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.