The Heisenberg equations of motion for a quantum particle of mass $m$ are deduced from the infinitesimal qr-number equations of motion for the particle. The infinitesimal qr-number equations, and hence the standard quantum mechanical equations, are related to the qr-number equations in much the same way as the equations of geometric optics are related to those of wave optics. The qr-number equations of motion for a quantum particle of mass $m$ describe the motion of a lump, given by on open set in the qr-number space of the particle, while the infinitesimal qr-number equations describe the motion of a point-like particle. The qr-number equations of motion are the Hamiltonian equations of motion for a classical particle of mass $m$ expressed in qr-numbers. The proof requires that the particle's position operators have only continuous spectrum and the force functions are smooth.The Heisenberg equations of motion for a quantum particle of mass $m$ are deduced from the infinitesimal qr-number equations of motion for the particle. The infinitesimal qr-number equations, and hence the standard quantum mechanical equations, are related to the qr-number equations in much the same way as the equations of geometric optics are related to those of wave optics. The qr-number equations of motion for a quantum particle of mass $m$ describe the motion of a lump, given by on open set in the qr-number space of the particle, while the infinitesimal qr-number equations describe the motion of a point-like particle. The qr-number equations of motion are the Hamiltonian equations of motion for a classical particle of mass $m$ expressed in qr-numbers. The proof requires that the particle's position operators have only continuous spectrum and the force functions are smooth.

arXiv:1901.08892Jan 19, 20191 citation

QUANTUM NUMBERS OF PARTICLES

Seitaro Nakamura

Progress of Theoretical PhysicsJan 1, 19650 citations

Complete solutions of particle in three dimensional box with variations in main quantum number

Bambang Supriadi, Lailatul Nuraini, A S R Maulani +3 more

Abstract Particle in a box is one of the applications of the Schrodinger equation. Schrodinger equation gives wave function which is used to determine the probability and expectation value of particle in a box. It is also can explain the energy levels of the particle. This study aimed to determine the probability, expectation values, and energy levels of a particle in a three-dimensional box with variations in main quantum numbers in each coordinate axis. The magnitude of the wave function and energy levels are influenced by the main quantum number and the width of the box. The results show that variations in the main quantum number influence the probability of particle in the three-dimensional box except for the width of the box <?CDATA $\frac{L}{2}$ ?> L 2 and L the probability of particle showing the same value in all variations of the main quantum number. Variations in the main quantum number also influence the expectation value of particle in a three-dimensional box except for the width of the box L the expectation value shows the same value in all variations of the main quantum number. All variations of the main quantum number also influence the magnitude of the energy level of the particle.Abstract Particle in a box is one of the applications of the Schrodinger equation. Schrodinger equation gives wave function which is used to determine the probability and expectation value of particle in a box. It is also can explain the energy levels of the particle. This study aimed to determine the probability, expectation values, and energy levels of a particle in a three-dimensional box with variations in main quantum numbers in each coordinate axis. The magnitude of the wave function and energy levels are influenced by the main quantum number and the width of the box. The results show that variations in the main quantum number influence the probability of particle in the three-dimensional box except for the width of the box <?CDATA $\frac{L}{2}$?> L 2 and L the probability of particle showing the same value in all variations of the main quantum number. Variations in the main quantum number also influence the expectation value of particle in a three-dimensional box except for the width of the box L the expectation value shows the same value in all variations of the main quantum number. All variations of the main quantum number also influence the magnitude of the energy level of the particle.

Journal of Physics Conference SeriesMay 1, 20200 citations

TOWARD A SET OF THE QUANTUM NUMBERS

Alexey Orlovsky

Aug 10, 20260 citations

Number-unconstrained quantum sensing

Morgan W Mitchell

Abstract Quantum sensing is commonly described as a constrained optimization problem: maximize the information gained about an unknown quantity using a limited number of particles. Important sensors including gravitational wave interferometers and some atomic sensors do not appear to fit this description, because there is no external constraint on particle number. Here, we develop the theory of particle-number-unconstrained quantum sensing, and describe how optimal particle numbers emerge from the competition of particle-environment and particle-particle interactions. We apply the theory to optical probing of an atomic medium modeled as a resonant, saturable absorber, and observe the emergence of well-defined finite optima without external constraints. The results contradict some expectations from number-constrained quantum sensing and show that probing with squeezed beams can give a large sensitivity advantage over classical strategies when each is optimized for particle number.Abstract Quantum sensing is commonly described as a constrained optimization problem: maximize the information gained about an unknown quantity using a limited number of particles. Important sensors including gravitational wave interferometers and some atomic sensors do not appear to fit this description, because there is no external constraint on particle number. Here, we develop the theory of particle-number-unconstrained quantum sensing, and describe how optimal particle numbers emerge from the competition of particle-environment and particle-particle interactions. We apply the theory to optical probing of an atomic medium modeled as a resonant, saturable absorber, and observe the emergence of well-defined finite optima without external constraints. The results contradict some expectations from number-constrained quantum sensing and show that probing with squeezed beams can give a large sensitivity advantage over classical strategies when each is optimized for particle number.

arXiv:1704.01293Quantum Science and TechnologyAug 17, 201712 citations

On quantum particles

Masao Nagasawa

Chaos Solitons & FractalsJun 1, 20024 citations

The Heisenberg equations of motion for a quantum particle of mass

m

are deduced from the infinitesimal qr-number equations of motion for the particle. The infinitesimal qr-number equations, and hence the standard quantum mechanical equations, are related to the qr-number equations in much the same way as the equations of geometric optics are related to those of wave optics. The qr-number equations of motion for a quantum particle of mass

m

describe the motion of a lump, given by on open set in the qr-number space of the particle, while the infinitesimal qr-number equations describe the motion of a point-like particle. The qr-number equations of motion are the Hamiltonian equations of motion for a classical particle of mass

m

expressed in qr-numbers. The proof requires that the particle's position operators have only continuous spectrum and the force functions are smooth.

\frac{L}{2}

?> L 2 and L the probability of particle showing the same value in all variations of the main quantum number. Variations in the main quantum number also influence the expectation value of particle in a three-dimensional box except for the width of the box L the expectation value shows the same value in all variations of the main quantum number. All variations of the main quantum number also influence the magnitude of the energy level of the particle.