koantum said:
Since you seem to know that the ket |L> is nothing but a shorthand notation for the projector |L><L|, you should also know that (pure) quantum states are 1-dimensional projectors.
(I'm saying many of the same things as hurkyl has. He is definitely on the right track.)
koantum
I suggest you revisit basic QM, and, best of all, go back to review Dirac's work in his QM book. Your statement is incorrect; states and projection operators are different.
koantum said:
Kets can be terribly misleading since they contain spurious information. The phase of a ket has no physical significance whatsoever. Since it does not appear in the corresponding projector, it's much safer to work with projectors.
Misleading?
Phases can occur in a projection operator, particularly in complicated angular momentum problems, compund states defined by means of 3-j symbols for example(Multipole radiation, for example.(We are, of course, talking relative phases. What spurious info is in a ket? If that's what you think, then the wavefunction has exactly the same problem
Tell me how to do the hydrogen atom with projectors. If you can get the hydrogen spectrum without states, but with projection operators, then I'll reconsider. In fact, I'll bet if you can do it, you will find yourself lecturing at CERN, Harvard, MIT, Stanford,... Such a calculation would astonish most of us -- unless it is one of the standard approaches in disguise, imitating Schrodinger or Heisenberg.
Perhaps I'm naive, even though I learned QM from a master, J. H. VanVleck. That, as is the case in most linear vector space matters, the wave function W(x) can be interpreted as a vector in Hilbert space. Dirac simply formulated a new notation, very clever indeed, very useful indeed, to aid in the basic QM. A projection operator is just that, an operator, which acts on state vectors And anyway, this is as basic as anything in QM -- see Landau and Lifschitz, Dirac, Kemble, Schiff, Cohen-Tannoudji, Messiah, Weinberg (Field theory), Condon and Shortley, Mott and Massey, Golberger and Watson, Schwinger, Feynman, Pauli, Wesskopf and on and on. You might even say that there's an overwhelming consensus, since the early 1930s on the mathematical structure of QM -- states and operators, that's all she wrote. And most of us who are, or have been in the physics trade use Dirac's ideas and notation -- like it's the best game in town.
I greatly regret that I must conclude from your confusion over operators and states, that you are just plain wrong, both the math and the physics. Even if you have taken a QM course, you should go back and review in detail that course. If your Prof suggested the nonsense that a state and a projection operator are the same thing, then that Prof. ought to be chastized. If you are going to criticize QM, at least understand QM well before you start.
I have a feeling that you might be dealing with a "sorta density" matrix approach. Does not change things at all. Also, in field theory we do use operators to create states, but, still, a state is a state, and an operator is an operator
koantum said:
Mystical = whatever you don’t understand?
Have a look at my recent post in the "pure and mixed" thread. There is a list that shows how the axioms of quantum mechanics can be derived. I haven’t mentioned it explicitly there, but the two rules (add absolute squares of amplitudes versus add amplitudes and take the absolute square of the sum) emerge naturally and are thereby fully explained.
Unless I'm badly mistaken, the sum of squares never was considered in QM, at least not by the founders. (The impetus for the square of the wave function to be interpreted (Born) as a probability density follows from general notions about wave intensity -- freshman physics, optics-- and or Poynting's Thrm -- E&M. Jackson's E&M text does a very nice job with semi-classical quantum E&M, in which a direct connection between (E*E + B*B)/2 as an energy density, and photons with energy h*Frequency. You say:
The "state of the electron" is a probability algorithm.
A state is not an algorithm -- at best it can be determined with an algorithm.
You say: Entanglement, like interference, is not a physical state or process but a mathematical feature of the quantum-mechanical probability algorithm.
How then do you explain entanglement in classical systems?
I learned my QM some 50 years ago, and taught QM 40 years ago. What I learned is still valid (cf, Schiff's and Kemble's texts). I could use my lecture notes today with minimal modification. Lot's of things have not changed. And, little has changed in the criticisms of QM. The plain fact is that much of the discussion in this thread could easily have taken place 25 or 30 years ago. The "anti-QMs" have brought little new to the table, and nothing that makes QM easier to use. The basic anti-QM arguments have not changed much over the past 50 years. On the other hand, the past 50 years have shown an explosion of successes -- from superconductivity to quarks. Telling isn't it.
I think that standard QM is here to stay. Even if it is superseded by a deeper theory, our QM will still be used to deal with atomic spectra, scattering, solid state, ... However, I recognize that QM might eventually be found wanting. Whoever makes that claim will, of necessity, be extensively familiar with QM theory and applications -- from atomic spectra to high energy particle physics -- not necessarily an encyclopedic knowledge, but enough to understand and communicate the problems and triumphs in the many fields in which QM is used -- get any of that wrong, and the game is over. QM is a huge subject, slit experiments compose a very small portion of QM.(For example, if you have problems with superposition, check out the K-meson system, in which the role of superposition is crucial.This system is of far more consequence than a two slit experiment.)Secondly it is of the highest importance to write with simplicity and to use precise rather than sloppy language.
To be very honest, my response was in no small measure due to the poor writing -- states=algorithm? The problem for you is that given such mistakes in writing, it is hard to take your ideas seriously. (I've said such things to my students, and to those who worked for me in the consulting business.) The problems generated by sloppy writing are compounded when the topic takes exception to standard notions.
That being said, I think your ideas are potentially interesting. While I think there's a lot of jive that goes in this Forum, I do not apply that epithet to you. If you can write your ideas in a compellingly clear fashion, then I'll be among the first to give you a fair hearing.
Once a professor, always a professor.
Regards, Reilly Atkinson