A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain $U$ functions defined for the same system with Boltzmann statistics. It is then shown that these $U$ functions can be evaluated in successive approximations in terms of a binary kernel $B$ which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain $U$ functions defined for the same system with Boltzmann statistics. It is then shown that these $U$ functions can be evaluated in successive approximations in terms of a binary kernel $B$ which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.

Physical ReviewMar 1, 1959210 citations

On the Formulation of Quantum Mechanics associated with Classical Pictures

Takehiko Takabayasi

By re-expressing the generalized Schrodinger equation in its coordinate representation in terms of amplitude and phase of the state vector, a quantum-mechanical change of a system can be made to correspond to an ensemble of classical motions of the system to which is added some internal potential. This ensemble may be taken as the statistical. This enables an alternative formulation of quantum mechanics with which is associated that classical picture. Such a formulation leads, however, if developed straightforwardly according to the picture, to a theoretical scheme quite different from that of ordinary quantum mechanics concerning the problems other than the equation of motion. Furthermore, this formulation is not applicable to Fermions when spin, or exclusion principle is taken into account. Bohm's recent renewed form of the statistical interpretation of quantum mechanics referring to this formulation is criticized in several points. On this formulation we can construct a singular ensemble which is shown to correspond to the transformation kernel of the wave function to infinitesimally later instant, and thus a connection between this formulation and that of Feynman's is taken out. Hydrodynamical picture formally equivalent to the statistical picture in one-body problems is also considered, and from such analogy certain formal generalization of Schrodinger equation is suggested.By re-expressing the generalized Schrodinger equation in its coordinate representation in terms of amplitude and phase of the state vector, a quantum-mechanical change of a system can be made to correspond to an ensemble of classical motions of the system to which is added some internal potential. This ensemble may be taken as the statistical. This enables an alternative formulation of quantum mechanics with which is associated that classical picture. Such a formulation leads, however, if developed straightforwardly according to the picture, to a theoretical scheme quite different from that of ordinary quantum mechanics concerning the problems other than the equation of motion. Furthermore, this formulation is not applicable to Fermions when spin, or exclusion principle is taken into account. Bohm's recent renewed form of the statistical interpretation of quantum mechanics referring to this formulation is criticized in several points. On this formulation we can construct a singular ensemble which is shown to correspond to the transformation kernel of the wave function to infinitesimally later instant, and thus a connection between this formulation and that of Feynman's is taken out. Hydrodynamical picture formally equivalent to the statistical picture in one-body problems is also considered, and from such analogy certain formal generalization of Schrodinger equation is suggested.

Progress of Theoretical PhysicsAug 1, 1952340 citations

Formulation of the quantum mechanical many-body problem in terms of one- and two-particle functions

L. Szasz

Physics LettersMay 1, 19633 citations

Foundations of quantum mechanics: Building on von Neumann's heritage

Gérard G. Emch

From 1927 to 1943, the mathematical and conceptual foundations of quantum mechanics, and their prolongations into the formulation of statistical mechanics, were either the direct subject of or one of the primary motivations for several publications by von Neumann. We contrast the reception received by the work done during the first 5 years (1927–1932) and that received by the monumental mathematical constructs wrought in the second period (1934–1943); we describe how the latter were to lead to the elaboration of a formalism generalizing the fundamental structures of quantum mechanics so that they can encompass and deal consistently with situations typical of the quantum many-body problems of statistical mechanics. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 379–387, 1997From 1927 to 1943, the mathematical and conceptual foundations of quantum mechanics, and their prolongations into the formulation of statistical mechanics, were either the direct subject of or one of the primary motivations for several publications by von Neumann. We contrast the reception received by the work done during the first 5 years (1927–1932) and that received by the monumental mathematical constructs wrought in the second period (1934–1943); we describe how the latter were to lead to the elaboration of a formalism generalizing the fundamental structures of quantum mechanics so that they can encompass and deal consistently with situations typical of the quantum many-body problems of statistical mechanics. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 379–387, 1997

International Journal of Quantum ChemistryJan 1, 19975 citations

A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain

U

functions defined for the same system with Boltzmann statistics. It is then shown that these

U

functions can be evaluated in successive approximations in terms of a binary kernel

B

which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.