Note that the problem you pose is non-realistic. If at a certain moment B is in a position eigenstate, $\delta (\vec r)$, at an extremely short time after , B can be everywhere is the universe with equal probability. You will see the effect of this, below.

But let's first calculate the force $<\vec F>$. In QM, the influence of between A and B goes as follows: let $\psi_A(\vec r)$ be the wave-function of the electron A, where the vector $\vec r$ connects A, wherever A is, with B.

Then the force of interaction is

$\vec F(\vec r) = -\frac {e^2 \vec r}{4 \pi \epsilon_0 |r|^2}$.

The average force between the two electrons is

$<\vec F> = \int d\vec r \int d\vec r' \psi_A^* (\vec r)\delta (\vec r') \frac {e^2 (\vec r - \vec r')}{4 \pi \epsilon_0 |\vec r - \vec r'|^3} \psi_A (\vec r) \delta (\vec r')$.

$=\int d\vec r d\vec r' \delta (0) |\psi_A (\vec r)|^2 \frac {e^2 \vec r}{4 \pi \epsilon_0 |\vec r|^2} = \delta (0)\int d\vec r |\psi_A^* (\vec r)|^2 \frac {e^2 \vec r}{4 \pi \epsilon_0 |\vec r|^2}$

So, we have a problem because the function $\delta (\vec r')$ has infinite norm. On the other side, if $\psi_A$ is spherically symmetrical, one gets $<\vec F> = 0$.
For the case that $\psi_A$ is not spherically symmetrical we have to replace the wave-function of B by another function, let's name it $\psi_B (r')$, highly localized around the point $\vec r' = 0$, but normalized. In that case

$<\vec F>=\int d\vec r d\vec r' |\psi^* _B (\vec r')|^2 |\psi_A^* (\vec r)|^2 \frac {e^2 (\vec r - \vec r')}{4 \pi \epsilon_0 |\vec r - \vec r'|^3}$,

and since $\psi_B (r')$ is highly localized around $\vec r' = 0$ we can approximate,

$<\vec F>=\int d\vec r d\vec r' |\psi^* _B (\vec r')|^2 |\psi_A^* (\vec r)|^2 \frac {e^2 (\vec r)}{4 \pi \epsilon_0 |\vec r|^3} = \int d\vec r |\psi_A^* (\vec r)|^2 \frac {e^2 (\vec r)}{4 \pi \epsilon_0 |\vec r|^3}$.

Now I return to the next moment after localization. The function $\psi_B (\vec r')$ will be practically zero, everywhere. So, in the before last equation we will get $<\vec F> = 0$.

Bohmian Quantum Mechanics doesn't make different predictions than any other interpretation.

The trajectories of Bohmian Mechanics are simply a particular choice of probability current. You could measure position at any time and associate a trajectory with that result and look at the path forwards or backwards.

The so called probability current was already there in any interpretation. And Bohmian Mechanics doesn't use the trajectories to make any new different predictions. Just like every other interpretation.

So it isn't about correctness. It's about how some people looked at pictures and said "that's a weird looking picture" as if that mattered. Now you can do experiments where the picture is more closely related to actual experimental results. So it seems less weird when there is data that has similarly shaped results.

But you could get those predictions even without saying the particles move on those trajectories. When you focus on the experimental results, all the different interpretations agree. So the results aren't evidence for any one over the others.

If someone wants to think that results just appear sometimes with certain frequencies and correlations (like Copenhagen does) then no evidence can ever refute that. And similarly you can make a theory where things act a certain way that produces the same results with certain frequencies and that doesn't show the theory is correct about how the things acted **other than the fact that you got the results you did with the frequencies you got.**

The story can look less weird when the pictures can line up with some experiments. But there will always be a boundary between results and the many many ways the universe could be that are consistent with those results. And nothing will distinguish between them. Which is fine. Use whichever is easier to compute, or teach, or remember, or catch mistakes, or make new discoveries, or modify into new theories. Or use different ones for different situations. Just don't think your evidence is more than it is.

It was never right to object that the trajectories look weird. Now it's a little bit easier to show people that was a wrong objection. But if they couldn't see that before then I'm not sure you've accomplished anything. People shouldn't get too excited about the parts of a theory that aren't used to make a prediction.

Does this paper show that Bohmian mechanics is correct and that the standard interpretation is not?

Again, different interpretations make the same predictions. In Bohmian mechanics you handle weak measurements and strong measurements the same way: by writing down the wave function of the combined system of subject and device and writing down the evolution as determined by the Hamiltonian of the joint system (which every interpretation does, so weak measurements aren't mysterious in the slighest) and then adding the one ingredient of Bohmian mechanics. Which is to consider a distribution of initial positions to consider special, and the streamlines of these initial positions evolves to give a distribution on final positions, and which of the separated packets this final position is in tells you which outcome to consider special.

If you post select your results, then you are just sorting the final results to line up with the kinds of trajectories Bohmian mechanics follows. You are still saying that the Schrödinger equation for the actual experimental setup describes the evolution of the actual system. Like any interpretation does.

Sure, interpretations other than Bohmian mechanics sometimes get lazier and don't write down the device part of the system and don't write down the Hamiltonian of the full system of device and subject. Because they want to use a hack to compute the frequency of the final results: a hack designed just for strong measurements. But that just means if you find a situation where their favorite hack doesn't work they will have to do it the full way. Which was never in doubt about being the correct way.

Keep in mind that Copenhagen doesn't make different predictions, many worlds is about as close to what the math says and Copenhagen merely asserts that one branch somehow magically survives when the others somehow somewhen magically disappear, but that's a nonprediction because it is untestable. Bohmian mechanics has the same branching as many worlds (because it also uses the Schrödinger equation and the Schrödinger equation branches for interactions of device and subject) but it asserts that one position in configuration space was *always* special and so as the branch separates, at most one branch becomes special. But the specialness of a branch changes nothing about the predictions. So like Copenhagen, its additional stuff is also merely a non prediction. Every interpretation is like that.

## 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

quantumFourier 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).