It is shown that in classical mechanics it is not possible to measure the body speed with absolute accuracy; at any accuracy, only the interval containing an infinite number of speed values can be established at best, while in reality only one physical speed value is true (current), and all remaining values of the interval are imaginary (non-existent in reality). It is shown that in quantum mechanics, in contrast to the classical case, all the values of speeds are true. It is noted that the comparison of classical and quantum approaches creates a rule according to which the relationship between the actual speed of an object and the measured continuous velocity spectrum is accompanied by the transformation of the quantifier of existence and uniqueness to the universal quantifier. It has been established that for a quantum free particle there is a monochrome wave function, and the wave packet exists only in the minds of researchers, which is equivalent to the fact that the unique speed value in the interval of speeds is true, and every other is imaginary.
It is shown that in classical mechanics it is not possible to measure the body speed with absolute accuracy; at any accuracy, only the interval containing an infinite number of speed values can be established at best, while in reality only one physical speed value is true (current), and all remaining values of the interval are imaginary (non-existent in reality). It is shown that in quantum mechanics, in contrast to the classical case, all the values of speeds are true. It is noted that the comparison of classical and quantum approaches creates a rule according to which the relationship between the actual speed of an object and the measured continuous velocity spectrum is accompanied by the transformation of the quantifier of existence and uniqueness to the universal quantifier. It has been established that for a quantum free particle there is a monochrome wave function, and the wave packet exists only in the minds of researchers, which is equivalent to the fact that the unique speed value in the interval of speeds is true, and every other is imaginary.
Based on the equation of the free movement of inert body built formal analogue of the wave function, which is the solution of the differential equation, almost identical to the Schrodinger equation for a free particle. It is shown that the inverse image of the wave function does not substantially coincides with the equation of rectilinear uniform motion. From a comparison of the formal analogue of the wave function and the classical wave equation to obtain an expression for the wave kinetic energy of an inert body. Show that a formal analogue of the wave function is a function monochrome, for which the phase and group velocities coincide and are equal velocity of the body. We consider two alternative variants of formation of the wave packet for a formal analogue of the wave function, according to which the body either moves simultaneously with all measurement speed range, or with a single.
The present study is concerned with the problem of realism in quantum mechanics. Contrary to the prevailing view that quantum mechanics forces us to abandon the concept of realism and reality, the author attempts to substantiate another standpoint. Quantum objects indeed exist in a different way from classical bodies. Before the act of measurement begins, one cannot attribute any specific properties to the quantum object observed, as they only occur in the course observation. The concept of observation, however, should not be interpreted as the inclusion of subjective traits in the description of nature. Observation records processes in space and time and establishes the fact of transition from the potential to the reality. The potential is described by a wave function, which in its meaning does not fall in the habitual space and time, whereas the actual is described in terms of a reduction of the wave function, i.e. the transition from a range of possible states to the single one, the observed. Whatever has been observed, in contrast, already belongs to ordinary space and time and can be detected. What follows is that, in quantum mechanics, it is the notion of reality which needs to be reconsidered, not the realism as such rejected. A reconsideration of this sort is conducted here within the framework of a double-modus picture of reality including both potential and actual being, where the quantum level of existence is distinguished from the observed modi of being as described by means of classical concepts. Such a scheme can be described in terms of Aristotelian metaphysics (the fact first pointed out by Heisenberg). The act of transition from potential to actual can be accounted for both in the language of Aristotle’s metaphysics and the language of modern science. In traditional metaphysics it is called the 'principle of individuation', while in modern science it is known as the extended Mach principle. It is argued that the development of these ideas leads to the rejection of Cartesian paradigm in which the existence of a material object always means an existence in space and time.
A quantum particle is often described as exhibiting dual particle-wave nature. A particle seems to be an object which obeys Newton’s laws/special relativity with a wave also being a phenomenon existing in Newtonian mechanics (special relativity). In other words the particle and wave are governed by x-t external variables which have infinite resolution. It was originally thought that x and t in addition to being external were absolute, but special relativity changed this idea. We have suggested in previous notes that a free quantum particle with energy E and momentum p creates its own space time i.e time intervals are proportional to 1/E and space to 1/p. This means no external clock is needed and different internal rulers and clocks exist for different E,p. It also suggests finite time and space intervals. In fact, an absolute is introduced namely A=0 = -Et+px with t and x being independent. This suggests that t→b/E and x→b/p. A is the free particle Lagrangian (both relativistic and nonrelativistic) with x and t being considered independent. Thus we suggest that the external x ruler and t clock of Newtonian mechanics represent an average of the internal picture and are also described by A, but with x/t=v and not x and t being treated as independent. In (1) we showed that having discrete x spacing with a running time (i.e. Netwon’s time variable) and insisting on spatial invariance leads to dA/dx partial = p. Thus instead of considering particle wave duality we suggest an underlying wave structure which creates its own spacetime intervals (1/E, 1/p), but may be mapped in certain situations to the external running x and t of Newton’s theory. For example, consider an electron (a quantum particle) with a fixed p moving over a distance much greater than its wavelength. It may be mapped to Newton’s external running x,t values with x=vt, but this is an average mapping into continuous x and t. In reality the situation is more complicated because there may be a Bohm-Aharonov effect occurring. Thus the underlying wave picture is in fact physically important and linked to interactions. Thus a self-ruler is not simply a measuring device - interactions may occur unlike in the case of the external ruler. Given that p defines a kind of spatial invariance approximated by a periodic function exp(ipx) (eigenfunction of -id/dx) and that Newton’s running time is not needed in a quantum bound state equilibrium, one may wonder what one should do in the case of a particle bound in a V(x) potential. In Newtonian mechanics V(x) suggests non-spatial invariance and v(x) velocity changes with x as well in contrast with exp(ipx) which is periodically invariant. In fact Newtonian mechanics (and Lagrangian/Hamiltonian as well) focus on d/dt i.e x(t). We have argued above that time intervals 1/E are independent of 1/p spatial ones. It may be noticed, however, that different p values lead to different wavelengths and so even though there is a kind of spatial invariance for each, when compared to each other this invariance is broken and superpositions of exp(ipx) may create local x probability densities. At the same they create spatial momentum densities because -d/dx (d/dx) 1/2m yields a kind of average kinetic energy when normalized. This seems to describe the x-p uncertainty relationship. In order to create spatial resolution (other than the built-in one in the wavelength) one needs to superpose different momenta, each of which are spatially invariant according to their periodic scheme exp(ipx). Thus one may create an average 1/2m <pp> and obtain a vrms. This may be described in terms of Newtonian external running x and t variables which have infinite resolution, but this vrms does not map to a physical state i.e to exp(i (mvrms) x) localized at x. Rather the various exp(ipx)’s which create vrms are the physical entities as may be seen by knocking a particle out of a bound state. The physical exp(ipx) momenta states have spatial invariance and so cannot be associated with a specific position in space. Only their superposition creates the notion of specific position in space, but then there is a distribution of p. This seems to be the basis for the x-p uncertainty principle. Thus we suggest that the particle of particle-wave duality is created through a mapping of a physical wave scenario to a continuous (infinite resolution) x-t scheme which in reality is an abstraction because each p,E free particle state has it sown internal clock-ruler i.e. 1/E and 1/p time and space units. Given that 1/E and 1/p have finite values the idea of dt and dx tending to 0 only makes sense as an abstraction which only holds in certain situations.
Classical physics is traditionally characterized as deterministic and founded on the real number continuum, while quantum mechanics is distinguished by its indeterminism and fundamental uncertainty relations. This apparent dichotomy, however, obscures a deeper continuity. We argue that both frameworks embody a common ontological principle: the physical prohibition of absolute precision. In classical physics, this prohibition is implicit, manifesting in information-theoretic bounds, the non-physicality of infinite-precision initial conditions (chaos theory), and the breakdown of determinism in singular configurations (Norton's dome). These phenomena reveal that the classical reliance on mathematical real numbers with infinite information content is an unphysical idealization. Quantum mechanics, by contrast, does not merely reflect this prohibition; it provides its precise quantitative expression. The Heisenberg uncertainty principle σxσp≥ℏ/2 and the non-commutative structure of observables are the exact mathematical formulation of the fact that physical information is fundamentally finite. The Planck constant ℏ is the universal scale that quantifies the minimal phase space volume—the “atom” of information—below which no further refinement is possible. Thus, quantum mechanics is not a separate theory but the completion of classical physics: it supplies the missing quantification of the principle that nature admits no infinite information density, and consequently, no absolutely precise physical quantities.
The paper deals with the motion of a Gaussian wave-packet in a parabolic potential field, contrasting the quantum-mechanical behaviour of the packet with that which follows from pure classical mechanics. The behaviour is entirely classical only if the initial phase of the space-wave-function is linear in the space coordinate; in which case the wave-packet is initially a ‘best-possible’ packet according to Heisenberg's uncertainty relationship. The paper concludes with a brief reference to the similar problem for a free particle. Note added in Proof, 16 February 1946. Through the courtesy of Mr M. S. Bartlett, we have been privileged to see the manuscripts of some work by Mr J. E. Moyal and himself, in which the motion of a probability wave-packet is discussed in more general terms than our own: and in which it appears that the motion of our wave-packet, which we have described as non-classical, is indeed actually classical provided that a certain correlation exists between momentum and position in the p, q-distribution function. With this knowledge we now find that classical and quantal predictions agree precisely provided that the initial classical probability function is given by this is a particular case of their more general results.
Conceptual difficulties still remain in quantum mechanics, even if they had already been identified by its inventors. This does not mean that the theory is not successful! The reality is quite the opposite: in fact, independently of these difficulties, quantum mechanics is certainly one of the most successful theories of all science. One can even consider that its ability to adapt to new situations is one of its most remarkable features. It continues to give efficient and accurate predictions while new experiments are performed, even in situations that the founding fathers had no way to imagine. Actually, there are very few theories that have been verified with the same accuracy in so many situations. Nevertheless, it remains true that conceptual difficulties subsist, and their discussion is the object of this chapter. As we will see, most of them relate to the process of quantum measurement, in particular to the very nature of the random process that takes place on this occasion. We have seen that, in most cases, the wave function evolves gently, in a perfectly predictable and continuous way, according to the Schrödinger equation; in some cases only (when a measurement is performed), unpredictable changes take place, according to the Von Neumann postulate of state vector reduction. Obviously, having two different postulates for the evolution of the same mathematical object is very unusual in physics. The notion was a complete novelty when it was introduced, and still remains unique, but also the source of difficulties – in particular logical difficulties related to the compatibility between the two different postulates.