* Seriously, Netflix, what the damn hell? I used to get two cycles per week. Now a turn-around takes ten business days. If you're not going to support your DVD service, then eliminate it. Don't do the typical American business thing where you screw the extant customer base with declining service until somebody shows up at your office with a Carcano. Tweeners whose mommy pays the streaming bill may snigger at us DVD players, but I would remind you we are the reason Reed Hastings is banging supermodels on a private jet.
But I digress.
More succinctly, I've been retooling older posts to play nicer with mobile devices and I came across my review of Thomas Jordan's Quantum Mechanics in Simple Matrix Form. The review segued into a derivation of energy quantization and it occurred to me breaking that out as a proper post might help bring it to light, or at least bring it to the attention of Google's search engine, which treats LabKitty the blog about as well as Netflix treats LabKitty the customer.
You never know what somebody might be looking to read on a Saturday night, is my point. And while this isn't my work -- it's Jordan's -- I did add boss figures and many (many) explanatory words. (Technically I guess it's Heisenberg's work, who by the way for any weeaboos tuning in today was one of the founding fathers of quantum mechanics and not the bald guy in that Starz show that makes you go W00T! on Reddit.)
Note to self: Man, whiskey really brings out AngryKitty.
I cannot lift you up from nothing. But if you agree to meet me halfway, I can show you how Heisenberg's matrix mechanics is used to derive Planck's energy quantization. Real actual quantum mechanics. The mathematical prerequisites are surprisingly modest, even for a Rutherfordian like LabKitty. You need to know something of (1) matrix algebra and (2) complex numbers (the i2 = -1 kind, not the hateful Cauchy integral conformal mapping kind). If you want to read the rest of Jordan's book, you will also need (3) vectors and (4) probability (the coin-flipping kind, not the hateful measure theory kind), but we will not use these in what follows.
You will also need to be familiar with some physics, or if not familiar than willing to take at face value the concepts we need to get the ball rolling. With this small investment, there is great payoff to be had if just a little imagination and thinking are applied.
C'est ne pas? QM begins with a blackbody, and so we shall too.
Planck's Blackbody Equation
We have an oven. Not the kind of oven things like pizza come out of; more like a block of metal with a cavity hollowed out of the middle and a spy hole drilled in the side so we can peer into the cavity. We heat up the block and measure the light that comes out of the spy hole.
Footnote: You must push out of your head the question of why anyone would think to study this. The blackbody is indeed odd duck in the canonical physics pantheon. Inclined plane, lever, pulley, mass-spring, prism, magnet, hunk of metal we heat up with a cavity hollowed out of the middle and a spy hole drilled in the side. One of these things is not like the others. However, our job is to explain physics, not physicists.
The blackbody is assumed to be in thermal equilibrium, which is something textbooks never bother to define, as if we're expected to have dropped from the womb knowing what it means. I think it means "constant temperature" albeit the temperature itself of no import. In your mind's eye you can see the cavity glowing orange at some temperature or red or white at some other temperature, but what is critical is that the glow is constant in brightness.
Imagine we capture the light coming out of the spy hole and run it though the appropriate piece of expensive gismology to measure the intensity present at each wavelength. If we plot the result, we would get something like this:
Now comes the problem. If you try to derive this plot using classical physics (to be honest, I don't know how to do that and I don't care) you get something that runs off to infinity at the shorter wavelengths (I've shown this as a red trace on the plot). The (wrong) equation you get is called the Rayleigh Jeans equation and the infinity weirdness is known as the ultraviolet catastrophe.
Max Planck fixed Rayleigh Jeans by making the revolutionary leap of logic that energy could only be absorbed or emitted in discrete chunks. This was not yet quantum mechanics, but you could see it from there -- more of a formula concocted to fit the experimental data. Indeed, Planck apparently viewed his result as a quick and dirty patch job, thinking he'd go back someday and fix things properly using classical mechanics. We now ask: Can his leap be derived from first principles?
The answer is yes. Schrödinger and Heisenberg both ran with the idea, independently creating an entirely new physics from which energy quantization appeared as a natural consequence. The rest, as they say, is history. Fast forward fifty years, and this mathematical Frankenstein has escaped the lab and requires a decade of physics graduate school to understand.
No wonder Rutherford hated these guys.
From Pictures to Equations
As the proceedings turn quantitative, there are physics tidbits I must assume if we are to keep the proceedings to a manageable length (yes, this is what I consider a manageable length shut up). In short, you need to recall the concept of an oscillator and the energy thereof. We consider the atoms in the cavity as oscillators (think: tiny masses on springs bobbing in simple harmonic motion). If we were working in sound, the oscillators would be analogous to tiny tuning forks, and we would hear a holy chorus (or perhaps an unholy one) were we to raise our ear to the spy hole. However, our oscillators traffic in light not sound, which adds a few more twists to the story.
Think of heating the cavity as jiggling the oscillators -- an energy input. The blackbody is in thermal equilibrium, so the oscillators must continually output an equal amount of energy (at least I think that's what "thermal equilibrium" means). The energy comes out in the form of light (hot things glow). More accurately, this comes out as a steady stream of photons.
The photon energy tells us the energy of the oscillator from whence it came. The energy of an oscillator is the sum of its potential and kinetic energy. These have standard formulae, established long before QM arrived and still valid after it did. Kinetic energy is 1/2 ⋅ m ⋅ v2. Potential energy is 1/2 ⋅ k ⋅ x2 where the "spring constant" for an oscillator, k, equals mv2 aka m(2πw)2 (this comes from v = √(k/m), which emerges in the solution of the governing ODE. The "v" here is really a Greek nu (frequency), not to be confused with v in 1/2 ⋅ m ⋅ v2 which really is v for velocity. And w should really be a Greek omega. Deal with it).
Putting all the pieces together, we have: oscillator energy = oscillator potential energy + oscillator kinetic energy. In symbols:
e = 1/2 ⋅ m ⋅ v2 + 1/2 ⋅ m(2πw)2 ⋅ x2
We recast this in terms of position (x) and momentum (p = mv) rather than position (x) and velocity (v) because Heisenberg discovered matrices for position and momentum not position and velocity:
e = (1/2m) ⋅ p2 + 1/2 ⋅ m(2πw)2 ⋅ x2
Here is our starting point. It seems reasonable the energy could be any value (within some range). Momentum and position are continuous, so an expression involving the sum of their squares should be also. Not so. As Planck proposed, and Heisenberg derived, and Jordan shows in QMiSMF, and LabKitty recounts below, this is not so. Rather, the oscillator energy can only take on a discrete set of values (one-mississippi, two-mississippi, and so on). Very strange.
If none of the above is intelligible to you at all, then we have reached an impasse. You're not a bad person, but perhaps you would be happier enjoying some other fine LabKitty product, like a cat cartoon, movie review, or funding rant. On the other hand, if I have not yet scared you off, then press on.
A Quantum of Solace
Till here, the physics has been classical. It is now time to bring the quantum thunder. Jordan spends many (many) pages gently introducing the rules of Heisenberg's game, much like Gandalf introduced the dwarves incrementally at the Great Hall instead of all at once, lest Beorn take on his Schrödinger's bear form and become inhospitable.
We don't have time for that. Instead, I will simply list the minimal subset of Heisenberg's postulates we need to continue. A two-way petting zoo, as it were, where you may gaze into the abyss as it gazes into you. When your brain begins to shout this is madness! you must respond this! is! quantum! The correctness of these ravings will be proved momentarily, when we apply them to the blackbody. Out will pour truth.
The Minimum Postulates of Heisenberg's Matrix Mechanics
1) Everything in existence is a matrix. This includes familiar physics things like position and momentum and energy, but also more exotic creatures like spin and magnetic moment and more, which will not concern us here. You probably think things like position and momentum and energy are numbers (or numbers arranged into a vector). That is a lie. The world shown to us by our eyes is just the shadow of puppets. The shadow is the value. The matrix is the puppet. Matrices (representations) appear in equations, numbers (values) appear in laboratory apparatus. The game is learning how to obtain one from the other.
Footnote: I will use uppercase letters to denote matrices and lowercase letters to denote ordinary numbers.
2) We can manipulate the matrices of quantum mechanics like we would any matrix. We can add them, subtract, multiply, invert -- more advanced treatments get into their eigenvalues (we will not). Any manipulation is valid as long as we follow the usual rules of matrix algebra. For example, we can only add or multiply two quantum mechanics matrices if their dimensions permit it.
3) A thing's matrix is not the thing, but properties of a thing's matrix tell us something about the thing. Some relations are intuitive, others are not. Sometimes you can work them out by writing down the components of a matrix explicitly and cranking through pages and pages of algebra. But if you want to sit at the cool kids' table, you don't work with components. Instead, you argue your case using high-level matrix properties until you corner the conclusion you seek like a helpless fox treed by the British aristocracy. Examples of useful high level matrix properties include symmetry and invertibility. The high level matrix properties we require today are embarrassingly modest:
3a) If A and B represent quantities having values a and b, then cA + B represents a quantity having the value c⋅a + b, where c is any constant.
3b) If A and B represent real quantities, then the matrix (A + iB) (A − iB) represents a nonnegative real quantity.
3c) If A and B represent real quantities and C represents a nonnegative real quantity, then the matrix (A + iB) C (A − iB) represents a nonnegative real quantity.
3d) The quantity represented by cI, where I is the identity matrix, has value c, where c is any constant.
Footnote: Jordan proves these claims in chapters 11, 13, and 14.
4) An equation of classical physics remains valid if we replace values appearing in the equation with their matrix representations (and replace constants with an appropriate constant matrix). We now have an expression relating representations instead of values. (Footnote: It's not clear to me if this is always true, but it's how Jordan attacks the oscillator problem.)
The final postulate has a rather different flavor than the others.
5) QP − PQ = ih/2π I, where Q and P are matrices representing momentum and position, respectively, I is the identity matrix, and the scalars are just constants you should recognize.
At its heart, this is a simple statement of noncommutability. Of the momentum and position matrices, but other versions of the equation exists using other representations. Since the RHS is never zero -- although h is numerically tiny -- it would appear matrices in quantum mechanics don't commute. Some do, just not ones like Q and P. If you ever get to the end of a calculation and find you have a Q and P that commute, you have made a mistake somewhere.
That being said, Heisenberg's equation doesn't apply to any old position and momentum. You can't apply the equation, say, to the position of a train leaving Sante Fe at 4 PM and the momentum of light leaving Betelgeuse. It describes quantities that exist in a single system.
A single system like an oscillator in a blackbody.
Footnote: You may be feeling gypped I have not derived this last equation. That's because it cannot be derived. Like F = ma or S = -k ln(W), it is something written into the fabric of the universe and we don't know how or why. We only know (i) Heisenberg sussed it out and (ii) all evidence suggests it's right.
Footnote: It may occur to you that I have not mentioned probability in these postulates. As you probably know, abandoning determinism for a probabilistic worldview is the foundational tenet of quantum mechanics. What gives? It turns out probability is hidden in commutability. Quantities that can be measured with certainty give rise to matrices that commute. Quantities that cannot be measured with certainty give rise to matrices that don't. Q and P do not commute because position and momentum cannot be measured simultaneously with certainty. This is Heisenberg's Uncertainty Principle. Important, yes, but we will not need it in what follows.
Heisenberg's Derivation of Quantized Oscillator Energy
You might think deriving one of the cornerstones of quantum mechanics using psychedelic mathematics would at least be fun, incomprehensible though it may also be. I hate to break the bad news, but it's mostly tedious. You really have to want it. I'm not trying to dissuade you; I'm saying this up front so you know we're in it together. If it helps, remember you're reading the very mind of God, in whatever form you chose to believe such exists. (I like to think Kevin Smith got it mostly right, but my God would have a'sploded Jason Mew's head too.)
Our entryway is Heisenberg's equation, QP − PQ = ih/2π I. I will quickly grow tired of dividing by 2π, so from now on I will write this as QP − PQ = ih I, where "h" is now the funky version people write with a little line through it. I don't know how to do that in HTML so you just get a plain h. Deal with it.
First, we rewrite the expression for the energy of the oscillator we obtained earlier using representation of the momentum (P) and representation of the position (Q) to form a representation of the energy (E). We obtain the matrix equation:
E = (1/2m) P2 + (1/2)mw2 Q2
Next, define two new matrices R and S:
R = Q − (i/mw) P
S = Q + (i/mw) P
The reason for this will become clear later. If you solve these equations for P and Q, you find
P = imw/2 (R − S)
Q = 1/2 (R + S)
Use these to write E in terms of R and S, then solve for RS. (I'm skipping a few lines of algebra to help move things along. It's just straightforward hack and slash):
E = (1/2)mw2 RS + 1/2 hw I
⇒ RS = 2/(mw2) [ E − 1/2 hw I ]
Now use them to write QP − PQ = ih in terms of R and S, and rearrange into something we'll need later:
RS − SR = −2h/mw I
⇒ RS = SR − 2h/mw I
⇒ RSS = SRS − 2h/mw S ⇒ (1/2)mw2 RSS
= (1/2)mw2 [ SRS − 2h/mw S ]
⇒ [(1/2)mw2 RS ] S
= S [ (1/2)mw2 RS ] − hw S ⇒ [ E − 1/2 hw I ] S
= S [ E − 1/2 hw I ] − hw S
⇒ ES = S(E − hw I)
Again, every h should have a little line through it.
Now, a final leap of intuition. We're going to examine a recursion relation for powers of RS. That is, consider:
Rn+1Sn+1 = RnRSSn
By Postulate 3b, RS represents a nonnegative real quantity (this is why R and S were defined the way they are). Since RS represents a nonnegative real quantity, we can show RnRSSn represents a nonnegative real quantity for any n by repeatedly applying Postulate 3c.
Footnote: I put that in bold because it might be important.
What else can we gleen from this recursion relation? We begin at the beginning:
R2S2 = RRSS
= R [RS] S
= R [ 2/(mw2) (E − 1/2 hw I) ] S
= R [ c (E − 1/2 hw I) ] S
The trick is to apply ES = S(E − hw I) to move the S on the right end of the bracketed expression to the left end, picking up an additional hw I term along the way. The quantity 2/(mw2) appears eleventy million times in what follows, so I defined c = 2/(mw2) so I don't have to keep writing it:
= R [ c (ES − 1/2 hw S) ]
= R [ c (S(E − hw I) − 1/2 hw S) ]
= R [ cS (E − hw I − 1/2 hw I) ]
= RS [ c (E − hw I − 1/2 hw I) ]
= c (E − 1/2 hw I) [ c (E − hw I − 1/2 hw I) ]
= c2 (E − 1/2hw I) (E − hw I − 1/2 hw I)
Next up: R3S3 = R2RSS2 = R2[RS]SS. This time you have to apply ES = S(E − hw I) twice because there are two S on the right end. I leave this as an exercise to the reader:
R3S3 = R2[RS]SS
= R2S2 c (E− 2hw I − 1/2 hw I)
= c2 (E − 1/2hw I)
(E − hw I − 1/2 hw I)
c (E− 2hw I − 1/2 hw I)
= c3 (E − 1/2hw I)
(E − hw I − 1/2 hw I)
(E− 2hw I − 1/2 hw I)
If I continued with higher powers until your patience ran out, you (hopefully) would notice the general pattern:
Rn+1Sn+1 = cn+1 Π k=0,1,2,...n [ E − (k+1/2)hw I ]
This is a matrix equation relating the representation on the LHS to the representation on the RHS. If this equation is valid, the rules of quantum mechanics say the value of the LHS is equal to the value of the RHS. If this equation is valid.
The leading constant cn+1 is just a positive number. But consider [ E − (n+1/2) hw I ]. By Postulates 3a and 3d, the value of this representation is [ e − (n+1/2) hw ], where e is the value of the quantity represented by E (i.e., the energy of the oscillator). We don't know what e is, but whatever e is, it's possible to find an integer n sufficiently large such that [ e − (n+1/2)hw ] is negative. This suggests that Rn+1Sn+1 can represent a negative quantity if n is sufficiently large (and generates an odd number of negative terms in the product). But we previously established that Rn+1Sn+1 can only represent a nonnegative quantity. We have a conundrum.
If it were 1923 and you ran into this conundrum for the first time, you would probably assume your theory was broken and head back to the drawing board. Instead, Heisenberg stuck to his guns. There is a way forward.
Consider the product Π k=0,1,2,...n [e − (k+1/2) hw]. This product will never be negative if we only allow e to have values (m+1/2)hw for some integer m. That is, the oscillator energy is quantized. If k < m, the terms are positive and all is well. If k > m, negative terms appear, but that doesn't matter because there will also be a term for k = m and that term will be zero, and the product of the terms will also be zero, and so we never obtain negative energy.
I have just dumped the punchline in your lap: The oscillator energy is quantized. QED.
Jordan includes a blurb from Heisenberg's autobiography in which he recounts the moment of his great discovery. Alone, in the middle of the night, on the island of Heligoland in the North Sea:
I became rather excited, and I began to make countless arithmetical errors. As a result, it was almost three o'clock in the morning before the final result of my computations lay before me. The energy principle had held for all the terms, and I could no longer doubt the mathematical consistency and coherence of the kind of quantum mechanics to which my calculations pointed. At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me.Many miles away, Erwin Schrödinger looks out the window of a comfortable apartment at the empty Berlin streets below, his pregnant mistress sleeping quietly in the next room.
Epilogue
In deriving Planck's result we never once looked inside P or Q, or any matrix at all. However, if you're still reading this (bless your heart), no doubt you are wondering just what the heck P and Q look like. I would be a cruel Juliet indeed to leave thoust so unfulfilled, and so in closing I will show them to you.
They are weird. Even in the mouth of all this madness they are weird. First, they are infinite dimensional. A postulate we did not cover is that if a quantity can have n values, then it is represented by an nxn matrix. An oscillator can have an infinite number of positions and momentums, hence P and Q are infinite dimensional.
Footnote: Working with an infinite matrix is not as terrible as you might think. It's like working with an infinite series -- you play with a finite chunk to grok patterns which (fingers crossed) allows you to suss the infinite result of interest.
So, here they are, your parting gifts. Q and P for the oscillator problem:
The constants out front have some h's and stuff. What's odd is the contents. They're Hermitian, which is something I guess. Beyond that, who knows. Jordan uses them to form R and S like we did earlier, but then takes things element-by-element and applies a few more postulates we didn't cover to derive quantized oscillator energy a second way. It's pretty cool actually, albeit a bit more involved.
The final weirdness may be that although Planck's result is fundamental to quantum mechanics, it isn't one of the harder things to derive using Heisenberg's matrix mechanics. Tedious, yes. Weird, hell yes. But there are far angrier things waiting in the darkness that require far more mastery of the machinery to tame. Why isn't nature straightforward and easier to understand? bemoaned Sin-Itiro Tomonaga, who would share a Nobel prize with Feynman and Julian Schwinger. Onegaishimasu. For all of the drooling Feynman fanbois out there gushing about how simple he made physics, I challenge any of them to explain renormalizable gauge theory.
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