The Temperature of an Atom

The concept ’temperature’ in classical thermodynamics describes an objective feature of reality, an objective property of matter.

A thermometer can tell us a certain temperature.

But it is difficult to define the temperature of an atom.

Actually, we cannot correlate this concept ’temperature of the atom’ with a well-defined property of the atom but have to connect it at least partly with our insufficient knowledge of it. We can correlate the value of the temperature with certain statistical expectations about the properties of the atom, but it seems rather doubtful whether an expectation should be called objective.

The concept temperature of the atom' is not much better defined than the concept mixing’ in the story about the boy who bought mixed sweets. In a similar way in quantum theory all the classical concepts are, when applied to the atom, just as well and just as little defined as the `temperature of the atom’; they are correlated with statistical expectations; only in rare cases may the expectation become the equivalent of certainty.

As in classical thermodynamics, it is difficult to call the expectation objective. One might perhaps call it an objective tendency or possibility, a potentia' in the sense of Aristotelian philosophy. In fact, I believe that the language actually used by physicists when they speak about atomic events produces in their minds similar notions as the concept potentia.'

So the physicists have gradually become accustomed to considering the electronic orbits, etc., not as reality but rather as a kind of `potentia.’ The language has already adjusted itself, at least to some extent, to this true situation. But it is not a precise language in which one could use the normal logical patterns; it is a language that produces pictures in our mind, but together with them the notion that the pictures have only a vague connection with reality, that they represent only a tendency toward reality.

The vagueness of this language in use among the physicists has therefore led to attempts to define a different precise language which follows definite logical patterns in complete conformity with the mathematical scheme of quantum theory. The result of these attempts by Birkhoff and Neumann and more recently by Weizsacker can be stated by saying that the mathematical scheme of quantum theory can be interpreted as an extension or modification of classical logic. It is especially one fundamental principle of classical logic which seems to require a modification. In classical logic it is assumed that, if a statement has any meaning at all, either the statement or the negation of the statement must be correct. Of here is a table or here is not a table,' either the first or the second statement must be correct. Tertium non datur,' a third possibility does not exist. It may be that we do not know whether the statement or its negation is correct; but in reality’ one of the. two is correct.

In quantum theory this law ` tertium non datur’ is to be modified. Against any modification of this fundamental principle one can of course at once argue that the principle is assumed in common language and that we have to speak at least about our eventual modification of logic in the natural language. Therefore, it would be a self-contradiction to describe in natural language a logical scheme that does not apply to natural language. There, however, Weizsacker points out that one may distinguish various levels of language.

One level refers to the objects — for instance, to the atoms or the electrons. A second level refers to statements about objects. A third level may refer to statements about statements about objects, etc. It would then be possible to have different logical patterns at the different levels. It is true that finally we have to go back to the natural language and thereby to the classical logical patterns. But Weizsacker suggests that classical logic may be in a similar manner a priori to quantum logic, as classical physics is to quantum theory. Classical logic would then be contained as a kind of limiting case in quantum logic, but the latter would constitute the more general logical pattern.

The possible modification of the classical logical pattern shall, then, first refer to the level concerning the objects. Let us consider an atom moving in a closed box which is divided by a wall into two equal parts. The wall may have a very small hole so that the atom can go through. Then the atom can, according to classical logic, be either in the left half of the box or in the right half. There is no third possibility: `

tertium non datur.’ In quantum theory, however, we have to admit — if we use the words atom' and box’ at all — that there are other possibilities which are in a strange way mixtures of the two former possibilities. This is necessary for explaining the results of our experiments. We could, for instance, observe light that has been scattered by the atom. We could perform three experiments: first the atom is (for instance, by closing the hole in the wall) confined to the left half of the box, and the intensity distribution of the scattered light is measured; then it is confined to the right half and again the scattered light is measured; and finally the atom can move freely in the whole box and again the intensity distribution of the scattered light is measured. If the atom would always be in either the left half or the right half of the box, the final intensity distribution should be a mixture (according to the fraction of time spent by the atom in each of the two parts) of the two former intensity distributions. But this is in general not true experimentally. The real intensity distribution is modified by the interference of probabilities'; this has been discussed before. In order to cope with this situation Weizsacker has introduced the concept degree of truth.'

For any simple statement in an alternative like The atom is in the left (or in the right) half of the box' a complex number is defined as a measure for its degree of truth.’ If the number is 1, it means that the statement is true; if the number is o, it means that it is false. But other values are possible. The absolute square of the complex number gives the probability for the statement’s being true; the sum of the two probabilities referring to the two parts in the alternative (either left' or right’ in our case) must be unity. But each pair of complex numbers referring to the two parts of the alternative represents, according to Weizsacker’s definitions, a `statement’ which is certainly true if the numbers have just these values; the two numbers, for instance, are sufficient for determining the intensity distribution of scattered light in our experiment. If one allows the use of the term

statement’ in this way one can introduce the term complementarity' by the following definition: Each statement that is not identical with either of the two alternative statements — in our case with the statements:the atom is in the left half’ or ` the atom is in the right half of the box’ — is called complementary to these statements.

For each complementary statement the question whether the atom is left or right is not decided. But the term not decided' is by no means equivalent to the term not known.’ Not known' would mean that the atom is really’ left or right, only we do not know where it is. But `not decided’ indicates a different situation, expressible only by a complementary statement.

This general logical pattern, the details of which cannot be described here, corresponds precisely to the mathematical formalism of quantum theory. It forms the basis of a precise language that can be used to describe the structure of the atom. But the application of such a language raises a number of difficult problems of which we shall discuss only two here: the relation between the different `levels’ of language and the consequences for the underlying ontology. In classical logic the relation between the different levels of language is a one-to-one correspondence.

The two statements, `The atom is in the left half’ and It is true that the atom is in the left half,’ belong logically to different levels. In classical logic these statements are completely equivalent, i.e., they are either both true or both false. It is not possible that the one is true and the other false.

But in the logical pattern of complementarity this relation is more complicated. The correctness or incorrectness of the first statement still implies the correctness or incorrectness of the second statement. But the incorrectness of the second statement does not imply the incorrectness of the first statement. If the second statement is incorrect, it may be undecided whether the atom is in the left half; the atom need not necessarily be in the right half. There is still complete equivalence between the two levels of language with respect to the correctness of a statement, but not with respect to the incorrectness. From this connection one can understand the persistence of the classical laws in quantum theory: wherever a definite result can be derived in a given experiment by the application of the classical laws the result will also follow from quantum theory, and it will hold experimentally.

The final aim of Weizsacker’s attempt is to apply the modified logical patterns also in the higher levels of language, but these questions cannot be discussed here. The other problem concerns the ontology that underlies the modified logical patterns. If the pair of complex numbers represents a statement' in the sense just described, there should exist a state’ or a situation' in nature in which the statement is correct. We will use the word state'

in this connection. The states' corresponding to complementary statements are then called coexistent states' by Weizsacker. This term coexistent’ describes the situation correctly; it would in fact be difficult to call them different states,' since every state contains to some extent also the other coexistent states.'

This concept of state' would then form a first definition concerning the ontology of quantum theory. One sees at once that this use of the word state,' especially the term `coexistent state,’ is so different from the usual materialistic ontology that one may doubt whether one is using a convenient terminology.

On the other hand, if one considers the word state' as describing some potentiality rather than a reality — one may even simply replace the term state’ by the term potentiality' — then the concept of coexistent potentialities’ is quite plausible, since one potentiality may involve or overlap other potentialities. All these difficult definitions and distinctions can be avoided if one confines the language to the description of facts, i.e., experimental results.

However, if one wishes to speak about the atomic particles themselves one must either use the mathematical scheme as the only supplement to natural language or one must combine it with a language that makes use of a modified logic or of no well-defined logic at all.

In the experiments about atomic events we have to do with things and facts, with phenomena that are just as real as any phenomena in daily life. But the atoms or the elementary particles themselves are not as real; they form a world of potentialities or possibilities rather than one of things or facts.