lördag 11 februari 2017

QM: Waves vs Particles: Schrödinger vs Born

From The Philosophy of Quantum Mechanics The Interpretations of QM in Historical Perspective by Max Jammer, we collect the following account of Schrödinger's view of quantum mechanics as wave mechanics, in full correspondence with realQM:
  • Schrödinger interpreted quantum theory as a simple classical theory of waves. In his view, physical reality consists of waves and waves only. 
  • He denied categorically the existence of discrete energy levels and quantum jumps, on the grounds that in wave mechanics the discrete eigenvalues are eigenfrequencies of waves rather than energies, an idea to which he had alluded at the end of his first Communication. In the paper "On Energy Exchange According to Wave Mechanics," which he published in 1927, he explained his view on this subject in great detail.
  • The quantum postulate, in Schrödinger's view, is thus fully accounted for in terms of a resonance phenomenon, analogous to acoustical beats or to the behavior of "sympathetic pendulums" (two pendulums of equal, or almost equal, proper frequencies, connected by a weak spring). 
  • The interaction between two systems, in other words, is satisfactorily explained on the basis of purely wave-mechanical conceptions as if the quantum postulate were valid- just as the frequencies of spontaneous emission are deduced from the time-dependent perturbation theory of wave mechanics as if there existed discrete energy levels and as if Bohr's frequency postulate were valid. 
  • The assumption of quantum jumps or energy levels, Schrödinger concluded, is therfore redundant: "to admit the quantum postulate in conjunction with the resonance phenomenon means to accept two explanations of the same process. This, however, is like offering two excuses: one is certainly false, usually both." 
  • In fact, Schrodinger claimed, in the correct description of this phenomenon one should not apply the concept of energy at all but only that of frequency.
We contrast with the following account of Born's view of quantum mechanics as particle statistics:
  • Only four days after Schrödinger's concluding contribution had been sent to the editor of the Annalen der Physik the publishers of the Zeitschrift fur Physik received a paper, less than five pages long, titled On the Quantum Mechanics of Collision Processes, in which Max Born proposed, for the first time, a probabilistic interpretation of the wave function implying thereby that microphysics must be considered a probabilistic theory.
  • When Born was awarded the Nobel Prize in 1954 "for his fundamental work in quantum mechanics and especially for his statistical interpretation of the wave function," he explained the motives of his opposition to Schrödinger's interpretation as follows: 
  • "On this point I could not follow him. This was connected with the fact that my Institute and that of James Franck were housed in the same building of the Göttingen University. Every experiment by Franck and his assistants on electron collisions (of the first and second kind) appeared to me as a new proof of the corpuscular nature of the electron."
  • Born's probabilistic interpretation, apart from being prompted by the corpuscular aspects in Franck's collision experiments, was also influenced, as Born himself admitted, by Einstein's conception of the relation between the field of electromagnetic waves and the light quanta.
  • In the just mentioned lecture delivered in 1955, three days before Einstein's death, Born declared explicitly that it was fundamentally Einstein's idea which he (Born) applied in 1926 to the interpretation of Schrödinger's wave function and which today, appropriately generalized., is made use of everywhere. 
  • Born's probability interpretation of quantum mechanics thus owes its existence to Einstein, who later became one of its most eloquent opponents.
We know that the view of Born, when forcefully missioned by Bohr, eliminated Schrödinger from the scene of modern physics and today is the text book version of quantum mechanics named the Copenhagen Interpretation. We understand that Born objected to Schrödinger's wave mechanics because he was influenced by Einstein's 1905 idea of a "corpuscular nature" of light and certain experiments suggesting a "corpuscular nature" of electrons. 

But associating a "corpuscular nature" to light and electrons meant a giant step back from the main advancement of 19th century physics in the form of Maxwell's theory of light as electromagnetic waves, a step back first taken by Einstein but then abandoned, as expressed by Jammer:
  • Born's original probabilistic interpretation proved a dismal failure if applied to the explanation of diffraction phenomena such as the diffraction of electrons. 
  • In the double-slit experiment, for example, Born's original interpretation implied that the blackening on the recording screen behind the double-slit, with both slits open, should be the superposition of the two individual blackenings obtained with only one slip opened in turn. 
  • The very experimental fact that there are regions in the diffraction pattern not blackened at all with both slits open, whereas the same regions exhibit strong blackening if only one slit is open, disproves Born's original version of his probabilistic interpretation. 
  • Since this double-slit experiment can be carried out at such reduced radiation intensities that only one particle (electron, photon, etc.) passes the appara- tus at a time, it becomes clear, on mathematical analysis, that the $-wave associated with each particle interferes with itself and the mathematical interference is manifested by the physical distribution of the particles on the screen. The wave function must therefore be something physically real and not merely a representation of our knowledge, if it refers to particles in the classical sense.
Summing up: 
  • Real wave mechanics in the spirit of Schrödinger makes a lot of sense, and that is the starting point of realQM.
  • Born's particle statistics does not make sense, and the big trouble is that this is the text book version of quantum mechanics.
How could it be, with these odds, that Born took the scene? The answer is the "obvious"  generalisation of Schrödinger's wonderful 3d equation for the Hydrogen atom with one electron with physical meaning, into the 3N-dimensional linear Schrödinger equation for an atom with $N > 1$ electrons, a trivial generalisation without physical meaning. There should be another generalisation which stays physical and that is the aim of realQM.

In the end Schrödinger may be expected to take the game because he has a most perfect and efficient brain, according to Born.

To get more perspective let us quote from Born's 1954 Nobel Lecture:
  • Einstein, De Broglie, and Schrödinger have unceasingly stressed the unsatisfactory features of quantum mechanics and called for a return to the concepts of classical, Newtonian physics while proposing ways in which this could be done without contradicting experimental facts. Such weighty views cannot be ignored. Niels Bohr has gone to a great deal of trouble to refute the objections. I, too, have ruminated upon them and believe I can make some contribution to the clarification of the position.
  • Schrödinger thought that his wave theory made it possible to return to deterministic classical physics. He proposed (and he has recently emphasized his proposal anew’s), to dispense with the particle representation entirely, and instead of speaking of electrons as particles, to consider them as a continuous density distribution or electric density. 
  • To us in Göttingen this interpretation seemed unacceptable in face of well established experimental facts. At that time it was already possible to count particles by means of scintillations or with a Geiger counter, and to photograph their tracks with the aid of a Wilson cloud chamber.
Born's argument against Schrödinger's wave mechanics in the spirit of Maxwell in favor of his own particle mechanics in the spirit of Newton, evidently was that a "tick" of Geiger counter or "track" in a cloud chamber both viewed to have "particle-like quality", can only be triggered by a "particle", but there is no such necessity...the snap of a whip is like a "particle" generated by a "wave"...

Born ends with:
  • How does it come about then, that great scientists such as Einstein, Schrö- dinger, and De Broglie are nevertheless dissatisfied with the situation? 
  • Of course, all these objections are levelled not against the correctness of the formulae, but against their interpretation.
  • The lesson to be learned from what I have told of the origin of quantum mechanics is that probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a concept, unjustified by experience, which we must eliminate to open up the road.

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