Can Statistical Mechanics Reproduce Quantum Mechanics?
I made some mistakes (now hopefully corrected) in the description of a statistical mechanics model in my last December essay Third Generation AI and the Ultimate Observer. It is truly very difficult to understand the difference between statistical mechanics (classical) and quantum mechanics probabilistic models without understanding the role of the fully deterministic part of the quantum model that lays the ground for its probabilistic predictions impossible for any classical probabilistic model to make. Yet that difference is pivotally important for the design of the most primary component of any truly intelligent entity. As of today I can’t explain that difference better than Erwin Schrödinger explained it almost ninety years ago in his seminal paper The Present Status of Quantum Mechanics.
Ҥ 3. Examples of Probabilistic Predictions.
All predictions, therefore, are as before about determining elements of a classical model, positions and velocities of point masses, energies, angular momenta, etc. But, unlike the classical theory, only probabilities of results can be predicted. Let us have a closer look at this. Officially, it is always the case that by means of a number of presently performed measurements and their results, the probabilities of results of other measurements, either performed immediately or after some time, are derived. How does this work in practice? In some important and typical cases it is as follows.
If the energy of a Planckian oscillator is measured, the probability that one finds a value between E and E’ can only be nonzero if the interval between E and E’ contains a value from the sequence 3πhν, 5πhν, 7πhν, 9πhν, • • • • •
For each interval which does not contain any of these values, the probability is zero, that is, other values are excluded. These numbers are odd multiples of the model constant πhν (h = Planck’s constant, ν = the oscillator frequency). Two things attract the attention. First of all, there is no reference to previous measurements; these are not necessary. Secondly, the statement certainly does not lack precision, on the contrary, it is far more accurate than any real measurement could ever be.
Another typical example is the value of the angular momentum. In Fig. 1, let M be a moving point mass, where the arrow represents the length and direction of its momentum (i.e. mass times velocity). O is an arbitrary fixed point in space, the origin of a coordinate system, say, not a point with physical meaning therefore, but a geometrical point of reference. In classical mechanics, the value of the angular momentum of M with respect to O is the product of the length of the arrow for the momentum and the length of the perpendicular OF.
In quantum mechanics the angular momentum behaves quite similarly to the energy of the oscillator. Again, the probability is zero for every interval that does not contain a value from the following sequence:
That is, only values from this sequence can appear. Again, this holds without reference to any prior measurement. And one can well imagine how important this precise statement is: much more important than the knowledge of which of these values actually occurs, or with what probability each value occurs in particular cases. Moreover, notice that the point of reference does not play any role: no matter where it is chosen, the result is always a value from this sequence. For the model, this claim makes no sense, for the perpendicular OF changes continuously as the point O is shifted, whereas the momentum arrow remains unchanged. We see from this example how quantum mechanics does make use of the model to read off which quantities can be measured and about which sensible predictions can be made, but on the other hand does not consider it suitable for expressing relations between these quantities. Does one not get the feeling that in both cases the essence of what can be said has been forced into the straightjacket of a prediction for the probability that a classical variable has one or another measurement value?
Does one not get the impression that this is in fact about fundamentally new properties, which have only the name in common with their classical counterparts? These are by no means exceptional cases; on the contrary, precisely the most valuable predictions of the new theory have this character. There are indeed also problems of a type for which the original description is approximately valid. But these are not nearly as important. And those that one could construct as examples where this description is completely correct, have no meaning. “Given the position of the electron in a hydrogen atom at time t=0; construct the statistics of the positions at a later time.” This is of no interest. It may sound as if all predictions are about the visual model. But in fact, the most valuable predictions cannot be easily visualized, and the most easily visualized characteristics are of little value.
§ 4. Can the Theory be Built on Ideal Quantities?
In quantum mechanics the classical model plays the role of Proteus. Each of its determining elements can in certain circumstances become the subject of interest and acquire a certain authenticity. But never all at the same time; sometimes these and sometimes those, but always at most half of a complete set of variables, which would provide a clear picture of the instantaneous state of the model. What happens in the meantime with the others? Are they not real at all, or do they perhaps have a fuzzy reality; or are they always all real, but is it simply impossible to have simultaneous knowledge about them as in Rule A of § 2 ?
The latter interpretation is extremely attractive to those who are familiar with the statistical viewpoint developed during the second half of the last century, especially if one realizes that it was this viewpoint that gave rise to the quantum theory, namely in the form of a central problem of the statistical theory of heat: Max Planck’s theory of thermal radiation, Dec. 1899. The essence of that theory is exactly that one almost never knows all determining elements of a system, but usually far fewer. To describe a real object at any given moment, one therefore uses not just one state of the model, but rather a so-called Gibbs ensemble. This is an ideal, i.e. imaginary, collection of states mirroring our restricted knowledge about the real object. The object then is supposed to behave in the same way as an arbitrary state from this collection. This idea has had tremendous success. Its greatest triumph was in those cases where not all states from the collection correspond to an identical observed behavior of the object. It turned out that the object in that case indeed varies in its behavior exactly as predicted (thermodynamic fluctuations). It is tempting equally to relate the often fuzzy predictions of quantum mechanics to an ideal collection of states, one of which applies in any individual case, but one does not know which.
The one example of angular momentum shows for all that this is impossible. Imagine that in Fig. 1, the point M is placed in all possible positions with respect to O, and the arrow of momentum has all possible lengths and directions. Consider the collection of all these possibilities. Then one can choose the positions and arrows in such a way that, in each case, the product of the length of the arrow and the length of the perpendicular OF has one of the allowed values with respect to the point O. But then they do not have allowed values with respect to other points O’ of course. The use of a collection of states therefore does not help.
Another example is the energy of an oscillator. In one case it has a perfectly determined value, e.g. the ground state 3πhν. The distance between the two point masses which make up the oscillator is then very undetermined. But, to relate this result to a statistical collection of states, the statistics of the distances should at least be bounded by that distance for which the potential energy attains the value 3πhν. This is not the case, however: even arbitrarily large distances are possible, albeit with rapidly decreasing probability. And that is not simply an inconsequential result of calculations, which can be ignored without seriously affecting the theory: it is, among other things, the quantum mechanical explanation of radioactivity (Gamov). There are infinitely many other examples. Notice that changes in time have not been considered. It would not help us if we were to allow the model to change in a very “unclassical” way, for example to “jump”. Even for a single moment in time, it does not work. At no time is there a collection of classical model states which can describe all quantum mechanical predictions. In other words: if we want to prescribe a given (unknown) state to the model at every instant of time, or, equivalently, prescribe certain values (not exactly known to us) to all determining variables, then there is no conceivable assumption about these values which would not contradict at least part of the quantum theoretical assertions.
This is not exactly what one might expect if one is told that the statements of the new theory are always imprecise compared with the classical theory.”
Source:
Die Naturwissenschaften 1935. Volume 23, Issue 48.
The Present Status of Quantum Mechanics
By E. Schrödinger, Oxford.
