## Tuesday, December 20, 2011 ... /////

### Assumptions in Heisenberg's 1925 paper

Hsirihs asked: 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.)

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 turned him intto 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 was failing for fundamental reasons. Such a helium atom with well-defined positions would be described by a chaotic (and totally aperiodic) 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. It may be measured from the frequencies of the photons.

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, merely 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 first new explicit quantum formulae for nontrivial predictions (beyond the spectrum of the Hydrogen atom that was "explained" by Bohr's toy model, something that had to be done properly just a little bit later).

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.

HS: Thanks for the detailed explanation. But I wanted to confirm the following - Heisenberg did not propose the indeterminacy of position/velocity due to some experimental results, rather, just as special relativity challenged the ad-hoc concept of time (which was used as a parameter for evolution of position, momentum and other quantities in classical mech), Heisenberg challenged the absolute determinacy of position/momentum (which were in turn parameters to describe fields, energy, etc.). And so in this sense it was a theoretical analogy to special rel? Is that correct?

LM: Thanks for your interest, @Hsirihs. I am not sure what to do with similar questions. All reasoning in physics ultimately does boil down to experimental results although one is often forced to process it by a long chain of reasoning. Lots of qualitative (and some quantitative) properties of the atoms were known in the mid 1920s and it was enough for Heisenberg to make his conclusions "otherwise by pure thought" and he was shown right. The analogy to special relativity is just a philosophy.

LM: There's no reason why certain things in atomic physics should be nicely analogous to a completely different context. Relativity was only mentioned because it showed that one's reasoning may easily get confined by invalid assumptions, by talking about things that aren't really real or objective (such as simultaneity of events). So Heisenberg, avoiding this mistake, made a new look at the microscopic world, and this led him to reformulate the dynamics of atoms in the new quantum way using (observable quantity) energy eigenstates and matrices – needed to get a qualitatively good theory.

LM: But the knowledge of the actual behavior of the real world of atoms (and emission of light) was very important for Heisenberg to find what he found. He was extremely good at it and he also knew what was wrong about various classical models one may have proposed. In turn, he could guess the right formulae that should replace the classical ones in those simple enough cases, and he was also able to "derive" these new formulae from a detailed and potentially universal formalism that was surprisingly new but mathematically natural. And most importantly, it was completely right.

LM: Let me say that your assumption that the experimental situation was totally inconclusive in the mid 1920s is entirely wrong. In fact, they have known pretty much everything about the atoms that we know today – the spectral lines and the transition rates were measured for many of them. It's a huge amount of data. And we know today that no "entirely new" kinds of experimental data were left, indeed. Heisenberg's goal was physics – to describe the actual known experimental data (frequencies of spectral lines and transition rates, among a few other related quantities) – and he found out that getting rid of nonphysics (unobservable mirage variables that are obliged to exist but, as it turns out, they don't) is needed to understand physics.

HS: I see... well definitely the helium model failure was a motivation as well. On a side note, instead of just studying quantum mechanics, I additionally intend to focus on such fundamental matters and questions underlying it. In other words I actually want to study the "Physics" of it, rather than just the mathematical framework blindly (excuse me if I'm being obscure), and understand how each aspect of the theory fits into the physical world. Do you have any suggestions as to how to go about it, and whether studying the original pioneering papers would help in this regard?

LM: Dear @Hsirish, apologies but I can't give you a recommendation on this matter because I don't understand what you mean by "physics rather than the mathematical framework blindly" and why the existing textbooks and courses in physics apparently don't satisfy you.

At least the new good textbooks contain the same things as the original sources plus the 85 years of hindsight; reading the original papers before things got settled is more important to study the history of physics rather than physics itself. And one of the great lessons by Einstein, Heisenberg, and others was to focus on the actual maths that does the predictions and not on words. If by "physics", you mean either "words" or "easy visualization of the usual kinds", then your approach is just misguided because the right laws describing the quantum phenomena depend on the maths and can't be accurately reproduced by words, especially not words with a classical connotation, or by visual models, especially not visual models that are intrinsically classical.

Quantum mechanics is conceptually new and the maths is totally essential to understand the new conceptual underpinningss of quantum mechanics. The maths isn't a slave that is serving some words or naive visual models; the maths is the master. It's the whole story. The task that faced Heisenberg and others was greater than the search for another classical theory. And this task could have only be solved by carefully abandoning classical fantasies that weren't necessarily right (and they were totally impossible, it turned out), and by carefully abandoning the related words and "hidden variables" that were believed by many to exist (but this belief wasn't experimentally justified and was ultimately shown by quantum mechanics to be invalid).