miércoles, 21 de marzo de 2018

Assumptions in Heisenberg’s 1925 paper



    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.


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? –  


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? – 


I meant that sometimes during the course of going through the mathematical formalism, it is possible to unknowingly ignore the physical interpretation of some steps taken, or the physical meaning behind some results. So as far as possible, I don’t want to ignore any of that. (Also which “good textbooks” are you referring to?) Again, sorry for the trouble. –


That’s very good you don’t want to ignore the physical interpretation and indeed, not too many words are being said about it in most cases. However, what is even more important than to appreciate the right physical interpretation of the formulae is to avoid a wrong interpretation of them – such as a classical or “visualizable” interpretation: none of it is ever right in QM. Certain things just don’t have any “easy to imagine” content and the calculated probabilities (and cross sections and allowed eigenvalues etc.) are really everything that is meaningful & “real” from a physical vantage point – 


The online reference to link Darrengol could be useful to see the problems H. was addressing at that time. Also a read of Sommerfeld paper, to see the “elipses” of the relativistic atom and how problematics they were, can be illuminating. –

Heisenberg’s paper is deriving its results from an assumption which is stated only obliquely in the paper, and which is central for all the conclusions. This assumption is explained more clearly on Wikipedia.

Heisenberg is dealing with the orbit of an electron in the atom. Let us assume that this orbit is precise, so that the electron has a position on the m-th Bohr orbit as a function of time is Xm(t). The motion is periodic, so you can Fourier transform this motion to get a Fourier series for the electron’s position


The quantity Xmn is the n-th Fourier coefficient of the m-th Bohr orbit. This quantity is associated with the frequency nω where ω=2π/T is the classical orbit (radian) frequency and T is the classical orbital period. Notice that the classical Fourier frequencies are multiples of a least common multiple, which is (2π times) the reciprocal period.

The fundamental reason Heisenberg rejects this description (which is very close to Bohr’s original idea, and developed by Kramers and Heisenberg) is the fact that these integer spaced frequencies nω are not observed in atomic transitions.

the frequencies that you do observe are the quantum frequencies, which are the difference in energy between the n-th Bohr orbit and the m-th Bohr orbit. There is a fundamental mismatch between the classical orbital description with its integer tower of frequencies and the observed electromagnetic wave emission of the atom, which has a completely different non-integerly spaced collection of frequencies.

The quantum frequencies are given by En−Em, the difference in energy of the n-th and m-th orbit, which however do become integer spaced when n and m are both large. In this limit, called the correspondence limit, En−Em=∂E∂J(n−m) where the partial derivative is of the classical energy with respect to the classical action variable J.

So in the correspondence limit, the classical orbit description is valid, because the frequencies you observe in atomic transitions match the frequencies you would deduce by Fourier transforming a sharp classical orbit.

But what about at smaller quantum numbers? Here Heisenberg makes a radical new assumption. He takes the quantities Xnm, which are the n-th Fourier coefficient of the m-th Bohr orbit, and says that they appear in quantum mechanics with the frequency En−Em, not with the frequency 2πnT! This idea is already present in Bohr to some extent, even in 1913 Bohr states that the transition from orbit n to orbit m should correspond to the classical Fourier component of motion somehow, but Bohr does not develop this idea fully, leaving it vague.

Heisenberg then states that if X_{nm} are quantum Fourier coefficients, then it is immediate that their time development should be


Here you can recognize the Heisenberg equation of motion for the matrix elements of X. This is required by the correspondence principle, to match the frequency of classical Fourier coefficients for large orbits. It is also incompatible with the picture of sharp orbits, because the X matrix elements no longer make integer-spaced towers which can be used to reconstruct a periodic classical orbit. Further, the coefficients with opposite frequencies are complex conjugates of each other Xmn=X∗nm, in the classical picture, it would be Xm,n=X∗m,−n.

Part of the difference is a trivial shifting: the classical n=0 point is shifted to n=m in the matrix description, just because the near-diagonal part is the classical motion, not the 0 column. This shifting is expressed by the correspondence rule that Xclm,n=Xm(m+n), where the left hand side is the classical Fourier coefficients, and the right hand side is the quantum matrix elements. But even with this shifting, the conjugation relations are off.

The complex conjugation in QM reflects along the diagonal of the matrix, it doesn’t reflect the horizontal row along a vertical line running down the middle. You can see how the classical limit emerges by looking at large m,m+p in the matrix, The reflection to m+p,m is p units away from the diagonal to the left, while the original position is p units to the right. So when the rows become continuous and the columns stay discrete, the complex conjugation relations reproduce those of classical mechanics on the Fourier coefficients.

But things are not quite right, because the stuff to the left of the midpoint in a given row is not the complex conjugate of the right. This means that if you try to write down the classical orbit as a function of time, you will fail, producing complex quantities which are not periodic, just some nonsense.

It is important to see Heisenberg’s intuition here— he was sure that the quantum Xmn is a complete description of the quantum motion, but it does not include the classical orbits. His conviction is that the orbit was not a part of the description, that it was a redundant classical idea that was no longer useful, and the fact that his description did not allow you to reproduce the orbit was a positive sign, not an incompleteness.

Other stuff in the paper

The next step is to derive the multiplication law. This is explained on Wikipedia, but it is pretty obvious from the classical law for multiplying Fourier series by convolution. The result is matrix multiplication.

Heisenberg then derives the on-diagonal part of the canonical commutation relations from some complicated radiation sum-rules he did with Kramers. The derivation on Wikipedia is more elementary, but uses essentially the same ingredients, without relying on Kramers-Heisenberg sum rules, and without doing ad-hoc tricks like differentiating with respect to n. The derivation of the on-diagonal canonical commutation relation is the main hurdle that makes this paper magical— it is difficult to follow, you need to do it a different way today.

Why uncertainty?

The uncertainty principle, although only explicitly formulated in 1927, is already present in 1925 to a large extent, except not stated in terms of complementary variables.

Heisenberg’s matrices only allow you to reconstruct a fuzzy orbit, it is only a classical periodic orbit to the extent the the frequencies are integer spaced. So for Heisenberg, the quantities which are “uncertain” are not uncertain yet in a statistical sense (that comes later, after Born’s interpretation of the wavefunction), but they are uncertain in the sense that they cannot be reconstructed in a quantum system.

Heisenberg would have said that the momentum is also uncertain, because the momentum fourier series cannot be reconstructed from the matrix elements of P. The energy would be certain, because the energy levels are precise in the description (ignoring back-reaction from the EM field emissions).

This is an artifact of the fact that Heisenberg was working in frequency space, so that the Hamiltonian was diagonal. In this picture, every quantity which does not commute with H would be considered uncertain, because it would necessarily have off-diagonal matrix elements that do not allow you to reconstruct it’s time variation precisely.

This concept of uncertainty is not the same as the 1927 uncertainty, which came after further developments clarified the notion of state. In 1925, Heisenberg wan’t sure how to describe the state, he could only describe the analogs of classical motion in the Bohr orbits.

So the notion of fuzziness of quantity in the 1925 paper should be considered an ill-definedness of the classical quantity as a function of time, not as a statistical statement about the values of observation of that quantity (at least not yet).

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