A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of $N$ fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and ${f}^{(2)}(\mathrm{ij})$ is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained $N$ one-particle equations for the $N$ one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of $N$ fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and ${f}^{(2)}(\mathrm{ij})$ is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained $N$ one-particle equations for the $N$ one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.

Physical ReviewOct 15, 196338 citations

Multiple Particles

Devon Bowen

Springer eBooksJan 1, 20260 citations

Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states

Charles H. Bennett, Stephen J. Wiesner

As is well known, operations on one particle of an Einstein-Podolsky-Rosen (EPR) pair cannot influence the marginal statistics of measurements on the other particle. We characterize the set of states accessible from an initial EPR state by one-particle operations and show that in a sense they allow two bits to be encoded reliably in one spin-1/2 particle: One party, ``Alice,'' prepares an EPR pair and sends one of the particles to another party, ``Bob,'' who applies one of four unitary operators to the particle, and then returns it to Alice. By measuring the two particles jointly, Alice can now reliably learn which operator Bob used.As is well known, operations on one particle of an Einstein-Podolsky-Rosen (EPR) pair cannot influence the marginal statistics of measurements on the other particle. We characterize the set of states accessible from an initial EPR state by one-particle operations and show that in a sense they allow two bits to be encoded reliably in one spin-1/2 particle: One party, ``Alice,'' prepares an EPR pair and sends one of the particles to another party, ``Bob,'' who applies one of four unitary operators to the particle, and then returns it to Alice. By measuring the two particles jointly, Alice can now reliably learn which operator Bob used.

Physical Review LettersNov 16, 19925,452 citations

Many particles

Benjamin W. Schumacher, Michael D. Westmoreland

Cambridge University Press eBooksMar 25, 20100 citations

Maxwell-Boltzmann Gas Pressure Ideas Applied to a Single Particle Yields exp(ipx)?

Francesco R. Ruggeri

A Maxwell-Boltzmann gas with no external potential V(x) has a constant spatial density. The value of this spatial density alone does not yield the pressure because this also depends on T through the ideal gas law Pressure = density(x) RT, where T is temperature (associated with a conserved average energy), but also the average single particle pressure. Taking d/density(x) of Pressure yields the single particle average pressure value RT, yet at the same time single particle pressure is a stochastic value evaluated as the average of constant v p where p is momentum and v velocity. The quadratic form of p arises, we argue, because one has more than one particle, thus v distinguishes between the particles and relative speed becomes important. In the case of a single particle (as in a single free quantum particle) moving at constant p (speed as well) there is again a constant spatial density as in the MB gas with no V(x). Given the value of the constant density 1/L (L=length), one has no information about momentum p. In the case of a single particle, impulse and not pressure is the important quantity, we argue, because there is no need for a v as there is no relative motion. One only needs to know impulse which is proportional to p. If one speculates that a free quantum particle may follow the model of pressure in an MB gas, one would require a function F(p,x) such that d/d something F = p where p is the single particle impulse. In the MB case, “something” is density, but this is not appropriate for a single particle as density consists of many particles. Without changing T, one may change pressure in an MB gas by changing density i.e. changing the x value of particles. For a quantum particle, we suggest that the conjugate variable to p is x. One may change x of the single particle. Furthermore an association between d/dx and p exists in Fermat’s Least Time Principle. T Thus we suggest d/dx F(p,x) = p where F is unknown both in form and meaning. Pressure, however, is a statistical quantity so F should also be a statistical quantity i.e. one could have d/dx P(p,x) / P(p/x) = constant p where P(p,x) is a kind of unnormalized probability. This suggests, however, that P(p,x) is an eigenfunction of d/dx and represents a kind of unnormalized probability. This, we argue, leads to the notion of exp(ipx) because one does not want a monotonically growing or decreasing probability in x for a constant p. The “constant” is then i. Thus, we argue that if one mimics the pressure behaviour of a MB gas, and applies its properties to a single particle moving at constant speed, one obtains an unusual unnormalized kind of probability exp(ipx) i.e. the quantum free particle wavefunction. At the same time, a constant spatial probability or density 1/L exists. The two are linked in that exp(ipx) exp(-ipx) =1. The overall idea is that the stochastic force i.e. the entropic pressure in an MB gas has an analogy even for a single particle with constant momentum.A Maxwell-Boltzmann gas with no external potential V(x) has a constant spatial density. The value of this spatial density alone does not yield the pressure because this also depends on T through the ideal gas law Pressure = density(x) RT, where T is temperature (associated with a conserved average energy), but also the average single particle pressure. Taking d/density(x) of Pressure yields the single particle average pressure value RT, yet at the same time single particle pressure is a stochastic value evaluated as the average of constant v p where p is momentum and v velocity. The quadratic form of p arises, we argue, because one has more than one particle, thus v distinguishes between the particles and relative speed becomes important. In the case of a single particle (as in a single free quantum particle) moving at constant p (speed as well) there is again a constant spatial density as in the MB gas with no V(x). Given the value of the constant density 1/L (L=length), one has no information about momentum p. In the case of a single particle, impulse and not pressure is the important quantity, we argue, because there is no need for a v as there is no relative motion. One only needs to know impulse which is proportional to p. If one speculates that a free quantum particle may follow the model of pressure in an MB gas, one would require a function F(p,x) such that d/d something F = p where p is the single particle impulse. In the MB case, “something” is density, but this is not appropriate for a single particle as density consists of many particles. Without changing T, one may change pressure in an MB gas by changing density i.e. changing the x value of particles. For a quantum particle, we suggest that the conjugate variable to p is x. One may change x of the single particle. Furthermore an association between d/dx and p exists in Fermat’s Least Time Principle. T Thus we suggest d/dx F(p,x) = p where F is unknown both in form and meaning. Pressure, however, is a statistical quantity so F should also be a statistical quantity i.e. one could have d/dx P(p,x) / P(p/x) = constant p where P(p,x) is a kind of unnormalized probability. This suggests, however, that P(p,x) is an eigenfunction of d/dx and represents a kind of unnormalized probability. This, we argue, leads to the notion of exp(ipx) because one does not want a monotonically growing or decreasing probability in x for a constant p. The “constant” is then i. Thus, we argue that if one mimics the pressure behaviour of a MB gas, and applies its properties to a single particle moving at constant speed, one obtains an unusual unnormalized kind of probability exp(ipx) i.e. the quantum free particle wavefunction. At the same time, a constant spatial probability or density 1/L exists. The two are linked in that exp(ipx) exp(-ipx) =1. The overall idea is that the stochastic force i.e. the entropic pressure in an MB gas has an analogy even for a single particle with constant momentum.

Apr 25, 20230 citations

Many particles

Jonathan Dimock

Cambridge University Press eBooksFeb 3, 20110 citations

A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of

N

fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and

{f}^{(2)}(\mathrm{ij})

is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained

N

one-particle equations for the

N

one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.