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.
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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]:
Plug in the operators for position and momentum in the position representation:
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:
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
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