The Link between Classical and Quantum Mechanics

Francesco R. Ruggeri

In quantum mechanics, it is often stressed that if one knows position (x) with complete certainty, then momentum (p) is completely uncertain. A similar result is applied to energy and time. It is also stressed that quantum mechanics is mathematically a wave type theory describing probability waves. In a previous note (1), we argued that one may obtain the Schrodinger equation from exp(i Action), where Action is the classical action, by varying x and t in v=x/t independently. A main idea of classical mechanics is that x and t are not independent, but are directly linked i.e. x(t). In this note, we argue that wave behaviour in quantum mechanics and in the wave equation for a photon decouples time and space which is the opposite notion of classical mechanics. One has partial derivatives (d/dx, d/dt) in an equation describing a problem, but there is a kind of decoupling of changes in x and t. There is still a relationship related to frequencies between the decoupled time and space functions as the wave-equation contains both d/dt d/dt and d/dx d/dx, but one does not have a scenario of x depending directly on t as in classical mechanics. This, we argue, leads to wave or a periodic type behaviour for both x and t. In the case of a potential V(x), there can be multiple x waves exp(ipx) stirred up. In the time-independent Schrodinger equation one does not even consider time (one has EW(x)) even though it is clearly present through exp(iEt), but x and t are not linked As a result, we argue that p and x are not linked because p=mv and if one had v(x), one could integrate dt = dx/v(x) to find an explicit relationship between x and t, as in classical mechanics, which negates the idea of wave mechanics. Thus, we argue that it is the “decoupling” of space and time that is important in quantum mechanics

A. Horzela

Emile Grgin, Aage Petersen

Partha Ghose

Nelson's stochastic mechanics links quantum mechanics to an underlying Brownian motion with the identification $\hbar = mσ$. Ghose's interpolating equation introduces a continuous parameter $λ$ that suppresses the quantum potential $Q[ψ]$ and yields a smooth transition between quantum ($λ=0$) and classical ($λ=1$) regimes. In this short note, we show that the Koopman--von Neumann (KvN) Hilbert-space formulation of classical mechanics emerges naturally as the $λ\to 1$ limit of this stochastic $σ$--$λ$ hierarchy. The KvN phase-space amplitude provides an operator representation of the classical Liouville equation, while the $λ$ parameter acts as a projection flow from the complex projective Hilbert manifold $\mathbb{C}P^n$ to its classical quotient $\mathbb{C}P^*/U(1)$, implementing phase superselection. This unified picture links quantum, stochastic, and classical dynamics within a single continuous framework.

Francesco R. Ruggeri

In classical physics, velocity follows from exact measurements of space and time intervals (i.e perfect resolution of both). Inertial mass follows from a resistance of an object to a force (push or pull). Momentum, however, which nonrelativistically is mv and should be associated with ideas of inertial mass, and space time resolution may be instead linked to the physical idea of an impulse. An impulse seems to be like a hit and does not appear to be linked directly to space time physically, although its consequences are i.e. a particle receiving the impulse moves in space and time. In classical mechanics, x(t) and so one considers either equally spaced dt intervals and corresponding dx(t) or vice versa (constant dx) if an integral only involves space dependent values. In quantum mechanics, one has two uncertainty principles, one related to E and t and the other to p and x. We argue that p is linked to the notion of an impulse which does not on the surface involve space or time. Given that p, even in nonrelativistic quantum mechanics, is mv there should at least be some built in notion of dx and dt. From the uncertainty principles one sees that dx (i.e. the wavelength) is linked to hbar/p and dt to hbar/E. Thus neither dt nor dx are constants, but depend on the motion. In the high energy quantum limit dt —> W(x)W(x) dx →1/v(x) dx where v(x) is classical velocity and W(x) the bound state wavefunction. The bound state wavefunction, however, applies to the envelope of WW (not the actual crests and troughs) and so the dx-dt scheme which emerges is a smoothed over one and not the actual dx -dp of quantum mechanics. We try to examine some of these ideas in this note.

Enrico G. Beltrametti, Slawomir Bugajski