Numbers: Their Tales, Types, and Treasures.

Chapter 11: Numbers and Philosophy

11.10.LIMITS OF THE MODEL OF NATURAL NUMBERS

Numbers do not always have a clear and unique meaning when applied to natural phenomena. Within quantum mechanics, the state of the physical system with a discrete energy spectrum is usually described in terms of a few “quantum numbers.” For example, one of the simplest quantum systems is the so-called harmonic oscillator. This is the quantum analog of a mass point attached to a spring—a particle bound by an attractive force that increases with the distance from the center of force. In quantum mechanics, the energy of the harmonic oscillator is quantized—that is, only integer multiples of an energy unit (a quantum of energy) can occur as the result of an energy measurement. When the harmonic oscillator is in a state with quantum number 5, this means that it has a total energy of 5 energy quanta (ignoring, for simplicity, the ground-state energy). Thus, measuring the energy effectively means to count the number of energy quanta belonging to a certain state of the harmonic oscillator. But the states with definite energy quantum numbers are, in fact, the exception. There are infinitely many others, which are superpositions of states with different quantum numbers. One could combine a state with quantum number 3 and a state with quantum number 4 into a new state, whose quantum number remains undetermined prior to an energy measurement (which would sometimes give the result 3 and sometimes the result 4). In the mathematical formalism of quantum mechanics, the energy is therefore not represented by a number (as in classical mechanics), but by a more complicated mathematical object, which is associated with all possible quantum numbers simultaneously. (In mathematical terms: the physical quantity is represented by a “linear operator” and its possible values, the quantum numbers, are represented by the so-called eigenvalues of the linear operator. In quantum mechanics, therefore, a physical quantity need not have a definite value but may have many different values at the same time.) In general, the number of energy quanta in a particular state is undetermined, and then, according to the standard interpretation of quantum mechanics, the precise value of the energy is not just unknown to us, but it does not exist as a precise value, only as a probability distribution of the possible values.

Strangely enough, in quantum mechanics of many particle systems, the property “number of particles” is also represented by a linear operator with many different possible values. Hence, the quantum system has states where the number of particles is not determined; a system could have 2 or 3 or 4 particles with equal probability. Creation and annihilation processes occurring with certain probabilities would soon turn a state with n particles into a new state, which is a superposition of states with different particle numbers. In such a state, the precise number of particles of the quantum system is undetermined and the physical system then simply does not have the property of containing a definite number of particles. At the level of elementary particles, the notion of number, thus, seems to lose much of its clarity. It is not even clear what it means to talk of a set of particles if the particles have no individuality and if their number is unsharp and undetermined.