I am not entirely clear as to what were the bases for Heisenberg's assumptions in his 1925 paper. He claims that one cannot consider relations between quantities that are unobservable "in principle", like the position and period of revolution of an electron.
To quote some text : "These rules (the abovementioned relations) lack an evident physical foundation, unless one still wants to retain the hope that the hitherto unobservable quantities may later come within the realm of experimental determination.
This hope might be justified if such rules were internally consistent and applicable to a clearly defined range of quantum mechanical problems."
My first query is why does he claim the position and period of an electron to be unobservable "in principle"? There was theoretically no reason (at THAT time) to doubt that these quantities could be measured, though certainly they were indeterminate practically.
Secondly, just because a theory dealing with those quantities is inconsistent, or not general enough, why does it imply that we cannot define or measure quantities that that theory deals with? We may be able to measure some quantities perfectly, but still formulate an incorrect theory around them.
Finally, is there any ad-hoc basis to decide what these "uncertain" quantities are? More specifically, how could Heisenberg pinpoint position of an electron as an uncertain parameter and not any other quantity (like some electric field, etc.)?
Thanks in advance. (by the way I'm studying the original paper solely to look more closely at the motivation for assumptions underlying the theory)
Answer
My first query is why does he claim the position and period of an electron to be unobservable "in principle"? There was theoretically no reason (at THAT time) to doubt that these quantities could be measured, though certainly they were indeterminate practically.
Werner Heisenberg obviously disagreed with this assumption of yours and it just happened that his ability to disagree made him a founder of quantum mechanics.
He has spent several years by trying to develop "quantized planetary" models of the helium atom etc. before he understood that this failing project is failing for fundamental reasons. Such a helium with well-defined positions would be described by a chaotic 3-body problem and there would be no way how it could be consistent with the known regular behavior of the helium atom (and other atoms and other coherent systems), including the sharp spectral lines.
So Heisenberg was able to see in 1925 something that you can't see now: that the electrons can't be going along any particular trajectories while they're in the atoms. Instead, what is observed is that they have a totally sharp energy from a possible list, the spectrum – something we can really observe via the photons that atoms emit or absorb. To conclude that electrons can't be going along particular classical trajectories in the atoms, he didn't have to wait for measuring apparatuses that would be sufficiently accurate. He was able to make this conclusion out of the available data by "pure thought", and he was right.
Secondly, just because a theory dealing with those quantities is inconsistent, or not general enough, why does it imply that we cannot define or measure quantities that that theory deals with? We may be able to measure some quantities perfectly, but still formulate an incorrect theory around them.
Many combinations of options would be possible in a generic hypothetical world and you're right that the combination of options you mentioned would be logically possible in another world but Heisenberg was talking about our world. He learned his message from special relativity that one shouldn't talk about things that can't be operationally defined – such as the simultaneity of events (which is observer-dependent) and tried to maximally apply this positivist mode of reasoning to the world of atoms. His analysis dictated that he may assume that the electron in the atom has a particular energy for a long time but it can't have a well-defined position or velocity. So he reformulated physics around the notion of the energy which is measurable and found out the first formulation of quantum mechanics in the energy eigenstate basis Heisenberg picture.
Finally, is there any ad-hoc basis to decide what these "uncertain" quantities are? More specifically, how could Heisenberg pinpoint position of an electron as an uncertain parameter and not any other quantity (like some electric field, etc.)?
You are mixing apples with oranges here. Heisenberg's paper wasn't discussing the electromagnetic field. It was discussing the general logical framework underlying physics and the examples he took were those from mechanics – rigid rotator and anharmonic oscillator – that were meant to be later generalized to a theory of atoms in particular just by a new choice of the potential energy formula.
There's no observable concept of "electric fields" in the description of an atom or anharmonic oscilator at all. Even in classical physics, one deals with functions of positions and momenta. He figured out that not all functions are equally observable: energy (a particular function of positions and momenta) is much more observable and stable.
The underlying logic he has developed was later (soon) applied to other systems in mechanics such as atoms and molecules as well as field theory such as electromagnetism. But the essence isn't in describing which degrees of freedom are there (they're kept as close to those in the corresponding classical theory as possible); the essence of quantum mechanics is in the totally new set of postulates and methods to make predictions.
He realized that the right goal wasn't just to find another classical theory just with some new degrees of freedom, which is the intrinsic, fundamental, and completely flawed assumption of your whole question from the beginning to the end. He realized that the new insights force physicists to formulate a completely new theory – and he (and others) has (have) already used the completely new term "quantum theory" for it – and he just did so, discovering some of the new explicit quantum formulae for nontrivial predictions (beyond the spectrum of the Hydrogen atom that was "explained" by Bohr's toy model).
You may repeat many times that a complete conceptual revolution in physics (switching from the classical to the quantum) wasn't needed and one should have only discussed new classical models with new variables (paying no attention to whether or not they may be actually observed) except that Heisenberg knew that it was needed and the months (and a few years) that followed his discovery made his assumption unquestionable.
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