People aren't perfect lab Summer 1974

Father's Day '77 and a 2-D complex vector space

I found an example from electrical engineering of a complex two-dimensional vector space.  If you already dislike math, you won't be too happy with these two pages from Circuits, Devices and Systems, © 1976, by Ralph J. Smith, used for a beginning EE class I took in the first summer term of 1977 at the University of Arkansas at Little Rock.

 

I'm not sure if engineers still prefer to use the letter j instead of the usual letter i in writing a complex number, but they preferred it back in the seventies, so that's what the j is doing here (below), representing the imaginary part of a complex number.  Vectors may be written in different equivalent ways, and one of those ways is to use complex numbers, which have a real (x-axis) part and an imaginary (y-axis) part.  For example, in the number 8.66 - j5  (see below) the real part is 8.66 and the imaginary part is 5. 

What this author says about electrical engineering is also true of quantum mechanics.  We have electronic devices (especially iPod, iPhone, iPad, etc) that depend on recently-discovered quantum effects, but in studying how these devices work, and in making the discoveries they are based on, we need abstract ideas.  (Very abstract, not to mention formal, ideas!  That's what I'm trying to make more sense of in these writings of mine.  Quantum theory is too abstract to understand as it stands.)

Yeh, those are my comments in the margin from 36 summers ago concerning TV ownership.

Our vector in this example, below, is called a phasor, a term used in physics as well as engineering, and also in Star Trek, but with a different meaning.  Electrical voltages and currents in alternating current (AC) circuits are generally not "in phase" with each other.  A phasor has a voltage or current as its "amplitude," and it has a phase angle ("phase" for short).  The variation in time of amplitudes and phases of different currents and voltages can be kept track of by the use of phasors.

The "Imaginary" and "Real" axes make this graph (above)  a complex two-dimensional vector space.  The variation in time of  60 Hz, by the way, is the standard frequency of household AC voltage.  Whatever the frequency is, Kenneth, it must be the same for both vectors in order for them to be represented in this simple way.

So, the amplitudes and the phase angles of the two vectors (above) are what are important. And here's the importance of my choosing this example:  Amplitude and phase are what are important in quantum mechanics. They take on strange new significance:  In quantum measurements, an "amplitude" becomes a "probability amplitude," and phase angles between vectors become those "delicate phase relationships" that are responsible for the weirdness of quantum superposition.

But how does this relate to Father's Day of 1977?   Well, I was enrolled in the EE class and reading and working problems in this textbook then.  Also I'd recently found a job as the 4-pm-to-midnight night watchman at the Old State House in Little Rock.  So that's a couple of possible reasons I'm remembering this Father's Day so well.

My family had lunch at The Embers (actually the Plantation Embers), a white-table-cloth restaurant in a better-than-average highway motel called the Pine Bluff Motel.  I usually got roast lamb with mint jelly when we ate there, so that's probably what I ate for lunch.  And my gift for Dad on that Father's Day was a copy of the New York Times Book Review.

 

This may be another reason I remember that day better than other Father's Days.  I had just discovered that the Book Review by itself was available at Publisher's Bookshop in Little Rock.  Back then, it was impossible in Little Rock to get a copy of the New York Times on the day it came out. It took until Tuesday or Wednesday for the Sunday NYT to arrive by truck and be available for sale in LR.

When I handed a copy of the Book Review to Dad at The Embers, I said it was not an expensive gift but also not an easy one to get.  He said something like "Is this last week's?"  I had the pleasure of saying no, it was the current issue.  Whether he knew about the early availability of the Book Review, I don't know.  But he sure didn't let on.  Dad and I had been to New York together (on our way to Rome) the previous October.  Not that he even read the NYT, but he was bookish.

He also got a kick out of the big headline for the leading book review in that June 19, 1977, issue:  "Newton, Kant, Ruskin, Virgins All."  It was a review of "The Book of Lists". The memorable title of that review could be another reason I recall that day and giving him that as a gift.

P.S.  And let's not forget that one scene in A Serious Man takes place at a restaurant called Embers.

 

Country Joe and the troll doll

In the scene in A Serious Man where Danny runs into his room when he hears F-Troop start on his television set, when everybody is supposed to be sitting shiva for the deceased Sy Ableman, and Larry comes to the door of Danny's room and reminds him that they're supposed to be sitting shiva--and as usual in the movie, the child just ignores the command or question of the adult--on the floor of Danny's room a group of record albums can be seen on the left, the first one in the stack being a Country Joe and the Fish album. A little ways away from the albums, a troll doll is lying on the floor. Ah, yes, I remember mine fondly.

Interference of probability amplitudes, not photons

"The things that interfere in quantum mechanics are not particles. They are probability amplitudes for certain events." So says Roy J. Glauber in a letter to the editor of the American Journal of Physics, printed in the January 1995 issue. Glauber was describing how Dirac's writing that each photon interferes only with itself, and thus interference between different photons can never occur, is not strictly correct. So there. The great ones can get it wrong too, and frequently do, so don't worry too much about your own little pultritudinal mistakes.

2-D complex vector space is still waiting in the wings to be written about...

Quantum mechanics wears a tuxedo

Classical electromagnetism and classical mechanics are rather intuitive and informal when compared to quantum mechanics.  In fact, so are  special and general relativity, which are updated versions of Newton's mechanics and Maxwell's electrodynamics (special) and of Newton's law of universal gravitation (general), and which are applicable to the large-scale structure of space and time. To put it differently, quantum mechanics, which applies to the sub-microscopic space and time scale, is strangely formal when compared to everything else in physics.

Why, you ask, is general relativity applicable only on the large-scale space and time stage?  Good question!  It would be great to think of gravity also acting on the teeny tiny scale, in particular if black hole theory could be invoked to explain some of the things now ruled over by quantum field theory!  This would be the reverse of what the quantum gravity gang is attempting to do.  It would remake the strong, weak, and electromagnetic forces in the image of the gravitational force, and the hail with virtual particles and complete reliance on gauge invariance!

In line with the opening pages of Sakurai, where he discusses quantum mechanical spin, we can start to discuss quantum mechanics in terms of a complex 2-D vector space.  Hoo boy. Why vector space?  Why 2-D?  Why complex?  Why why why?

On a side note, if you take the "i" off Sakurai, you have Sakura, which, according to my Level 2 Adult Piano book, translates as Cherry Blossoms, the name of a traditional and rather haunting little Japanese melody.

Looking at the emphasized words (see below) in the first chapter of Baym, you can get an idea of quantum theory's strict formality.  One underlying trait of QM is the separation of the observer and the observed.  A wave function (not too formal a word) or state vector (the preferred formal description) evolves in time deterministically as described by the Schrödinger equation--until (cymbal clash) a measurement occurs.

A measurement is a quantitative observation--that is true in all of physics.  But the quantum measurement problem brings statistics into play when a measurement is made.  An observer is not necessarily a person but could be any device capable of not only causing a measurement to be made but also recording the measurement. 

To give an overview of how QM is related to some other fields of theoretical physics, I'll quote what David J. Griffiths says in the preface of his live-cat-on-the-front-dead-cat-on-the-back textbook Introduction to Quantum Mechanics:

Unlike Newton's mechanics, or Maxwell's electrodynamics, or Einstein's relativity, quantum theory was not created--or even definitively packaged--by one individual, and it retains to this day some of the scars of its exhilarating but traumatic youth.  There is no general consensus as to what its fundamental principles are, how it should be taught, or what it really "means."  Every competent physicist can "do" quantum mechanics, but the stories we tell ourselves about what we are doing are as various as the tales of the Scheherazade, and almost as implausible.  Niels Bohr said, "If you are not confused by quantum physics then you haven't really understood it"; Richard Feynman remarked, "I think I can safely say that nobody understands quantum mechanics."

 

Next time, the quantum formalism of 2-D complex vector space...

Baym's words of emphasis

Emphasized words (those in italics) in Chapter 1 of Lectures on Quantum Mechanics, by Gordon Baym:

Polaroid filter correspondence principle exactly probability complex basis superposition principle orthonormal basis probability amplitude transformation matrix eigenvector eigenvalue angular momentum spin operator for the photon definitely definitely expectation value eigenvalue eigenstate operator completeness relation probability amplitude superposition interference never indistinguishable probability amplitudes or amplitude amplitude exactly interference completely unpolarized or not mixed state mixed case pure case extraordinary ray ordinary ray optic axis transition amplitude transition probability change unitary.

The movie cat equation described by Asher Peres

The "Live Cat Dead Cat" superposition equation that appears in A Serious Man (see my Back to the Cat and Back of the Cat blog entry from last year) also shows up in the November 1975 issue of the American Journal of Physics--the only other place I've seen it--in a short paper by a fellow named Asher Peres.  I don't know if he was (he died in 2005) related to the current president of Israel, Shimon Peres, but that is not our concern of the moment.  (However: he wasn't.  See Wikipedia article about him.)

Our concern is partly that I just stumbled upon this article and equation while looking over some old discarded issues of the AJP that I picked up when I worked as a lab assistant at Austin Community College in the 1990s.  Our other concern is for the physics involved, and whether these old 1975 statements would be legitimate today.  What statements?  Oh, just that dead cats might be made alive again (the question of whether this could happen exactly nine eight times is not addressed). 

So, first we have Asher publishing his 1974 AJP paper (which I don't have a copy of, yet) called “Quantum Measurements are Reversible,” then we have P. J. van Heerden of Polaroid Research Laboratories, Cambridge MA, in the November 1975 AJP, on pages 1014 and 1015, complaining about this particular statement in Peres’ 1974 paper:  “An ensemble of identical systems is in some state, but a single system has no state.”  P. J. v H.  retorts “I believe that this statement is incorrect,” and he goes on to give an example he claims shows that “single systems do have eigenstates,” which is the title of his paper. 

So then in response Asher pipes up in the following paper, “A single system has no state,” on pages 1015 and 1016, with, among other responses to P. J. v H.'s paper, this paragraph:  “Most puzzling is van Heerden’s statement, ‘one can even think of an experiment exhibiting the interference pattern between the cat alive and the cat dead.’  If such an experiment could indeed be performed, then the phase θ in the state

ψ = 2^-1/2[ |live> + exp(iθ)|dead>]

would be meaningful.  One could then resuscitate dead cats in the following way:  Take an ensemble of dead cats and measure on each one of them the projection operator on state ψ.  In 50% of the cases, the state of the cat will become ψ.  Now measure whether the resulting cat in state ψ is alive or dead. In 50% of the cases, it will turn out alive.  I did not say this is impossible, but only that I don’t know how to construct the ψ-measuring machine.”

Were the Coen Bros friends with Asher Peres?  Asher's homebase was Technion-Israel Institute of Technology, in Haifa, so it's unlikely that the C brothers and A. Perez crossed paths.  Ethan and-or Joel may have heard second hand through the Jewish physicists' grapevine about the outrageous idea of reviving dead cats using the superposition equation for the live and dead states with the extra ingredient of a phase factor multiplying the dead "ket." 

Sorry, that's a stupid pun, but an accurate one, because our old untalkative pal, PAM Dirac, invented a terminology that is now universally used in quantum mechanics where |a> is called a ket and <a| is called a bra.  Bras and kets, which are vectors, often get multiplied together, most often with the bra vector first.  Whadda ya got then?  Bra-ket, or bracket!  And also, you have "bras" appearing in a pointedly bra-like way in the equations of physics. 

Asher Peres spent some time, on sabbatical, at the University of Texas at Austin, in 1979 or 1980.  His Physical Review paper from 1980, called "Can We Undo Quantum Measurements?" appears in a book I have (Quantum Theory and Measurement, published in 1983 and edited by J. A. Wheeler and W. H. Zurek).  I haven't made much sense of this paper yet, due partly to my ignorance of the subject and partly to the fact that it isn't very well written.  The latter happens a lot in physics papers, and books.  Or does it just happen a lot in general?  Yeh, it may not just be your stupidity that keeps you from understanding something!  Maybe the article or book is just poorly written, or written in such a pedestrian manner as to be sleep inducing.  Stephen Hawking has confessed to not doing a very good job with the writing in A Brief History of Time, probably the most purchased yet unread physics book of all time.  Most others who don't write well don't bother with the confessions.  


(My profile lists science and math books that I think are good.)

Anyway, at the end of that 1980 paper Peres says: "I am very grateful to J. A. Wheeler for the warm hospitality of the University of Texas and to E.C.G. Sudarshan for many stimulating discussions.  The final version of this paper has benefitted from comments by J. S. Bell (CERN) and A. Ron (Technion)."

I'll be discussing John S. Bell and his famous theorem later.  A well-written article by a fellow named Travis Norsen concerning Bell's theorem just recently appeared in the American Journal of Physics.  The AJP is not for publication of professional original results, which appear in other journals--you know, those "professional" journals such as Physical Review Letters or Nature or countless others (and also of course online at open access sites)--but is instead at the level of "pedagogy."  But Norsen's article does break new ground nevertheless, I think.