THEN & NOW

# Einstein, Ehrenfest, and the quantum measurement problem

Article first published online: 8 FEB 2013

DOI: 10.1002/andp.201300708

© 2013 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Issue

## Annalen der Physik

Special Issue: Ultrafast Phenomena on the Nanoscale

Volume 525, Issue 1-2, pages A15–A19, February 2013

Additional Information

#### How to Cite

Unna, I. and Sauer, T. (2013), Einstein, Ehrenfest, and the quantum measurement problem. Ann. Phys., 525: A15–A19. doi: 10.1002/andp.201300708

#### Publication History

- Issue published online: 8 FEB 2013
- Article first published online: 8 FEB 2013

- Abstract
- Article
- References
- Cited By

A joint paper by Albert Einstein and Paul Ehrenfest, published just weeks after the results of the Stern-Gerlach experiment became known, shows with remarkable clarity and prescience the unsurmountable difficulties that the experiment posed for any classical interpretation. With a focus on the measurement process, rather than on the underlying theoretical alternatives for this experimentum crucis, the authors almost anticipate what would later be recognized as a central conceptual difficulty of quantum mechanics, i.e., the quantum measurement problem.

### 1. Introduction

- Top of page
- 1. Introduction
- 2. Outline of Einstein's and Ehrenfest's Paper
- 3. Correspondence related to the Einstein-Ehrenfest paper
- References

In mid-May 1922, Albert Einstein wrote to Max Born:

But the most interesting these days is the experiment by Stern and Gerlach. The alignment of the atoms via radiation and without collisions is (according to present methods of considering the problem) not understandable. Such an alignment should, by the rules, take more than 100 years. Ehrenfest and I did a little calculation of it. [[1], Doc. 190]

Some thirteen years later Max Born wrote in his book “Atomic Physics”:

[...] Stern and Gerlach's experiment is perhaps the most impressive evidence we have of the fundamental difference between classical and quantum mechanics. [[2], p. 127]

For many authors and teachers the Stern-Gerlach experiment has become the starting point for teaching quantum mechanics and a paradigm of quantum measurement (see, for example, [3], [[4], pp. 2–6], [[5], §1–5, pp. 14–18], and preceding papers by A. Peres). It is remarkable that Einstein and Ehrenfest sensed this right away.

Stern and Gerlach, themselves, were fully aware of the importance of their results. Their seminal publication, “The experimental proof of the directional quantization in a magnetic field” [6], states clearly that the results provide a direct experimental proof of this quantization.

As is well known (e.g., [[7], sec. IV.3], [8-10]), the experiment (see Fig. 1) showed that a beam of silver atoms passing through an inhomogeneous magnetic field ends up in two narrow beams (see Fig. 2), one of which has the atomic magnetons aligned, the other antialigned, with the strong (10, 000 gauss) magnetic field. This was contrary to the classical prediction of a continuous broadening of the beam in accordance with the random distribution of the emitted single atomic magnetic moments.

Einstein and Ehrenfest's “little calculation” dealt with the measurement process. They were not concerned about which is the correct theory confirmed by the experiment. They were troubled by the actual process through which the measurement proceeded. At that time the common wisdom (in great part promoted by Einstein's 1916/1917 papers [[11], Docs. 34, 38]) taught that any quantum transition between states has to take place via radiation or collision processes. This experiment was the first which posed a real problem for this way of understanding the measurement process, as we hope to clarify in the following discussion.

Our subject is a paper by A. Einstein and P. Ehrenfest entitled: ”Quantum Theoretical Comments on the Experiment of Stern and Gerlach” [12], [[1], Doc. 315]. It was published a couple of months after the paper by Stern and Gerlach. It was received on 21 August 1922, while the Stern-Gerlach paper was received 1 March 1922. This paper may have been the first which pinpointed down the major problem of quantum measurement, which later became known as the wave-function collapse.

The Einstein-Ehrenfest paper was probably conceived during Einstein's visit with Ehrenfest in Leyden which took place between 29 April and 13 May 1922. Discussions and voicing doubts continued, however, after Einstein's return to Berlin via correspondence.

An additional important source enabling us to catch glimpses of the struggle Einstein and Ehrenfest went through in preparing this important paper is Paul Ehrenfest's personal diary.

The importance of this paper and the discussions of its authors around it lies, as already mentioned, in discovering and pointing out a major problem of quantum mechanics. As we shall see, although the authors are not able to understand what is going on in this experiment, their discussions sharpened the problem to a point that, with hindsight, can be identified as the quantum measurement problem.

### 2. Outline of Einstein's and Ehrenfest's Paper

- Top of page
- 1. Introduction
- 2. Outline of Einstein's and Ehrenfest's Paper
- 3. Correspondence related to the Einstein-Ehrenfest paper
- References

After briefly describing the experiment Einstein and Ehrenfest pose the question: “Obviously, the urging question comes up how the atoms attain this orientation.”

During their passage through the deflecting magnetic field no collisions occur. So long as one neglects radiation processes, the atoms perform Larmor precession around the direction of the magnetic field. If the direction of the magnetic field changes slowly relative to the speed of the precession, no change in the angle of the precessing motion will take place. The alignment required by quantum theory (0 and π in the present experiment) therefore *requires* an external influence such as radiation or collision.

But how quickly can the directional change due to the influence of the radiation (of room temperature) take place? The transition time between one quantum state to another one can be estimated—at least to order of magnitude—by using the corresponding classical model. In our case of a precessing atom with magnetic moment, the model would be the radiation of a conically rotating magnetic dipole. But then the alignment time (in a field of 10, 000 Gauss) becomes of the order of 10^{11} seconds if only emitted radiation is considered. If one takes into consideration also the room-temperature radiation (“positive and negative radiation absorption” [[11], Doc. 38]) the time reduces to 10^{9} seconds. These times have orders of magnitudes which cannot be relevant at all to the experiment, since here the entire time of flight available for the alignment process is less than 10^{−4} seconds.

We find a detailed calculation of the alignment time in Ehrenfest's diary (items #5712, 5714, after 20 May 1922, see Fig. 4).

The radiation rate of a magnetic dipole μ precessing in a magnetic field *H* is given by:

- (1)

where cm/s is the velocity of light; erg/Gauss is the Bohr Magneton, is the precession frequency, which for a magnetic field of 10, 000 Gauss is s^{−1} (see Ehrenfest's diary item #5698, 22 April 1922).

The energy which has to be emitted or absorbed by the atom in order to align or antialign with the magnetic field is of the order ,

i.e., in our case: erg. Inserting the above values for , we get:

- (2)

Thus, the alignment time, Θ, will be

- (3)

Taking into account the thermal radiation of the surrounding room temperature, we use the Einstein coefficients to calculate the ratio between the rates of absorption and emission, i.e.:

- (4)

(Rayleigh-Jeans approximation). Thus, the alignment time reduces to seconds, i.e., of the order of a hundred years.

If one wants to avoid this difficulty, one is led to two alternative assumptions:

- The real mechanism is such that the atoms can never enter a state which is not already totally quantized.
- Under quick influences, situations result which violate the quantum principles as related to orientation; the alignment required by the quantum rules through emission and absorption of radiation is accomplished at an exceptionally greater speed than the transitions between quantum states.

Einstein and Ehrenfest elaborate on the difficulties to which each of these alternatives leads.

Alternative A leads to the following assumption: Even very weak fields are decisive for immediate orientation after the collision (i.e. the effect of strong fields). The magnetic axis of the atom will follow completely any change of the magnetic field, even if the change is much quicker than the Larmor precession of the atom, just as in the case of a slow (adiabatic) change. This can then be generalized: Any quick change of external conditions of a mechanical system should keep it in the same state as it would under a similar, infinitely slow (adiabatic) change. But this assumption necessarily implies the violation of the mechanical equations. In a detailed footnote the examples of a quick shortening of the cord length of a pendulum and of the quick rotation of a magnetic field around an atom are given. In the first example, conservation of energy will be violated, in the second example, conservation of angular momentum no longer holds.

According to alternative B the magnetic axis of each atom is randomly oriented after a collision relative to the direction of the weak magnetic field at that point. The alignment, parallel or antiparallel, is accomplished by emission or absorption of infrared radiation. However, these transitions from non-quantum to quantum states have transition probabilities which are many orders of magnitude higher (≈10^{13}) than transitions from quantum to quantum states. The adjustment of quantum states requires the possibility of emitting and absorbing radiation. Thus, it creates a principal difference between purely mechanical and systems that emit or absorb radiation. For example, should the rotation axis of a symmetrical heavy top reach quantum alignment with the gravitation field only if it carries proper electric charges? If one wants to generalize hypothesis B, as related to the alignment in quantum states, i.e., for example to crystal grating oscillations or a rotating molecule, should one allow these to align spontaneously into quantum lines only with proper electric charges? This will lead to an obvious contradiction with experimental evidence as related to specific heats, e.g., of diamond and of gaseous *H*_{2}.

The discussed difficulties show how unsatisfactory both alternatives are for the interpretation of the results found by Stern and Gerlach.

### 3. Correspondence related to the Einstein-Ehrenfest paper

- Top of page
- 1. Introduction
- 2. Outline of Einstein's and Ehrenfest's Paper
- 3. Correspondence related to the Einstein-Ehrenfest paper
- References

On 16 May, 1922—three days after Einstein left Leyden—Ehrenfest sent a letter to Einstein in which he discussed a hypothesis advanced by G. Breit which is similar to assumption A in the final paper. Ehrenfest considered Breit's original idea “nonsense.” He pressed Einstein to provide his quick and clear reaction to this hypothesis and to say how it should, eventually, be formulated [[1], Doc. 191].

Two days later—on 18 May, 1922—Einstein answered, rejecting Breit's assumption completely [[1], Doc. 193].

On 23 May, 1922, Einstein wrote again, responding to a letter by Ehrenfest which, unfortunately, cannot be found. He reacted to a new hypothesis of Ehrenfest—his “Schock” hypothesis. (We do not know what this hypothesis is.) Although it appealed to Einstein when he first received it, he now had a different opinion, which, however, he was not able to justify rigorously. To us it sounds again similar to assumption A in the published paper. Einstein stated clearly that their calculation of the alignment time is irrelevant here, since it assumes transition between quantum states via radiation, contrary to the assumption presented now. Einstein pointed out in some detail the serious difficulties posed by this assumption. At the end of his letter, he voiced doubts whether “our note” should be published at all, due to the controversy between them. “If we are unable to be clear enough so that we two can agree, let us leave it.” [[1], Doc. 200].

On 30 July, 1922, Ehrenfest promised Einstein to send him the manuscript, print-ready, for publication in *Zeitschrift für Physik*. He explained:

I am sending it to you because I added 5 lines right at the end that would take a weight off my chest but that you perhaps want to discard. Please let it stand if you ever can bring yourself to do so. The rest of the entire manuscript is exactly as we had discussed. [[1], Doc. 316]

These final lines concern a cryptic remark about Bohr's interpretation.

About three weeks later Einstein reported having sent the paper to Karl Scheel, the managing editor of *Zeitschrift für Physik*. His reaction to the added lines is typical:

I left your last sentence standing unchanged even though I did not understand the suggestion at all. I can't understand why the alignment in a magnetic field should be better understood by assuming that the quantization is in principle imprecisely realized. It would be good if you could make it plausible for me, just to appease my literary conscience. [[1], Doc. 329]

The special role of measurement in quantum mechanics was, according to accepted history, first pointed out in 1929 by Mott [13] and accentuated by von Neumann in his *Mathematical Foundations of Quantum Mechanics*, 1932 [14]. Einstein and Ehrenfest recognized how problematic the concept was as soon as Stern and Gerlach first confronted the physics community with a genuine quantum measurement.

### References

- Top of page
- 1. Introduction
- 2. Outline of Einstein's and Ehrenfest's Paper
- 3. Correspondence related to the Einstein-Ehrenfest paper
- References

- 1The Collected Papers of Albert Einstein. Vol. 13. The Berlin Years: Writings & Correspondence, January 1922–March 1923 (Princeton: Princeton University Press, 2012)., , , and (eds.),
- 2Atomic Physics (New York: Stechert, 1936).,
- 3 ,
- 4Modern Quantum Mechanics (Menlo Park, CA: Benjamin/Cummings, 1985).,
- 5Quantum Theory: Concepts and Methods (Dordrecht: Kluwer, 1993).,
- 6 and ,
- 7The Historical Development of Quantum Theory. Vol.1, Pt. 2 (New York: Spinger-Verlag, 1982).and ,
- 8 ,
- 9Toennies, J. Peter and Schmidt-Böcking, Horst and Friedrich, Bretislav and Lower, Julian C., Annalen der Physik 523, 1045–1070 (2011).
- 10Der Stern-Gerlach-Effekt. Genese, Entwicklung und Rekonstruktion eines Grundexperimentes der Quantentheorie 1916–1926, Ph.D. Thesis, Johann Wolfgang Goethe-Universität Frankfurt, 2011.,
- 11The Collected Papers of Albert Einstein. Vol.6. The Berlin Years: Writings, 1914–1917 (Princeton: Princeton University Press, 1996)., , and (eds.),
- 12 and ,
- 13 ,
- 14Mathematische Grundlagen der Quantenmechanik (Julius Springer, 1932).,