arXiv:1002.0310 · Physica Scripta

From qubits and actions to the Pauli–Schrödinger equation

Salomon S Mizrahi

Published Jul 1, 2009• 10 citations

Abstract

Here I show that a classical or quantum bit state plus one simple operation, an action , are sufficient ingredients to derive a quantum dynamical equation that rules the sequential changes of the state. Then, by assuming that a freely moving massive particle is the qubit carrier, it is found that both, the particle position in physical space and the qubit state, change in time according to the Pauli–Schrödinger equation. So, this approach suggests the following conjecture: because it carries one qubit of information the particle motion has its description enslaved by the very existence of the internal degree of freedom. It is compelled to be described no more classically but by a wavefunction. I also briefly discuss the Dirac equation in terms of qubits.

DOI: 10.1088/0031-8949/2009/t135/014007

Related papers

Perceptions of the Schrodinger Equation

Spyros Efthimiades

The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.

arXiv:1307.6108Bulletin of the American Physical SocietyJul 23, 20130 citations

Quantum mechanical motion of classical particles

V. D. Rusov, Konstantin Alexandrovich Lukin

The Schrödinger equation for the particle wave function introduced via its action was derived from Newton equation for a point-like particle moving under effect of both potential force and fluctuation-dissipative environment, provided considering only stable motion of the particle. The model considered assumes existence of wave function as a physical field rather than just a mathematical abstraction.

Journal of Physics Conference SeriesMay 10, 20120 citations

The Relation between Classical Mechanics and Wave Mechanics

Yoshio Nishiyama

Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.

Progress of Theoretical PhysicsNov 1, 19631 citation

Quantum Physics and Quantum Computation

Andreas Wichert

Classical mechanics is usually valid at the macro scale. The changes in the position and the momentum of the object over time are described by the Hamiltonian equation of motion. The wave function in quantum mechanics, if unobserved, evolves in a smooth and continuous way according to the Schrodinger equation, which is related to the Hamiltonian equation of motion. This equation describes a linear superposition of different states at time. A quantum mechanical description of a physical system is related to a probabilistic representation; it is described by a vector in Hilbert space. This description extends the two- or three-dimensional Euclidean space into spaces that have any finite or infinite number of dimensions. The state of a compound system is projected to the subspace that corresponds to the observed state and the vector representing the state is renormalized to the unit length. An observable describes a subspace for some dimensions with a special case of one dimension.

Chapman & Hall/CRC eBooksNov 28, 20230 citations

Quantum equations from Brownian motion

B. S. Rajput

The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.

Canadian Journal of PhysicsFeb 1, 20111 citation

Hamilton's Equations Within a Quantum Energy Eigenstate?

Francesco R. Ruggeri

In previous notes (1) we argued that the free particle action A (either nonrelativistic or relativistic) may have v (velocity) written as x/t and x and t varied independently to yield: dA/dx partial = p and dA/t partial = -E with A=-Et+px, the Lorentz invariant classical Action = t Lagrangian. We further argued that one may write eigenvalue equations i.e. -idexp(iA)/dx partial = p exp(iA) and idexp(iA)/dt partial = Eexp(iA) and call exp(iA) a type of probability. We noted that if a potential is present one may use P(p/x)= a(p)exp(ipx)/W(x) [W(x)=wavefunction = Sum over p a(p)exp(ipx)] and establish a conservation of energy equation KE(x) + V(x) = E such that there exists the same constant E at each x. We described this in two ways. First E represents binding energy so at each x one has the same binding energy i.e. each x point maps to the same E as part of the mass of an overall moving system. Secondly, we argued that E was a parameter indicating that equilibrium existed at each x point, much like temperature T. In this note we wish to consider a different idea suggested in (2), namely that if one has a quantum state: W(x)= Sum over n a(n,t) Wn(x) this may be written as: <i |H| j> a*(i,t)a(j,t) = <W H W>.. If Hamilton’s equations for complex variables are applied to these then: i d/dt partial a*(i,t) = d/da(i,t) partial { <W H W> } ((A)) (2) argues this is the time-dependent Schrodinger equation. In other words, probability weights of the Schrodinger equations (which are complex) obey Hamilton’s classical equation. In this note, we point out that the formalism dA/dx partial = p already introduces operators into classical physics, so it follows that H is an operator which may act on states. We consider the idea of ((A)) as applied to a free quantum particle moving. Next, we apply ((A)) to a quantum bound state and argue that a classical Hamilton’s equation applied to d/dt a(p) partial may describe equilibrium. Finally ((A)) may be applied to a sum of eigenstates, but this is already shown in (2).

Feb 26, 20220 citations