Certain features of Dirac’s notation and a physical analog

Important Update on 2015.03.20:

tl;dr version: Don’t bother with this post. It’s in error.

Long version:

On second [and third…] thoughts, I think that this post has turned out to be just bad. (I am being serious here.) Regardless of whatever seeds of some good or promising ideas there may be in it (and I do think there are some), there also are far too many errors or wrong ideas in it, and the errors make the overall description just plain wrong.

If you are interested in knowing which ones I now think are bad or very bad, drop me a line. That is, should you decide to read this post at all, in the first place—something I won’t recommend. The only reason I am keeping this post is to keep a record of how crazy QM can sometimes get to get, especially to me. [Yes, even if I have published a paper on some aspects of the foundations of QM.]

Yet, if you still choose to go through this post, then I would say: OK, go through it, finish reading it, and then come back to this point once again, and think about points like these: (i) Why two chambers? Ideally, there should be only one chamber. (ii) Does the system really model a complex-valued vector and its conjugate correctly? Answer: no. (iii) Does the system model the vector-matrix-vector multiplication right? Answer: no. (iv) Does it even model the multiplication? Answer: no, not really. (v) There also are other inconsistencies.

Of course it’s a fact that as far as QM is concerned, I don’t get to discuss ideas with any one—there is absolutely no informal tossing of ideas back and forth with any one—no fleshing out (or thrashing out) of ideas at the blackboard, gaining clarity as you go on explaining them to someone else (say to a student), nothing. … So, things do get a bit crazy. … Yesterday, I met an engineer friend, and thus had my very first chance to speak with anyone else about the ideas of this post. I could not discuss the QM aspects of it because he hasn’t studied it, but I could at least discuss phasors and conjugates, vectors and matrices, Fourier transforms and waves, etc. I told him the kind of error I thought I was making, and asked him to confirm it. Frankly speaking, he was not sure. He could give me a benefit of doubt because of symmetries, though, being an informal discussion (over a small drink), we let it go at that. But whatever he happened to mention also brought phasors into full focus for me. That was enough to confirm my suspicions. … Finally, today, I decided to put on record the bad points, too.

No, I will not give up attempting to model the Dirac notation via some easily understandable physical analogs. And if I get to something right, I will sure post about it.

That way, these days, I hardly even look at QM (except for browsing of others’ blogs now and then). I am mostly thinking or reading or working something about my other researches—water conservation, CFD, FEM, etc. So, it will be a long while before I could possibly take out some time to get down to thinking about the Dirac notation and all, as my primary thinking goal. And, it can only be after that, that if I at all get something about it consistently right, I could post something about it.

All that I am saying, in the meanwhile, is that no matter how many seeds of some workable ideas this post might otherwise have, the system description in this post is in error. It is bad—bad, even as an analogy. Treat it that way.

Let me not bother with this post any further.

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[Note: I have added a significant update (more like an extension) on 2015.03.19]

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This post follows my browsing of Piotr Migdal’s guest post on John Baez’ blog, here [^], yesterday. Migdal’s aim is make QM simple to understand. He somehow begins with Dirac’s notation, and rapidly comes to stating this formalism:

E = \langle \psi | H | \psi \rangle

I read through about half of this post, and then rapidly browsed through the remaining part, before returning to this formalism and begin thinking a bit about it. … After all, he was doing something about presenting the QM ideas as simply as possible, you know…

Then, an analogy struck me. It’s based on my ideas of QM, of course—remember those pollen grains and the bumping particles and all that stuff which I had written a couple of months ago or so? (On second thoughts, here it is: [^].)

Anyway, let me share with you the analogy that struck me today. If you find something objectionable with it, sure feel free to drop me a line.

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A physical system with a gas-filled cylinder:

Consider a cylinder with two pistons, one at each end, and a rigid, impermeable but movable partition in the middle. Assume that the system is frictionless.

Suppose that both the chambers of the cylinder are of the same length and that both are filled with the ideal gas to the same pressure—some sufficiently low pressure.

Now suppose that the piston on the right hand side (RHS for short) is moved to and fro at a constant angular frequency \nu, a certain maximum displacement A, and a certain initial phase \theta_0. This motion can be specified using a phasor, i.e. a complex number; the phasor rotates in the CCW sense in the abstract phasor plane.

The RHS piston imparts momentum to the gas molecules in the right chamber. The generated sound waves hit the central partition, impart it the momentum, and thus tend to make it move back and forth as well.

But suppose we wish to ensure that the partition in the middle remained stationary. How could we accomplish this goal?

If you were allowed to move the piston on the left, in precisely what way would you move it so that the central partition remained motionless at all times?

Obviously, you would have to move the LHS piston in such a way that its frequency and maximum amplitude are the same as for the RHS piston, viz., the same values as \nu and A. However, the initial phase of the phasor for the LHS piston must be made  -\theta_0 (opposite to that of the RHS piston), and the sense of rotation of the phasor for the LHS piston must be made CW (whereas that for the RHS piston had the CCW sense).

If the pistons were to be linked to the central partition via ideal continuous springs, then the central partition would always remain perfectly standstill.

However, if instead of springs, a gas is used for filling the chambers, then since a gas is made of only a finite number of discrete molecules, the transmission of momentum to the central partition acquires a discrete character. Further, if the molecules are randomly distributed (in terms of either positions, momenta, or both), then the momentum transmission acquires a stochastic character.

As a result, the partition does not remain perfectly standstill at all times, but undergoes a small, random, vibratory motion.

In the terminology deployed by QM, the position of the partition is said to be, you know, uncertain.

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Update the next day (on 2015.03.19)

Let me rapidly note down a few additional points (some of which should be very obvious to many):

(i) Irregular pulses instead of a regular (single) sine wave:

The motion of the RHS piston doesn’t have to be perfectly sinusoidal. Even if the motion is a rather irregular wave (as is the case when one side of a drum is banged), such a motion can always be analysed via the Fourier transform. In other words, |\psi\rangle now has several basis components of different frequencies. Doesn’t matter; just make sure that for each frequency component, the LHS piston too perfectly opposes the motion.

(ii) A system with parallel grooves:

For illustration via a working physical model (or for implementation in a C++ program), I think it could be better to think of the following situation.

Suppose there are ten (/hundred/thousand) straight-line grooves smoothly carved into a horizontal platform. All the grooves are of same width and lie parallel to each other. Suppose, there are several ball-bearing balls placed in each groove (the number per groove may or may not be constant). At the initial time, the balls are placed at randomly different distances. Instead of the RHS piston, we now have a rigid plunger normal to the grooves; it simultaneously moves through the same distance over all the grooves—something like a comb going over some parallel scratches. The middle partition and the LHS piston, too, of course are something like this “comb.” The balls represent the gas molecules. This mechanism makes the one-dimensionality of motion (positions and momenta) inescapable. You can figure out the rest. (For instance, ask yourself what role does the initial speed of a ball has? Does it imply anything towards an independent frequency component, energy, basis vector? Can all balls in a given groove have random initial positions but the same initial speeds, with balls from different grooves differing in speeds? Etc.). You can more easily implement a software program than a build a physical model, to study the behaviour.

(iii) Trying something for the quantum discreteness:

If you wish to go even further, think of having side-walls parallel to and outside of the extreme grooves, and suppose that these walls carry some serrations. Suppose also that the middle partitioning “comb” carries a small ball and a spring (lying in the plane of the comb) in such a way that the comb successively halts only in the valleys of the serrations, The middle partition thus snaps in at discrete positions, say, 0, \pm \Delta x, \pm 2 \Delta x, \cdots, etc., thereby imparting the motion of the partition something like a discrete character.

Finally, if you must have something to stand in for that H symbol, think of a system with two symmetrically placed middle partitions instead of just one—say, one each at \pm x. This gives rise to a system of three chambers. For a system with the ideal gas, insert a sensitive thermometer in the central chamber. It will measure the level of the kinetic energy contained within the central chamber. …

Honestly, though, at least to me, this idea looks like an overkill. After all, the entire system still remains only classical. It merely serves to highlight some of the features of QM—not all.

(iv) What all these systems are good for:

Realize, all the above models are purely classical. None is fully quantum. They do, however, help simplify and bring out certain features of QM.

As far as I am concerned, even a simple C++ program with just two chambers (or parallel grooves with just one partition) might be enough—it will still bring out the the discrete and stochastic momentum-transmission events, and the 1D random walk undergone by the middle partition.

And even this simple a system should bring out many more features of the quantum formalism pretty well… Features like: the necessity of complex numbers in the Dirac notation, the necessity to define the row vectors with complex conjugates, the idea of basis vectors for the column and row vectors, etc.

This is good enough. It is much better than letting your ideas float in an abstract Dirac sea the thin air—thereby making you susceptible for recruitment by many quantum interpretations [^]. The chance that irrational ideas have to grab or overpower your mind is inversely proportional to the clarity which you derive about even simple-looking, basic, concepts. Even a partial clarity can be sometimes good enough. I mean not some half-baked knowledge, but a full clarity on some aspects of a very complex phenomenon. You can always build on it, later.

Bye for now. In the next post, I will return to some notes from my studies of the micro-level water resources engineering.

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A Song I Like:
(Marathi) “jaambhuL pikalyaa jhaaDaakhaali…”
Music: Hridaynath Mangeshkar
Lyrics: N. D. Mahanor
Singer: Asha Bhosale

[E&OE]

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