Letters to the Editor

On Nikolsky’s Article “Principles of Quantum Mechanics”

V. A. Fock, Leningrad

In his article “Principles of Quantum Mechanics”* K. V. Nikolsky attempts to construct quantum mechanics entirely on the basis of statistics and to derive the quantum-mechanical apparatus (wave functions, operators, Hilbert space), considering exclusively probabilities and mathematical expectations (means) for various physical quantities.

Let us consider the basic requirement that such a construction must satisfy. From quantum mechanics we know that all probabilities and mathematical expectations are expressed in terms of wave functions and operators, and moreover an essential role is played not only by the modulus but also by the phase of the wave function; this phase is a quantity determined from experiment with accuracy up to an additive constant (one and the same for all states under consideration). Therefore any logical construction of quantum mechanics must derive from the initial postulates the properties of the wave function, including its phase.

If, as K. V. Nikolsky does, one proceeds from considering probabilities for various physical quantities, then first of all one would have to indicate a method for determining the phase of the wave function from these probabilities, and also to indicate the conditions that the probabilities must satisfy in order that the problem of determining the phase have a solution. Quite apart from the artificiality of such a formulation of the question, this problem is extremely difficult mathematically and, so far as can be seen, has no single-valued solution. However, K. V. Nikolsky does not even pose the problem of determining the phase, but seems to forget about the phase. On p. 545 he says that the wave functions \(c(k_n, l^*)\) and \(d(s_n, l^*)\) are determined by the equations

\[ w(k_n, l^*) = |c(k_n, l^*)|^2; \qquad w(s_n, l^*) = |d(s_n, l^*)|^2 \]

(where \(w\) is the probability), whereas these equations obviously determine only their moduli. A few lines below he says: “We express probabilities as the squares of the moduli of certain auxiliary numbers (functions), which remain arbitrary.” But from quantum mechanics it is known that the wave functions \(c(k_n, l^*)\) and \(d(s_n, l^*)\) are not only not arbitrary, but are expressed one through the other. Moreover, even if one considers not both functions \(c(k_n, l^*)\) and \(d(s_n, l^*)\) together, but only one of them, for example the first, it too will not be arbitrary, but must satisfy a number of conditions; for example, the set of quantities \(c(k_n, l^*)\) must form a closed system (the “argument” \(l^*\) and the “index” \(k_n\)).

* Uspekhi Fizicheskikh Nauk, 16, 537, 1936.

If the words quoted by K. V. Nikolsky are understood in the sense that he wants to express probabilities parametrically through auxiliary (wave) functions, then he is faced with the task of deriving and investigating the properties of these auxiliary functions. But this task is equivalent to the independent postulation of Hilbert space and the entire apparatus of quantum mechanics, i.e. precisely what he wants to avoid.

Let us return to the arguments of K. V. Nikolsky and understand by \(c(k_n, l^*)\) one of an infinite set of functions with a given square of the modulus. With the same right we could take, instead of \(c(k_n, l^*)\), the quantity

\[ c(k_n, l^*) e^{i f(k_n, l^*)}, \]

where \(f\) is a completely arbitrary real function of two arguments. Nikolsky expands the function \(c(k_s, l^*)\) into a series in an arbitrarily closed system of functions \(\varphi(n_\mu, l^*)\). In view of the arbitrariness in the choice of the variable phase \(f(k_s, l^*)\), neither the moduli nor the phases of the expansion coefficients can have any physical meaning. [Nikolsky denotes these coefficients by \(c(n_\mu, k_s)\) or \(c(k_s, n_\mu)\), probably wishing thereby to show that they are analogous to the quantities \(c(k_s, l^*)\), i.e. that the squares of their moduli give certain probabilities; although this is obviously not so, we shall not dwell on this.] In particular, it in no way follows that the quantities \(c(k_s, n_\mu)\) form an orthogonal system.

Next Nikolsky considers the mean value of the quantity \(K\), defined by the formula

\[ \text{mean value } K = \sum_{\mu=1}^{\infty} k_\mu \left| c(k_\mu, l^*) \right|^2 \]

and writes it in the form

\[ \text{mean value } K = \int dn\, \bar{\varphi}(n, l^*) \int K(n,n') \varphi(n', l^*)\, dn', \]

where \(K(n,n')\) denotes the sum

\[ K(n,n') = \sum_{\mu=1}^{\infty} k_\mu c(k_\mu,n)\,\overline{c(k_\mu,n')}. \]

Nikolsky regards this sum as the kernel of the operator for the physical quantity \(K\). Thus, he believes that from the mathematical expectation of \(K\) (i.e. from the diagonal element of the matrix for \(K\)) he has derived the entire operator \(K\) (i.e. all elements of the matrix for \(K\)). This is obviously impossible, since for given diagonal elements the remaining elements of the matrix may be arbitrary.

Further, on p. 549, Nikolsky says that the operator having the kernel

\[ K^2(n,n') = \sum_{\mu=1}^{\infty} k_\mu^2 c(k_\mu,n)\,\overline{c(k_\mu,n')} \]

represents the twice repeated operation \(K\). This would be true if the \(c(k_\mu,n)\) represented a closed system of functions, but for \(c(k_\mu,n)\) as defined by Nikolsky this is obviously false, since his \(c(k_\mu,n)\),

are subject only to the very general condition that the sum of the series

\[ c(k_s,l^*)=\sum_{\mu=1}^{\infty} c(k_s,n_\mu)\,\varphi(n_\mu,l^*) \]

have the prescribed modulus. Thus, a direct contradiction is obtained here.

Further contradictions arise for Nikolsky when he begins to consider jointly operators for two quantities \(K\) and \(R\). According to his definition, \(K\) is one of an infinite set of operators possessing the given diagonal elements of the matrix \((l^*K l^*)\). Similarly, \(R\) is one of an infinite set of other operators. Since these operators have been chosen at random from an infinite set of equivalent ones, there can be no question of any definite “mutual properties” following from their definition, for example, commutativity or noncommutativity of these operators. Therefore all of Nikolsky’s arguments concerning the calculation of mean quadratic deviations are devoid of any logical foundation.

We have examined in detail § 3 of K. V. Nikolsky’s article, which contains its principal ideas. We have convinced ourselves that, although formulas analogous to Nikolsky’s formulas occur in quantum mechanics, there they follow from the basic assumptions about wave functions and operators. In Nikolsky’s article, however, these formulas are completely unsubstantiated. But if we accept them even as postulates, then they will already contain all the basic hypotheses of quantum mechanics. Meanwhile Nikolsky believes that his formulas do not depend on these hypotheses and constitute a method independent of physics (p. 541). This is the further error of K. V. Nikolsky.

As a result, the reader of K. V. Nikolsky’s article receives the impression that quantum mechanics is merely a kind of statistics. Yet the most important achievement of modern physics should be recognized precisely in the fact that it has learned to observe elementary physical processes, individual particles. Quantum mechanics gives a formulation of the laws to which these elementary processes are subject. In the language of quantum mechanics the result of a single measurement can be written down. Therefore, although the predictions given by quantum mechanics have, in the general case, a statistical character, one should not forget that quantum mechanics is by no means reducible to statistics.