[Physics] Assumptions in Heisenberg’s 1925 paper

quantum mechanics

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)

Best Answer

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 $X_m(t)$. The motion is periodic, so you can Fourier transform this motion to get a Fourier series for the electron's position

$$ X(t) = \sum_n e^{in\omega t} X_{mn} $$

The quantity $X_{mn}$ is the n-th Fourier coefficient of the m-th Bohr orbit. This quantity is associated with the frequency $n\omega$ where $\omega=2\pi/T$ 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 ($2\pi$ 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 $n\omega$ 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 $E_n - E_m$, 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, $E_n - E_m = {\partial E\over \partial J} (n-m)$ 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 $X_{nm}$, which are the n-th Fourier coefficient of the m-th Bohr orbit, and says that they appear in quantum mechanics with the frequency $E_n - E_m$, not with the frequency $2\pi n \over T$! 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

$$ X_{nm}(t) = e^{i (E_n - E_m) t} X_{nm}(0) $$

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 $X_{mn} = X_{nm}^*$, in the classical picture, it would be $X_{m,n} = X_{m,-n}^*$.

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 $X^{\mathrm{cl}}_{m,n} = X_{m(m+n)}$, 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 $X_mn$ 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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