Re: [math-fun] Dimensional analysis (was Re: Gimbal lock??)
Brad Klee <bradklee@gmail.com> wrote:
If you haven't gone to college, shouldn't you be more inclined to argue with the entitlement crowd? What gives them the right to vote and not you?
Good question. Thanks for asking it. My answer is complicated, and has little to do with math or even physics, so I will respond off-list, probably this weekend. (Anyone else who wants to be CCd on my reply, please email me. Thanks.)
The gist of Ralston's argument is that classical quantities energy and mass are not useful in the quantum regime, and that they should not appear in formal equations of quantum mechanics, such as Schroedinger equation.
My understanding is that mass and energy are equally useful concepts at every scale. The only major issues I can think of offhand are whether mass and rest mass should be distinguished from each other, and whether the energy of the vacuum can really be usefully defined as zero.
Now if we measure photoelectric nu2 on a Hydrogen standard nu1, we get photoelectric band gap E2 relative to hydrogen standard E1, *without mentioning Planck's constant*.
For any concept, no matter how useful, we can always work around it, using circumlocutions. In some cases, which things are considered fundamental is completely arbitrary. In other cases, working around a lack would be as awkward as abolishing money and using barter instead. I think the Planck constant is somewhere between those two extremes.
That is to say, the photo-electric effect is another version of the Rydberg Hydrogen experiment (with a hell of a lot more uncontrolled variables and uncertainties. Also with less of an intelligible level structure.).
We could certainly use the Rydberg constant in place of the Planck constant, but most equations would be more unwieldy.
Once the hangup with photoelectric effect is sidestepped, there is still often an objection about electron mass. This is a concept that makes some amount of sense in terms of the oil drop experiment.
Are you sure? I thought Millikan was balancing the force from the electron's *charge* in a known electric field against the force from the *oil drop's* mass in a known gravitational field. You can't get the electron's mass from that. Measurement of how much a moving isolated electron's trajectory is bent by a known electric or magnetic field gives you the electron's charge-to-mass ratio. (And, as a side effect, invents television.) And if you already know the charge, it gives you the electron's mass. Assuming those two experiments were the only ways of measuring those quantities, the electron's charge-to-mass ratio would always be known to a higher precision than its mass. Similarly, each planet's GM (product of the gravitational constant and the planet's mass) is known to a much higher precision than either G or M alone. I assume NASA uses each planet's GM for navigation. But that's no reason to deprecate the concept of a planet's mass. Or of an electron's rest mass.
Again, this is not a direct measurement of electron mass, it is an inference of electron mass from classical physics involving gravitational and electric fields. Thus the extracted parameter is a mess of other constants besides m_e.
I didn't know there was any issue about an electron's rest mass (other than, as with every measured quantity, the measurement having some error). There is an issue about its charge being larger close up, apparently without limit. It can be modeled as an extreme negative charge surrounded by a cloud of positive charge which almost but not quite cancels it out. I can understand objecting to the concept of the classical electron radius, which comes from assuming the electron's rest mass "comes from" the energy it would classically take to assemble infinitely many infinitesimal like charges (whose total charge sums to that of an electron) from an infinite distance into a small radius. There's a unique value for which the numbers balance, and that's the classical electron radius. Very likely it has no physical significance whatsoever.
Section 2 of the paper does an admirable job showing that Planck's constant (1900) was over-promoted relative to the fine structure constant (Sommerfeld 1916), and and the Compton wavelength (1923). It all drives to the conclusion of equation (24) that: including Planck's constant and electron mass introduces an extra, perhaps delusional, degree of freedom.
I'm not convinced that it's meaningful to count degrees of freedom in a system in which things can't vary.
In electron experiments, we only need Compton wavelength to set scale, and fine structure constant to search through the perturbation hierarchy (alpha as an expansion parameter is discussed in most QM textbooks).
Perhaps we can do without the Compton wavelength too, and only count dimensionless parameters as degrees of freedom.
This analysis of quantum pre-history doesn't exactly answer the question about generality.
It would certainly be nice if we could contact another advanced civilization and compare notes. I would bet that they would also have the concept of the Planck constant. And the concept of prime numbers. I'd love to be proven wrong. I'm reminded of the Hugo and Nebula winning "Story of Your Life," a 1998 science fiction story by Ted Chiang. A race of aliens turn out to be familiar with variational principles in physics, but completely ignorant of cause-and-effect explanations, not just in physics, but also in daily life. (In 2016 it was made into a movie, "Arrival.")
PS. QM has "Symplectic symmetry", so position and momentum are treated as fully equivalent variables.
They're equivalent in some senses, but not in the sense of being the same thing. Whether a person is guilty of trespassing depends entirely on his position, and not at all on his momentum.
The Schroedinger EQ itself can be written in either position or momentum basis, and nothing much changes.
The uncertainty in one is the Fourier transform of the uncertainty in the other (if you ignore signs -- the FT of an FT is not the original function, but its negative). Similarly with looking at radio signals in the time domain vs. the frequency domain. An FT simply rotates a waterfall display by 90 degrees. But that doesn't mean that time and frequency are the same thing.
Anyhow, the Planck constant would still exist, though it would become a mere conversion factor, the number of hertz in a joule, ...
Of course I meant the number of joules in a hertz. I'll agree that it's just as technically accurate to say the Titanic broadcast its distress call at 500 kilohertz or at 0.00033 yoctojoules. (Sorry, there's no prefix smaller than yocto.) Actually, both are ahistorical, as they called it 600 meters. Or rather metres, as it was a British ship. Sorry I'm rambling. It's past my bedtime.
Hi Keith, Thanks for the correction on electron mass. My argument was loose on that point. In general, mass / charge comes from similar mass spectrometry experiments--the most accessible of which is probably the deflection of faucet water by a static charge. However, an approach which needs two classical experiments to determine m_e makes the argument about "mess of other constants" even stronger. Ralston's article also mentions penning traps on page 32-33 (more material that could possibly be given earlier). One talking point is that the most precise determination of e/m_e still deals with a "small cloud" rather than an "single electron wave". I would never recommend using Rydberg constant in place of Planck's constant, and strongly disagree that Rydberg constant or Compton wavelength could be eliminated from the theory. Up to powers of alpha, only one of the two is necessary, but we must have at least one of these two, as soon as we have dispensed with Planck h and electron mass. There is a strong analogy between Electron diffraction through a crystal and light diffraction through a thin slit grating. This analogy is discussed pragmatically in Ch. 5 of: http://www.diyphysics.com/book/book-contents/ As you probably know, light wavelength is a property that controls the width of a far-field diffraction pattern, so we need a full, continuous spectrum of wavelengths to describe light diffraction experiments ( which are easy to perform using: cd's, dvd's, blu-rays, butterfly wings, or a notecard w/ a small puncture + laser diodes ). For matter waves, as far as we know, there is no spectrum, but instead quantum values. These quantum wavelengths also show up in S.E. in place of masses, once they have been eliminated with Planck h. As for your comment about trespassing, I think you missed the mark. The S.E. determines wavefunctions and their evolution. You can have a momentum space wavefunction, and still measure position as long as you choose the correct form of the position operator. This goes back to the original discussion... Position space vs. Momentum space, the wave vectors & adjoint wave vectors are dimensionless, with all dimensions carried on the operator, a "quantum metric", if you will. Say that we know either distribution |q) or |p) in either position or momentum space, i.e. we know either the position or the momentum of a defendant in a bound state. We can then build an observable Q, meaning "in the room where it happened", and calculate (q|Q|q) or (p|Q|p). However this would not be good evidence for a classical trial, because of the probabilistic interpretation of quantum mechanics. It's reasonable to assume that the wavefunction has a tail "outside of the room where it happened", and this is enough to prove "reasonable doubt". If we are talking about a trial in present times, I'm afraid there's an argument to say that even Classical Mechanics is out the window, and that the only thing that matters to judge and jury is the relative strength or weakness of some particular identity values. --Brad PS. I did not, but should, read "story of your life"; however, my most popular ever article was written after the movie "Arrival", more than 20,000 views, ha ha ha: https://community.wolfram.com/groups/-/m/t/1034626 We'll have to see what happens with Dune. On Tue, Feb 4, 2020 at 11:28 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Brad Klee <bradklee@gmail.com> wrote:
If you haven't gone to college, shouldn't you be more inclined to argue with the entitlement crowd? What gives them the right to vote and not you?
Good question. Thanks for asking it. My answer is complicated, and has little to do with math or even physics, so I will respond off-list, probably this weekend. (Anyone else who wants to be CCd on my reply, please email me. Thanks.)
The gist of Ralston's argument is that classical quantities energy and mass are not useful in the quantum regime, and that they should not appear in formal equations of quantum mechanics, such as Schroedinger equation.
My understanding is that mass and energy are equally useful concepts at every scale. The only major issues I can think of offhand are whether mass and rest mass should be distinguished from each other, and whether the energy of the vacuum can really be usefully defined as zero.
Now if we measure photoelectric nu2 on a Hydrogen standard nu1, we get photoelectric band gap E2 relative to hydrogen standard E1, *without mentioning Planck's constant*.
For any concept, no matter how useful, we can always work around it, using circumlocutions. In some cases, which things are considered fundamental is completely arbitrary. In other cases, working around a lack would be as awkward as abolishing money and using barter instead. I think the Planck constant is somewhere between those two extremes.
That is to say, the photo-electric effect is another version of the Rydberg Hydrogen experiment (with a hell of a lot more uncontrolled variables and uncertainties. Also with less of an intelligible level structure.).
We could certainly use the Rydberg constant in place of the Planck constant, but most equations would be more unwieldy.
Once the hangup with photoelectric effect is sidestepped, there is still often an objection about electron mass. This is a concept that makes some amount of sense in terms of the oil drop experiment.
Are you sure? I thought Millikan was balancing the force from the electron's *charge* in a known electric field against the force from the *oil drop's* mass in a known gravitational field. You can't get the electron's mass from that.
Measurement of how much a moving isolated electron's trajectory is bent by a known electric or magnetic field gives you the electron's charge-to-mass ratio. (And, as a side effect, invents television.) And if you already know the charge, it gives you the electron's mass.
Assuming those two experiments were the only ways of measuring those quantities, the electron's charge-to-mass ratio would always be known to a higher precision than its mass. Similarly, each planet's GM (product of the gravitational constant and the planet's mass) is known to a much higher precision than either G or M alone. I assume NASA uses each planet's GM for navigation. But that's no reason to deprecate the concept of a planet's mass. Or of an electron's rest mass.
Again, this is not a direct measurement of electron mass, it is an inference of electron mass from classical physics involving gravitational and electric fields. Thus the extracted parameter is a mess of other constants besides m_e.
I didn't know there was any issue about an electron's rest mass (other than, as with every measured quantity, the measurement having some error). There is an issue about its charge being larger close up, apparently without limit. It can be modeled as an extreme negative charge surrounded by a cloud of positive charge which almost but not quite cancels it out.
I can understand objecting to the concept of the classical electron radius, which comes from assuming the electron's rest mass "comes from" the energy it would classically take to assemble infinitely many infinitesimal like charges (whose total charge sums to that of an electron) from an infinite distance into a small radius. There's a unique value for which the numbers balance, and that's the classical electron radius. Very likely it has no physical significance whatsoever.
Section 2 of the paper does an admirable job showing that Planck's constant (1900) was over-promoted relative to the fine structure constant (Sommerfeld 1916), and and the Compton wavelength (1923). It all drives to the conclusion of equation (24) that: including Planck's constant and electron mass introduces an extra, perhaps delusional, degree of freedom.
I'm not convinced that it's meaningful to count degrees of freedom in a system in which things can't vary.
In electron experiments, we only need Compton wavelength to set scale, and fine structure constant to search through the perturbation hierarchy (alpha as an expansion parameter is discussed in most QM textbooks).
Perhaps we can do without the Compton wavelength too, and only count dimensionless parameters as degrees of freedom.
This analysis of quantum pre-history doesn't exactly answer the question about generality.
It would certainly be nice if we could contact another advanced civilization and compare notes. I would bet that they would also have the concept of the Planck constant. And the concept of prime numbers. I'd love to be proven wrong.
I'm reminded of the Hugo and Nebula winning "Story of Your Life," a 1998 science fiction story by Ted Chiang. A race of aliens turn out to be familiar with variational principles in physics, but completely ignorant of cause-and-effect explanations, not just in physics, but also in daily life. (In 2016 it was made into a movie, "Arrival.")
PS. QM has "Symplectic symmetry", so position and momentum are treated as fully equivalent variables.
They're equivalent in some senses, but not in the sense of being the same thing. Whether a person is guilty of trespassing depends entirely on his position, and not at all on his momentum.
The Schroedinger EQ itself can be written in either position or momentum basis, and nothing much changes.
The uncertainty in one is the Fourier transform of the uncertainty in the other (if you ignore signs -- the FT of an FT is not the original function, but its negative). Similarly with looking at radio signals in the time domain vs. the frequency domain. An FT simply rotates a waterfall display by 90 degrees. But that doesn't mean that time and frequency are the same thing.
Anyhow, the Planck constant would still exist, though it would become a mere conversion factor, the number of hertz in a joule, ...
Of course I meant the number of joules in a hertz.
I'll agree that it's just as technically accurate to say the Titanic broadcast its distress call at 500 kilohertz or at 0.00033 yoctojoules. (Sorry, there's no prefix smaller than yocto.) Actually, both are ahistorical, as they called it 600 meters. Or rather metres, as it was a British ship.
Sorry I'm rambling. It's past my bedtime.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
-
Brad Klee -
Keith F. Lynch