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

Europhysics Letters (EPL)

Centre-Of-Mass Motion in Many-Particle Systems. Application to Atoms and to Nuclei

Roger F. Prat

Published Feb 14, 1992• 0 citations
Publisher

Abstract

The centre-of-mass position and momentum of an ( N + 1)-particle system are treated as superselection rules. As a consequence, particle positions and momenta do not obey Heisenberg commutation rules. A Schrödinger equation for the internal state is obtained in terms of the ( N + 1)-particle original positions. Application to atoms justifies the fixed-nucleus approximation as consistent with the classical CM premises. Application to nuclear structure gives at once the Schrödinger equation for the intrinsic states. The present theory allows system states to be altogether localized, translation invariant with respect to a laboratory frame, and with a nonzero total momentum.

DOI
10.1209/0295-5075/18/1/004

Related papers

Centre of mass motion in non-relativistic quantum electrodynamics

W P Healy

Considers aspects of the non-relativistic quantum theory of radiation that occur when the coordinates and momenta of all material particles (nuclei as well as electrons) are treated as dynamical variables. The formalism gives the complete scheme of atomic field equations for an ensemble of ions, free electrons, atoms or molecules and comprises the multipolar Hamiltonian for systems composed solely of electrically neutral atoms or molecules. It also deals with effects due specifically to motion of the nuclei, such as the Doppler effect or the recoil that occurs when photons are emitted, scattered or absorbed.

Journal of Physics A Mathematical and General•Feb 1, 1977•19 citations

Center-of-Mass Motion in Many-Particle Systems

David J. Ernst, Carl M. Shakin, Roy M. Thaler

We have reexamined the problem of constructing a nuclear ground-state wave function consistent with the requirement of translational invariance, beginning with a wave function which does not have this property. A class of transformations is constructed which transforms the initial-state vector into an eigenstate of total momentum. The particular transformation of this class which leads to the lowest intrinsic energy is given.

Physical Review C•Mar 1, 1973•18 citations

The Schrödinger Wave Equation

Daniel D. Pollock

The approach is to start with some of the elementary ideas of classical mechanics and to translate these concepts to their equivalents in quantum mechanics. This will be done by changing the classical components of the total energy of particle into their quantum mechanical equivalents and arriving at the Schrodinger equation. The conjugate variables, position and momentum, in the Hamiltonian functions can be determined precisely for classical mechanical applications. This chapter deals with the Schrodinger model of the behavior of single electrons as affected by potential wells. The wave mechanic approach is extended to provide an outline of the interactions of many electrons with nuclei. In terms of the Pauli exclusion principle, the four quantum numbers of the electron states must be different. As additional electrons are added, the states available for occupation can be filled only according to the rules governing the quantum numbers.

CRC Press eBooks•Oct 1, 2020•0 citations

Quantum Mechanics and Center of Mass Considerations?

Francesco R. Ruggeri

In the literature (1), the Schrodinger equation: {p1p1/2m + p2p2/2m + V(x1-x2)} W(x1,x2,t) = id/dt partial W(x1,x2,t) may be transformed using P=p1+p2 and (p=m1p2-m2p1)/M and x=x1=x2 where M=m1+m2 ((1)). This leads to a bound Schrodinger equation in x and a “center of mass” equation” in P i..e Hcm =PP/2M which depends on m1 and m2, but not binding energy. . In a previous note, we argued that a free particle’s classical action A (relativistic or nonrelativistic) may be obtained by setting v=x/t and treating x and t as independent (2). This leads to dA/dx partial=p and dA/dt partial = -E. One may write eigenvalue equations: -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = E exp(iA). A solution is A=-Et+px with E,t,p and x independent. -Et+px is a Lorentz invariant and this form is used in both relativistic and nonrelativistic quantum mechanics (although E and p have different values). We argue that a free particle rest mass mo may actually involve internal quantum physics, yet the “system” is replaced with a parameter mo and exp(-iEt+px) used to describe it. One may try to apply this approach to a bound system, but exp(-iEt+px) is a statistical probability like object. Thus one needs a statistical theory. We have suggested using an ensemble of exp(-iEt+px) with E=En for all p to indicate a resonance. Then: { Sum over p pp/2m a(p)exp(ipx) } / {Sum over p a(p)exp(ipx) } + V(x) = En ((2)) This is the Schrodinger bound equation. We argue that W=Sum over p a(p)exp(ipx) should be real and this implies that Sum over p a(p) p exp(ipx) = 0 i.e. average momentum for this type of average is 0. As a result, one may create a 4-vector using { 0, mcc + En} and use the Lorentz boost approach to find the momentum and energy in a moving frame. One may then take the nonrelativistic limit to find kinetic energy i.e. the full center of mass kinetic energy. We argue that this is equivalent to considering exp(-iEt+ipx) for the overall bound system which is now translating in space. We argue that ((2)) is a different approach to ((1)) in that binding energy, which may be significant with respect to mcc, should be part of the overall energy of the translating bound system which is called the “center of mass” system in the literature.

•Mar 26, 2022•0 citations

On the role of Spin in Quantum Mechanics

Salvatore Esposito

From the invariance properties of the Schrodinger equation and the isotropy of space we show that a generic (non-relativistic) quantum system is endowed with an ``external'' motion, which can be interpreted as the motion of the centre of mass, and an ``internal'' one, whose presence disappears in the classical limit. The latter is caused by the spin of the particle, whatever is its actual value (different from zero). The quantum potential in the Schrodinger equation, which is responsible of the quantum effects of the system, is then completely determined from the properties of the internal motion, and its ``unusual'' properties have a simple and physical explanation in the present context. From the impossibility to fix the initial conditions relevant for the internal motion follows, finally, the need of a probabilistic interpretation of quantum mechanics.

arXiv:quant-ph/9902019•Feb 5, 1999•6 citations

On the angular momentum of a system of quantum particles

O Chavoya-Aceves

The properties of angular momentum and its connection to magnetic moment are explored, based on a reconsideration of the Stern-Gerlach experiment and gauge invariance. The separation of angular momentum of a quantum system of particles into orbital angular momentum plus intrinsic angular momentum is reconsidered, within the limits of the Schrödinger theory. A proof is given that, for systems of more than two particles, unless all of them have the same mass, the possibility of having eigenvalues of the form $(n+1/2)\hbar$ is not excluded for the operators of angular momentum of the centre of mass and the internal angular momentum. \noindent\textbf{PACS: 03.65.Ta} angular momentum, magnetic moment, spin crisis

•Jan 1, 2014•0 citations
The centre-of-mass position and momentum of an ( N + 1)-particle system are treated as superselection rules. As a consequence, particle positions and momenta do not obey Heisenberg commutation rules. A Schrödinger equation for the internal state is obtained in terms of the ( N + 1)-particle original positions. Application to atoms justifies the fixed-nucleus approximation as consistent with the classical CM premises. Application to nuclear structure gives at once the Schrödinger equation for the intrinsic states. The present theory allows system states to be altogether localized, translation invariant with respect to a laboratory frame, and with a nonzero total momentum.
Considers aspects of the non-relativistic quantum theory of radiation that occur when the coordinates and momenta of all material particles (nuclei as well as electrons) are treated as dynamical variables. The formalism gives the complete scheme of atomic field equations for an ensemble of ions, free electrons, atoms or molecules and comprises the multipolar Hamiltonian for systems composed solely of electrically neutral atoms or molecules. It also deals with effects due specifically to motion of the nuclei, such as the Doppler effect or the recoil that occurs when photons are emitted, scattered or absorbed.
The approach is to start with some of the elementary ideas of classical mechanics and to translate these concepts to their equivalents in quantum mechanics. This will be done by changing the classical components of the total energy of particle into their quantum mechanical equivalents and arriving at the Schrodinger equation. The conjugate variables, position and momentum, in the Hamiltonian functions can be determined precisely for classical mechanical applications. This chapter deals with the Schrodinger model of the behavior of single electrons as affected by potential wells. The wave mechanic approach is extended to provide an outline of the interactions of many electrons with nuclei. In terms of the Pauli exclusion principle, the four quantum numbers of the electron states must be different. As additional electrons are added, the states available for occupation can be filled only according to the rules governing the quantum numbers.
In the literature (1), the Schrodinger equation: {p1p1/2m + p2p2/2m + V(x1-x2)} W(x1,x2,t) = id/dt partial W(x1,x2,t) may be transformed using P=p1+p2 and (p=m1p2-m2p1)/M and x=x1=x2 where M=m1+m2 ((1)). This leads to a bound Schrodinger equation in x and a “center of mass” equation” in P i..e Hcm =PP/2M which depends on m1 and m2, but not binding energy. . In a previous note, we argued that a free particle’s classical action A (relativistic or nonrelativistic) may be obtained by setting v=x/t and treating x and t as independent (2). This leads to dA/dx partial=p and dA/dt partial = -E. One may write eigenvalue equations: -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = E exp(iA). A solution is A=-Et+px with E,t,p and x independent. -Et+px is a Lorentz invariant and this form is used in both relativistic and nonrelativistic quantum mechanics (although E and p have different values). We argue that a free particle rest mass mo may actually involve internal quantum physics, yet the “system” is replaced with a parameter mo and exp(-iEt+px) used to describe it. One may try to apply this approach to a bound system, but exp(-iEt+px) is a statistical probability like object. Thus one needs a statistical theory. We have suggested using an ensemble of exp(-iEt+px) with E=En for all p to indicate a resonance. Then: { Sum over p pp/2m a(p)exp(ipx) } / {Sum over p a(p)exp(ipx) } + V(x) = En ((2)) This is the Schrodinger bound equation. We argue that W=Sum over p a(p)exp(ipx) should be real and this implies that Sum over p a(p) p exp(ipx) = 0 i.e. average momentum for this type of average is 0. As a result, one may create a 4-vector using { 0, mcc + En} and use the Lorentz boost approach to find the momentum and energy in a moving frame. One may then take the nonrelativistic limit to find kinetic energy i.e. the full center of mass kinetic energy. We argue that this is equivalent to considering exp(-iEt+ipx) for the overall bound system which is now translating in space. We argue that ((2)) is a different approach to ((1)) in that binding energy, which may be significant with respect to mcc, should be part of the overall energy of the translating bound system which is called the “center of mass” system in the literature.
From the invariance properties of the Schrodinger equation and the isotropy of space we show that a generic (non-relativistic) quantum system is endowed with an ``external'' motion, which can be interpreted as the motion of the centre of mass, and an ``internal'' one, whose presence disappears in the classical limit. The latter is caused by the spin of the particle, whatever is its actual value (different from zero). The quantum potential in the Schrodinger equation, which is responsible of the quantum effects of the system, is then completely determined from the properties of the internal motion, and its ``unusual'' properties have a simple and physical explanation in the present context. From the impossibility to fix the initial conditions relevant for the internal motion follows, finally, the need of a probabilistic interpretation of quantum mechanics.
The properties of angular momentum and its connection to magnetic moment are explored, based on a reconsideration of the Stern-Gerlach experiment and gauge invariance. The separation of angular momentum of a quantum system of particles into orbital angular momentum plus intrinsic angular momentum is reconsidered, within the limits of the Schrödinger theory. A proof is given that, for systems of more than two particles, unless all of them have the same mass, the possibility of having eigenvalues of the form (n+1/2)ℏ(n+1/2)\hbar(n+1/2)ℏ is not excluded for the operators of angular momentum of the centre of mass and the internal angular momentum. \noindent\textbf{PACS: 03.65.Ta} angular momentum, magnetic moment, spin crisis