The Certainty and Uncertainty Principles
By Doug Sweetser | July 5th 2011 12:43 AM | 15 comments | Print | E-mail | Track Comments

Trying to be a semi-pro amateur physicist (yes I accept special relativity is right!). I _had_ my own effort to unify gravity with other forces in...

View Doug's Profile
The game of quantum mechanics is rigged, but I will let you in on the rules. The wave function has all the information usually written as complex values. Operators are used to grab out different values from the wave function. Once you agree to those ground rules, the uncertainty principle and its Silent Bob twin the certainty principle are set in stone. It is the product rule of calculus that does it all. It is not the Wizard of Schrödinger behind the curtain, it is Isaac Newton or his not so silent contemporary Gottfried Leibniz.

The Uncertainty Principle gets all the press. Philosophers can construct a career thinking precisely about fuzziness. They will advance farther thinking fuzzily about fuzziness. I am not a fan of philosophy for the simplest of reasons: no school of philosophy has led to a change in an equation. Equations are the bottom line in physics. In business, things that don’t help the bottom line are sold off. Philosophers cannot be divested so I let them play by themselves. Physics is the art of the deep one liner.
Click or skip a reading of this blog:
The wave function maps the odds of some system of being in a particular state into the complex numbers. The wave function is already naked and in bed with probability, so uncertainty is certain. Squaring the probability amplitude is the way to make it real to an observer.

Why is there a need for the complex numbers? The reason can be found in the wonderful interference properties that make quantum mechanics weird. The math for the interference pattern of a coherent source that passes through two slits requires complex numbers. That is reason enough for me. A far more sophisticated analysis was worked out in 1936 by Birkhoff and Von Neumann. That paper was too technical for me to digest. I relied on an executive summary by Stephen Adler. He said they said quantum mechanics can be done over the real, complex numbers, or quaternions. Adler even wrote the book, “Quaternionic Quantum Mechanics”, \$265 bucks on Amazon, ouch, use a library. It was Adler who claimed that quantum mechanics over the reals would not have the interference effects. His research project has not completely panned out (and I will skip any further discussion of it here).

Why square complex numbers, whatever they are? Readers of this series of blogs will hopefully recall I have an unusual take on complex numbers: the real part is time while the imaginary part is space (a definition which extends to other 4-vectors like energy and 3-momentum). Observers camp out at the same address, zero, zero, zero, zero, no matter what the choice of coordinates. When they see something it is at a later time t, but the same spacial location, or (t, 0, 0, 0). Squaring the probability amplitude give the odds the observer will see the event while camped out at the spacial origin which travels with the observer.

Operators wear a hat in the nomenclature. What an operator does in detail depends on the choice of the representation. In the position representation, the position x operator, \hat{x}, plucks out the value along the x axis. The momentum operator gets more complicated, using a derivative. Sometimes people work in the momentum representation. In that case momentum is easy, while position requires more calculus.

I must be open about three fetishes I have. First, I like to write equations in a dimensionless form. By being dimensionless, equations result in plain old numbers so subject to all the tools of mathematics. Trig functions live on a diet of pure numbers (think of their Taylor series). Second, I work with quaternions which contain 3 imaginary basis vectors, so I don’t have to write i ever. It is great to have i built into the algebra. Third, I always write the wave function. This is part of my dot all i’s, cross all t’s rules of accounting: if an operator only makes sense operating on a wave function, then write out said wave function. I won’t go into my forth fetish, but no small farm animals are harmed in the process.

A commutator of two operators position and momentum in the x direction is [corrected a spurious factor of 1/2]:
$[\hat{X}, \hat{P_x}] \psi = (\hat{X} \hat{P_x} - \hat{P_x} \hat{X}) \psi \quad eq. ~ 1$

Plug in the operators for position and momentum in the position representation:
\begin{align*}[\hat{X}, \hat{P_x}] \psi &= -x \frac{\partial \psi}{\partial x} + \frac{\partial (x \psi)}{\partial x} \quad eq. ~ 2\\ &= -x \frac{\partial \psi}{\partial x} + x \frac{\partial \psi}{\partial x} + \psi \frac{\partial x}{\partial x} \\ &= \psi \end{align*}
Notice this is an exact relation. It is non-zero because of the product rule known to Newton and Leibniz.

Here was a fun realization I had while writing this blog. I was doing something with the position and momentum operators, and there was no inequality. Did I mess up? Nope. The uncertainty principle uses only one particular ordering of the two operators. The basic idea is that the square of the position operator and the square of the momentum operator are going to be equal to or greater than the square of the commutator. At its smallest, the uncertainty principle for position in x and momentum along x equals the commutator. Yet the uncertainty principle could be bigger. The funny factor of a half comes from the commutator too. Sages in quantum mechanics emphasize commutators over the uncertainties: the former are exact, the latter have lower bounds equal to the former.

In this blog, I will call [X, Px] the uncertainty commutator. The fact that it is not equal to zero leads to the uncertainty relation between position along an axis and momentum along the same axis.

Repeat this exercise for position along the x axis and momentum along the y axis:

\begin{align*} [\hat{X}, \hat{P_y}] \psi &= -x \frac{\partial \psi}{\partial y} + \frac{\partial (x \psi)}{\partial y} \quad eq. ~ 3\\ &= -x \frac{\partial \psi}{\partial y} + x \frac{\partial \psi}{\partial y} + \psi \frac{\partial x}{\partial y} \\ &= 0 \end{align*}
Because the coordinates are orthogonal, this commutator is equal to zero. That is what people call this: orthogonality. That sounds big and fancy and utterly unrelated to the uncertainty principle. Compare equations 2 and 3. The only thing that changes is the differential element, a dy steps in for dx.

In this blog, I will call [X, Py] the certainty commutator. The fact that it is equal to zero means we can measure both to arbitrary accuracy.

I have done nothing new in this blog. What I strive to do is push ideas closer together so my brain can chew and swallow a smaller idea. The greater than sign found in the uncertainty principle is an indication people don’t want to teach about commutators. Here, the focus shifts to commutators. Conjugate variables like x position and momentum along x mean that their expressions in terms of calculus will have a non-zero product rule. That sounds unfuzzy, a good thing.

The next time you get trapped by a philosopher type ready to go on a half hour ramble about the uncertainty principle, ask him how he explains certainty in pairs of measurements that appear just as often in quantum mechanics. It is a safe bet he will never mention the product rule of calculus.

Doug

Snarky puzzle. Where’s my cow bell? I want more cow bell. People feel lost without their hbar. Add it back to eq. 1. Discuss the silliness of the exercise.

Bonus problem: Make the relationship between position and momentum operators complete. Figure out what is missing - the minus sign in the commutator should give a big clue. When added back in, the greater than sign can be resigned to bad methods in education that will continue for years to come.

Next Monday/Tuesday: Julian Barbour and Me

"I am not a fan of philosophy for the simplest of reasons: no school of philosophy has led to a change in an equation." Apparently, you do not understand that epistemology, a branch of philosophy, underlies all mathematics. Equations are simply models that try to predict reality; they are not reality themselves. Equations are always preceded by simplifying assumptions. Data is always, always, imprecise, so you can never have perfect knowledge of existing conditions on which to base your predictions, so your predictions cannot be perfectly accurate. So, actually, philosophy underlies every equation ever written.

I am indifferent to whether yours (philosophy) is bigger than mine (mathematical physics). My question to someone more schooled than I in philosophy is precise: since the 1950's, has any philosophy ever altered an equation that appears in the canons of mathematical physics?

Physicists are quite precise about imprecision. That is much of what the data modeling in big atom smashers is all about. They formally explore variations in the assumptions of their models. Those folks work with code and super computer networks, not epistemology.

Reality is reality. Equations are equations. Equations are not reality. OK. There are limitations to equations. OK. "So, actually, philosophy underlies ever equation ever written." Let's grant you that because it is interesting. So what? How are every equation every written altered? It give philosophers something to do perhaps, but since I can just assume it is true for a new equation I write down, I don't have to give it any attention. I do have to pay attention to factors of 1/2. I am going to stick with Door Number 1.
" My question to someone more schooled than I in philosophy is precise: since the 1950's, has any philosophy ever altered an equation that appears in the canons of mathematical physics?"

If you had not said "since the 1950s" I would've said David Hume. Einstein said that Hume's arguments against the a priori inspired SR more than other scientists (But he also studied the philosophy of Poincare and Mach); allowing him to overcome the unfounded commonsense assumptions about how the worldmust be. At the time, most of the best scientists had acknowldged philosophical prejudices that guided their work (Bohr and the Copenhagen Interpretation, etc.). It is difficult to say what phiosophy influenced the creation of an equation. It would require historical anaylsis, but everyone has a philosophy that guides their work whether they realize it or not. They have assumptions about how things are in nature (a metaphysics) and what should and shouldn't be relevent to investigations (an epistemology). Your philosophy is an anti-philosophy philosophy.

Catch-22.
Wigner spoke of the unreasonable effectiveness of math in the natural sciences. I think it was Weinberg who spoke of the unreasonable ineffectiveness of philosophy in the natural science. But I'm afraid the philosophy of which Weinberg spoke is a physicist's version or interpretation of philosophy, not a philosopher's version. The former is often a caricature of the latter, which is why it would be a good thing if physicists just stuck with the math and the minimum required to relate it to their lab equipment.

Example: Kant has been blamed by physicists (who never read him) and philosophers of science (who did not understand him) for delaying the discovery of non-Euclidean geometry. Why? Because he supposedly claimed that Euclidean geometry was a priori certain. What Kant's arguments actually establish is that any geometry is a priori certain; so he might as well be interpreted as making room for non-Euclidean geometries in physics. (Later, after the discovery of non-Euclidean geometries, Poincaré added to this that one geometry could not be more true than another; it could only be more convenient.)
everyone has a philosophy that guides their work whether they realize it or not. They have assumptions about how things are in nature (a metaphysics) and what should and shouldn't be relevent to investigations (an epistemology).
Absolutely. I would have said that science cannot be divorced from metaphysics because it presupposes a metaphysical framework in which questions are formulated and answers are interpreted. The appropriateness of the questions and the correctness of the interpretations is not something that can be tested empirically, though one set of questions or one interpretation can make more sense than another. Needless to say, what makes sense to me may not make sense to you. Depending on one's intellectual development, this either leads to war or makes life more interesting.
In Anthony Zee's great video QFT lectures (available online), he tell his students that prejudices about the nature of the world and physics are important to. They are useful and should be developed. This seems to me to be saying nothing more than a physicist should have a philosophy.

I knew that Einstein was interested in philosophy, but I didn't appreciate how important it was to him until I read Isaacson's biography.

"The theory of relativity suggests itself in positivism. This line of thought had a great influence on my efforts, most specifically Mach and even more so Hume, whose Treatise on Human Nature I studied avidly and with admiration shortly before discovering the theory of relatvity." -Einstein (from Walter Isaacon's "Einstein" page 82)

You've got a miscreant factor of 1/2 in your commutator (eq 1)
Equation 1 was modified, and added the comment [corrected a spurious factor of 1/2]Thanks.
"Mathematical physics" has been founded on the philosophical work of many of the past's thinkers, e.g. Francis Bacon, Sir Isaac Newton, and, as another commenter suggested, David Hume. Add to this René Descartes and many other mathematicians who were also philosophers. The entire structure of modern mathematics is an evolution of epistemology and metaphysics. What equations have changed since 1950 as a result of philosophy? Bogus criterion. (Newton's work in kinematics and calculus is invalid because it hasn't changed since 1950? Nonsense!). Those equations which have been adapted to new discoveries were adapted because of the underlying philosophical/epistemological principles inherent in the entire field of mathematics.

What equations have changed since 1950 as a result of philosophy? Bogus criterion.
People usually reject my well defined tests, particularly ones they fail. By the way, that is fine, you should not feel insulted in any way. All it does is clearly define our differences. I accept you do not accept my criterion. This is directed less at you whose opinion I don't care if I change, and more at someone else reading these comments.

Go back far enough in our culture's history, and the words we use today don't quite make sense. Sir Isaac Newton did not have a Ph.D., and I believe he referred to his work as Natural Philosophy, not physics. Unlike our high achievers in physics today, he did not stay on the subject, doing much work in other subjects, ending up at the mint. Stephen Hawking might have Newton's chair at Cambridge, but I doubt he could switch to head of the mint.

I am a fan of John Steward Mill, Mr. Utilitarianism. That has nothing to do with this post, nor with the correction I made to the commutator.

I don't mean to imply that Newton's work is somehow invalid. That would be nonsense.

This reminds me of a line of logic used by Steven Weinberg in his book "Dreams of a Final Theory: The Search for the Fundamental Laws of Nature". He said that all things are made up of subatomic particles, so subatomic physics is the most fundamental theory in science. So true, so useless. I never needed to know the difference between a meson and a hadron when ligating DNA together in eppendorf tubes for molecular biology studies of leprosy. I have not reflected on any issues between philosophers when trying to get Mathematica to confirm or deny the line of logic in a collection of equations I had worked out by hand.
I am not a fan of philosophy for the simplest of reasons: no school of philosophy has led to a change in an equation. Equations are the bottom line in physics. In business, things that don’t help the bottom line are sold off.
It is not the business of philosophy to change equations. It is the business of philosophy to make sense of them. There is more to physics than equations, for instance the tricky business of making sense of equations. Physics is not economics. (Rhetorical sleight-of-hand: pretend that "bottom line" means the same in both physics and economics.)
Readers of this series of blogs will hopefully recall I have an unusual take on complex numbers: the real part is time while the imaginary part is space (a definition which extends to other 4-vectors like energy and 3-momentum)....
Your physical interpretation of the complex numbers is not a philosophy? Don't make me laugh.
I am indifferent to whether yours (philosophy) is bigger than mine (mathematical physics).
If you think of mathematical physics as a philosophy, it looks as if you don't know the meaning of philosophy.
I don't have to give it any attention. I do have to pay attention to factors of 1/2.
That of course is true.
Complex numbers are used in a variety of ways by a variety of users. In modelling electrical quantities, for example, the "real" part represents resistance or "real" power, while the "imaginary" part represents reactance or "reactive" power. In other contexts, they may represent time relationships between various phaser or vector quantities. In electromagnetic field theory, two dimensions are not enough, so field quantities may be represented with eigenvalue vector sets, i, j, k etc. What this is intended to say is there is no one correct meaning for complex numbers; the applications are numerous. Such is the nature of mathematical modelling tools: they can be applied many ways, so long as the underlying "rules" of operators are applied consistently.
On my bookshelf at home is a volume titled The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, by Max Jammer, who apparently fully understands the point I've been trying to drive home: mathematical physics by any name is still very much founded in philosophy.

"Bogus Criterion"

Since 1950 physics hasn't made any major conceptual paradigm shift analogous to the shift from classical to quantum mechanics. If a shift like that occurs again, it will be probably be achieved by a philosophically minded physicist. One that questions the metaphysics of our knowledge (for example, assumptions about the flow of time or causality that are assumed by our present equations. Consider Julian Barbour or V. Stengers pictures of a "timeless universe." These ideas are based on philosophical reflection; not merely fiddling with the equations.) Good philosophy that is rational (i.e. not postmodernism) could very well help you. And there isn't a sharp line between science and philosophy. Consider the question, "what is a measurement?"

One of the first books on physics I read (after Rudy Rucker's "Geometry and the Fourth Dimension" was Reichenbach's Philosophy of Space and Time. It was excellent and I can't imagine it was harmful to my intellect or a waste of time. Such work shouldn't be dismissed lightly.

Since 1950 physics hasn't made any major conceptual paradigm shift analogous to the shift from classical to quantum mechanics. If a shift like that occurs again, it will be probably be achieved by a philosophically minded physicist.
Guess this will not be me (a safe bet, since almost no one makes that significant a contribution to science). I did study Thomas Kuhn in high school. It was a fun read, fun to think about. Here I am a few decades later, at least making a public effort to do research in physics. I have rejected his model for scientific change. Instead, I use science to do science. Several people will claim that is a philosophy, but I don't think philosophers deserve the credit, scientists do.

One of the more creative elements of science is evolution. That is the main driver behind my method. In evolution, you are mostly like your parents, right down to transfer RNAs used decode messenger RNAs into proteins. You are not quite like your parents, a bit is scrambled in the process. If something works, then do it again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and again and agin and again and again and again and again and again and again and again and again, without cheating (I did type that last sentence, not a repeated cute and paste). Most people don't appreciate the scale of repetition in evolution. In my research, I have tried to rewrite equations using quaternions again and again. There are variations made in the process. And I go back, and most people get bored quickly and move on.

New physics requires new math. Newton co-invented calculus to do classical physics. Einstein used the Lorentz group in 4D spacetime (it took more mathematical folks to hang those labels on his work). General relativity is non-Euclidean geometry at work. Quantum mechanics uses Hilbert spaces. Feynman diagrams are used to do perturbation expansions. Chaos theory (post 1950) uses fractals. The only reason I have non-zero odds of succeeding is because I have never seen analytical animations before, and they have grand potential I am willing to bet my time and money on (oops, I have little time or money, so it goes).

One can read lots about the flow of time. I gave my answer in an earlier blog:
$R=(-t^2+x^2/c^2+y^2/c^2+z^2/c^2, 2 \, t \, x, 2 \, t \, y, 2 \, t \, z) / (t^2+x^2+y^2+z^2)$
If t >>> x, y, z:

$R\approx (-1, \vec{0})$

Is the idea well-developed enough to win over the world? Nope. It is precise, and I can play with it. I will deal with Julian Barbour's nice insights, but silly answers next week.
"New physics requires new math... Einstein used the Lorentz group in 4D spacetime (it took more mathematical folks to hang those labels on his work). General relativity is non-Euclidean geometry at work."

But it was Einstein's personal philosophy about how the world should be (Mach's Principle and General Covariance) that lead him to the mathematics of Differential Geometry. He learned, from his friend Grassmann (who also helped him pass his classes in college by providing Einstein with notes for the classes he skipped) that Geometry was capable of forming invariant equations using tensors. Maybe, he wouldn't have thought to use Differential Geometry if he didnt have this metaphysics. That was what he needed. But there isn't a clear distinction between philosophical and scientific research. Minkowski's work showing that Special Relativity is described by a flat space also inspired the conjecture that curved space may be gravitational. I wouldn't consider his work philosophical.

"Guess this will not be me (a safe bet, since almost no one makes that significant a contribution to science)"

Indeed, the world needs what Kuhn would call "Normal Scientists" who operate within the confines of their era. But it's fun to speculate too (realizing that you will probably be wrong).

" I have rejected his model for scientific change"

Isn't it slightly better than the classic explanation of scientific process? Observation, Theory, Experiment, Repeat. When you look closely at history it seems more complicated than that. But some Kuhnians appear to be saying there is really no scientific process at all. Absurd.

You might enjoy Reichenbach's The Direction of Time.

I'll check that blog post out.