Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Potentials as the Result of Unobserved Degrees of Freedom

C. K. Iddings, M. Teague

Published Jun 1, 1971• 1 citation
Publisher

Abstract

Following a proposal of Hertz, we show that the one-dimensional motion of a point mass in a potential is dynamically equivalent to free particle motion of the particle in a curved space of two dimensions, where one of the coordinates is cyclic and is not observed. We consider both the classical and quantum mechanical cases.
DOI
10.1119/1.1986245

Related papers

Relating the free particle with the harmonic oscillator

G. F. Torres del Castillo

We show that by means of a coordinate transformation in the extended configuration space the problem of a free particle can be related to that of a harmonic oscillator in classical mechanics and in quantum mechanics.

Revista Mexicana de Física•May 1, 2025•1 citation

Nonrelativistic Dynamics of the Point with Internal Degrees of Freedom and Spinning Electron

А. Н. Тараканов

Nonrelativistic formalism is developed, which allows describing systems with internal degrees of freedom in the scalar potential field U, which is a function both on relative coordinates and time, and on relative speed and accelerations. The equation for energy, which is an integral of motion when U satisfies to certain differential condition, is derived for the general case. For a free mass point all solutions of the equations of motion in the center-of-inertia reference frame, moving with constant velocity, are obtained. As a result, the center of mass follows along a complicated trajectory round a direction of motion of the center of inertia. This motion can be interpreted as Zitterbewegung. It is shown that equation of moments is a consequence of the equation of motion. The general equation of motion for a spin which can be interpreted from positions of a classical mechanics is written down. Application of the obtained results to the electron leads to new conception of electrical charge, sign of which corresponds to a sign of spin polarization.

•Jan 1, 2012•4 citations

Zitterbewegung as purely classical phenomenon

А. Н. Тараканов

Nonrelativistic formalism is developed, which allows describing systems with internal degrees of freedom in the scalar potential field $U$, which is a function both on relative coordinates and time, and on relative speed and accelerations. The equation for energy, which is an integral of motion when $U$ satisfies to certain differential condition, is derived for the general case. For a free mass point all solutions of the equations of motion in the center-of-inertia reference frame, moving with constant velocity, are found. As a result, the center of mass follows along helical and more complicated trajectories round a direction of motion of the center of inertia. This motion can be interpreted as trembling movement (Zitterbewegung). On this basis a conclusion is done that Zitterbewegung has purely classical origin, arising even in a nonrelativistic case if internal degrees of freedom are taken into account. The general equation of motion for a spin which can be interpreted from positions of a classical mechanics is written down. Application of the obtained results to the electron leads to new conception of electric charge, sign of which corresponds to a sign of spin polarization.

arXiv:1201.4965•Jan 24, 2012•1 citation

Potential Energies of Identical Motion in Classical and Quantum Mechanics

David B. Berkowitz, Peter D. Skiff

Potential energies are found for which the stationary-state motion of a particle in a static force field is the same according to the Schrödinger equation as in classical mechanics.

American Journal of Physics•Nov 1, 1972•13 citations

Bound Quantum Mechanics, Classical Mechanics and Flow/Flux Scheme

Francesco R. Ruggeri

In a previous note (1), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) A(x,t) turns out to be the classical action tL(v) (L=Lagrangian) with x/t=v and the above hold for both the relativistic and nonrelativistic cases. We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time). In (2) we argued that ((1a)) and ((1b)) are consistent with p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv) (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is: (E-V(x)) = .5mvv. The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e. id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt (linked to exp(-iEt+ipx) and periodic probability motion (spin) and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin). Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.

•Jan 6, 2022•0 citations

Scattering

Abstract Besides the bounded, periodic classical motion of particles in potentials as shown in Fig. 5.2 (for energy E1 ), there is also the possibility of unbound states which are not localized in space and not repetitive in time; these correspond to particles incident on the potential and which subsequently “bounce “ off (energy E2 in Fig. 5.2) or temporarily change their speeds as they go over the potential (energy E3 ). Classical mechanics, which solves for the exact trajectories in either case, makes little distinction between the two types of motion aside from some tech- nical details,

•Apr 13, 2006•0 citations
Nonrelativistic formalism is developed, which allows describing systems with internal degrees of freedom in the scalar potential field U, which is a function both on relative coordinates and time, and on relative speed and accelerations. The equation for energy, which is an integral of motion when U satisfies to certain differential condition, is derived for the general case. For a free mass point all solutions of the equations of motion in the center-of-inertia reference frame, moving with constant velocity, are obtained. As a result, the center of mass follows along a complicated trajectory round a direction of motion of the center of inertia. This motion can be interpreted as Zitterbewegung. It is shown that equation of moments is a consequence of the equation of motion. The general equation of motion for a spin which can be interpreted from positions of a classical mechanics is written down. Application of the obtained results to the electron leads to new conception of electrical charge, sign of which corresponds to a sign of spin polarization.
Nonrelativistic formalism is developed, which allows describing systems with internal degrees of freedom in the scalar potential field UUU, which is a function both on relative coordinates and time, and on relative speed and accelerations. The equation for energy, which is an integral of motion when UUU satisfies to certain differential condition, is derived for the general case. For a free mass point all solutions of the equations of motion in the center-of-inertia reference frame, moving with constant velocity, are found. As a result, the center of mass follows along helical and more complicated trajectories round a direction of motion of the center of inertia. This motion can be interpreted as trembling movement (Zitterbewegung). On this basis a conclusion is done that Zitterbewegung has purely classical origin, arising even in a nonrelativistic case if internal degrees of freedom are taken into account. The general equation of motion for a spin which can be interpreted from positions of a classical mechanics is written down. Application of the obtained results to the electron leads to new conception of electric charge, sign of which corresponds to a sign of spin polarization.
In a previous note (1), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) A(x,t) turns out to be the classical action tL(v) (L=Lagrangian) with x/t=v and the above hold for both the relativistic and nonrelativistic cases. We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time). In (2) we argued that ((1a)) and ((1b)) are consistent with p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv) (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is: (E-V(x)) = .5mvv. The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e. id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt (linked to exp(-iEt+ipx) and periodic probability motion (spin) and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin). Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.
Abstract Besides the bounded, periodic classical motion of particles in potentials as shown in Fig. 5.2 (for energy E1 ), there is also the possibility of unbound states which are not localized in space and not repetitive in time; these correspond to particles incident on the potential and which subsequently “bounce “ off (energy E2 in Fig. 5.2) or temporarily change their speeds as they go over the potential (energy E3 ). Classical mechanics, which solves for the exact trajectories in either case, makes little distinction between the two types of motion aside from some tech- nical details,