{"id":4656,"date":"2016-04-07T17:28:59","date_gmt":"2016-04-08T00:28:59","guid":{"rendered":"http:\/\/www.wall.org\/~aron\/blog\/?p=4656"},"modified":"2017-04-18T08:05:45","modified_gmt":"2017-04-18T15:05:45","slug":"quantum-mechanics-iii-wavefunctions","status":"publish","type":"post","link":"https:\/\/www.wall.org\/~aron\/blog\/quantum-mechanics-iii-wavefunctions\/","title":{"rendered":"Quantum Mechanics III: Wavefunctions"},"content":{"rendered":"

[Fixed typo in Schrodinger’s equation below\u2014AW]<\/strong><\/p>\n

Previously I talked about interference<\/a>, the chief weird thing about QM that makes it different from Classical Mechanics.\u00a0 You have to think about complex-valued “amplitudes”, from which you derive (real-valued) probabilities.\u00a0 From this you can also derive the notion of a Hilbert Space<\/a> of states.\u00a0 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.<\/p>\n

Now let’s consider a more complicated system: a single particle moving around in empty space.\u00a0 There are infinitely many states, because space is a continuum.\u00a0 Hence we need to use an infinite dimensional <\/em>Hilbert space.\u00a0 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.<\/p>\n

There are many possible choices of basis, but two of them are particularly nice.\u00a0 You can choose to either express the system as superposition of position<\/em> states, or as a superposition of momentum<\/em> states, but you can’t specify both at the same time, because they are two different bases of the Hilbert Space!\u00a0 This is the origin of the Heisenberg Uncertainty Principle<\/a>.<\/p>\n

Of course, since the position and momentum are continuous variables, the probability of having any particular<\/em> exact value of position or momentum is always 0.\u00a0 So we have to generalize the framework slightly and talk about amplitude densities.\u00a0 <\/em>(However, there are other choices of basis where you don’t have to do that).<\/p>\n

An amplitude density is an amplitude per unit square-root-of-volume<\/em>.\u00a0 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.\u00a0 The amplitude density is more commonly called the\u00a0wavefunction<\/em> of the particle.\u00a0 So the wavefunction can be written as a function of position: \\(\\Psi(x,\\,y,\\,z)\\), or<\/em> as a function of momentum: \\(\\Psi(p_x,\\,p_y,\\,p_z)\\), but not both at the same time.\u00a0 However, if you know one of them, you can calculate the other one by a Fourier transform<\/a> (should you be lucky enough to know what that is).<\/p>\n

If you have multiple particles, you shouldn’t think that each particle has a separate wavefunction.\u00a0 Instead, you use a single<\/em> wavefunction which depends on the positions (or momenta) of all of the particles.\u00a0 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.\u00a0 Hence the wavefunction for 2 particles actually is a function living in a 6 dimensional space!\u00a0 (More generally, the wavefunction of \\(N\\) particles will live in \\(3N\\)-dimensions, assuming that space is 3 dimensional.)<\/p>\n

(Given these two particles, it might be<\/em> 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.\u00a0 But in general, there’s lots of wavefunctions you could write down which do not<\/em> factorize in this way.\u00a0 This allows the particles to be correlated in strange ways not allowed by classical physics.\u00a0 We call this phenomenon entanglement<\/em>.)<\/p>\n

Now the two particles might be either different type, or the same type.\u00a0 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.\u00a0 (E.g. all electrons have identical properties, and all photons also have identical properties.)<\/p>\n

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”.\u00a0 So there should be a symmetry of the wavefunction if we switch the two particles.\u00a0 (Remember, in QM we have interference<\/a> 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.)\u00a0 There are two different ways to implement this symmetry.\u00a0 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.\u00a0 This sensible approach is taken by identical\u00a0bosons<\/em>, which includes particles such as photons, gluons, gravitons, mesons, He-4 nuclei, and so on.<\/p>\n

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\u00a0fermions<\/em> such as electrons, quarks, neutrinos, protons, neutrons, and He-3 nuclei.\u00a0 (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.)<\/p>\n

So photons are strictly identical, while electrons are almost identical, but you get a minus sign if you switch them.\u00a0 But remember, the overall phase of a QM system doesn’t matter.\u00a0 So you won’t actually notice anything weird if you definitely<\/em> switch two fermions.\u00a0 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.<\/p>\n

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\\).\u00a0 This is a special case of the Pauli Exclusion Principle<\/a>, which is the reason for the Periodic Table.\u00a0 Since electrons can be either “spin up” or “spin down”, you can only put 2 distinct electrons in each energy level of an atom.\u00a0 Then the energy levels get full, and you have to put the electrons into higher energy shells.<\/p>\n

Bosons, on the other hand, are gregarious and love<\/em> to be in the same place.\u00a0 Or rather, to speak less anthropomorphically, their probability to be in the same place is greater than you would expect from classical probability theory.\u00a0 This is what makes lasers (a bunch of photons all in the same state) practically possible.<\/p>\n

I’ve mentioned “spin” several times, but I haven’t actually said what it is.\u00a0 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).\u00a0 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<\/em>.\u00a0 But it works in a weird way.\u00a0 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\\)).\u00a0 The maximum possible angular momentum along any axis is called the “spin” of the particle.<\/p>\n

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.\u00a0 (You can prove this rule mathematically in QFT, but it requires Special Relativity<\/a> and some additional physical assumptions.)<\/p>\n

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).\u00a0 You only get to specify the spin along one<\/em> axis, say the vertical one.\u00a0 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.\u00a0 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.\u00a0 The (2 complex dimensions = 4 real dimensions) space of possible electron spins is called a spinor<\/em>.<\/p>\n

A spinor needs to be rotated by 720\u00ba (2 full circles) to get back to its original state.\u00a0 Yes, you read that right.\u00a0 If you only rotate it by 360\u00ba (1 full circle) then it comes back to itself with an extra minus sign in the amplitude.\u00a0 Just like when you switch two electrons.\u00a0 They’re just perverse that way.<\/p>\n

<\/em>Most of the fundamental bosons are spin 1, so their polarization is given by a vector<\/em>, as in the case of the photon which we discussed last time<\/a>.\u00a0 Vectors get a minus sign when you rotate them by 180\u00ba, and return back to the way they were after 360\u00ba, just like you were taught in school.<\/p>\n

However, the Higgs boson (which gives mass to most of the other fundamental particles) is a spin-0 or scalar<\/em> field.\u00a0 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 90\u00ba, and goes to itself under 180\u00ba.\u00a0 Its polarization is described by a matrix<\/a>, but let’s not get into that here.<\/p>\n

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.\u00a0 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.\u00a0 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)\\).\u00a0 (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.\u00a0 But the explanation of this involves relativity and gauge symmetry and a bunch of other things from QFT…)<\/p>\n

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.\u00a0 This is the basis where the energy<\/em> of the system takes on a definite value.\u00a0 In this basis, the only thing that changes is the phase of each energy state.\u00a0 The phase changes with time at a speed proportional to the energy.<\/p>\n

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.\u00a0 (You think of this an operator<\/em>, a gadget which acts on the wavefunction to get another wavefunction.\u00a0 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.)\u00a0 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.\u00a0 This is related to the Hamiltonian approach<\/a> in Classical Mechanics.\u00a0 As you take the \\(\\hbar \\to 0\\) limit, you recover classical mechanics.<\/p>\n

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<\/a>, where you assign to each history an amplitude proportional to \\(e^{iS\/\\hbar}\\), where \\(S\\) is the action.\u00a0 This approach is actually more closely related to the picture I started with in part I<\/a>!<\/p>\n

In this sense, Quantum Mechanics is a fulfillment of Classical Mechanics, just as (in Christian doctrine) the New Covenant fulfills the Old Covenant.\u00a0 That is, the new model justifies the quirky, previously-inexplicable features of the old model, in terms of a more basic (yet also<\/em> more mysterious) set of ideas.\u00a0 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.<\/p>\n

Incidentally, there’s a very important flaw in what I’ve told you so far.\u00a0 Generally speaking, in modern physics it’s better to think of the universe as being made of fields<\/a>, not particles!\u00a0 This is the subject of Quantum Field Theory.\u00a0 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.).\u00a0 Consider a scalar field \\(\\Phi(t,x,y,z)\\), which is basically a function of the spacetime points.\u00a0 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.\u00a0 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<\/em>!\u00a0 The “particles” are then quantized excitations associated with different modes of this field.\u00a0 (The relationship between QFT and the QM of multiple particles should not be obvious from what I’ve said so far…)<\/p>\n

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.\u00a0 Since this happens all the time in the real world, that’s kind of important!<\/p>\n

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.\u00a0 (Perhaps I shouldn’t have given the photon as an example, because it always travels at the speed of light and is never nonrelativistic.)<\/p>\n","protected":false},"excerpt":{"rendered":"

[Fixed typo in Schrodinger’s equation below\u2014AW] Previously I talked about interference, the chief weird thing about QM that makes it different from Classical Mechanics.\u00a0 You have to think about complex-valued “amplitudes”, from which you derive (real-valued) probabilities.\u00a0 From this you can also derive the notion of a Hilbert Space of states.\u00a0 We discussed the space […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-4656","post","type-post","status-publish","format-standard","hentry","category-physics"],"_links":{"self":[{"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/posts\/4656","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/comments?post=4656"}],"version-history":[{"count":56,"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/posts\/4656\/revisions"}],"predecessor-version":[{"id":4917,"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/posts\/4656\/revisions\/4917"}],"wp:attachment":[{"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/media?parent=4656"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/categories?post=4656"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wall.org\/~aron\/blog\/wp-json\/wp\/v2\/tags?post=4656"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}