} = -\frac{i}{\hbar} H \Psi.$$Thus, if you know what the formula for the energy is, you can predict the dynamics of the wavefunction as time passes.  This is related to the Hamiltonian approach in Classical Mechanics.  As you take the \(\hbar \to 0\) limit, you recover classical mechanics.

There is also a “path integral” picture due to Feynman, related to the Lagrangian or “Least Action” approach to physics mentioned at the same link, where you assign to each history an amplitude proportional to \(e^{iS/\hbar}\), where \(S\) is the action.  This approach is actually more closely related to the picture I started with in part I!

In this sense, Quantum Mechanics is a fulfillment of Classical Mechanics, just as (in Christian doctrine) the New Covenant fulfills the Old Covenant.  That is, the new model justifies the quirky, previously-inexplicable features of the old model, in terms of a more basic (yet also more mysterious) set of ideas.  Concepts such as action, energy, momentum, the associated conservation laws, and so on, all follow naturally from the interference of wavefunctions over space and time.

Incidentally, there’s a very important flaw in what I’ve told you so far.  Generally speaking, in modern physics it’s better to think of the universe as being made of fields, not particles!  This is the subject of Quantum Field Theory.  The idea is that we should really think of the universe as being made of some finite number of types of fields (e.g. the electron field, the photon/EM field, the quark field, etc.).  Consider a scalar field \(\Phi(t,x,y,z)\), which is basically a function of the spacetime points.  If we want to keep track of the amplitude for any possible configuration of the field, then we really need our wavefunction to be a function of all possible configurations of \(\Phi\) at one moment of time.  Morally speaking (i.e. I am about to make certain dreadful oversimplifications) this means that the state of the universe at one time is something more like:$$\Psi(\Phi(\vec{r})),$$in other words the wavefunction is a function of functions!  The “particles” are then quantized excitations associated with different modes of this field.  (The relationship between QFT and the QM of multiple particles should not be obvious from what I’ve said so far…)

QFT gets kind of complicated, but the advantages are that 1) it is easier to make it compatible with Special Relativity, and 2) it allows one to consider situations where particles are created and destroyed, e.g. an electron can emit or absorb a photon.  Since this happens all the time in the real world, that’s kind of important!

But as long as you’re dealing with situations where the particles are all going much slower than the speed of light, and none of them decay into other particles, you can use QM as described above.  (Perhaps I shouldn’t have given the photon as an example, because it always travels at the speed of light and is never nonrelativistic.)