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Journal of Mathematical Physics

Formulation of the Many-Body Problem for Composite Particles. III. The Projected Hamiltonian

Marvin D. Girardeau

Published Aug 1, 1971• 27 citations
Publisher

Abstract

A previously developed representation for many-atom problems is extended by incorporation of all effects of interatomic exchange of electrons and nuclei into a ``projected Hamiltonian.'' An expression for the projected Hamiltonian is derived which is exact on the two-atom subspace, and hence incorporates those exchange effects which are important at low densities. By the use of the center-of-mass separation of atomic wavefunctions, this expression for the projected Hamiltonian is transformed into a representation in terms of atomic field operators ψν(r) and ψν†(r), where r is the center-of-mass position of an atom and v stands for all of its internal quantum numbers.

DOI
10.1063/1.1665808

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SYSTEMS OF IDENTICAL PARTICLES

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If we consider ( ) 2,kr tΨ G to be the probability density for finding, at time t, particle 1 at particle k at and particle N at ,,1 … Gr …G ,kr ,Nr G a normalization condition analogous to Eq.(29.2) holds as (35.2) 1V* =ΨΨ∫ d The integral is defined in a space with 3N dimensions called the configuration space of the system, where a configuration point is given by 3N position coordinates 1 1 1, , , , , , , , , ,k k k N N N ,x y z x y z x y z… … and the volume element is NNNkkkNk dzdydxdzdydxdzdydxdddd ………… 1111 VVVV == Physical observables of the system will be represented by Hermitian operators, obtained from the classical representation in terms of the position and momentum variables as indicated by Postulate 2. Thus the many-particle Hamiltonian results from Eq.(2.32) as ( ) (∑ ∑ = ≠ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ++∇−= k rrUtrU m H ,, 2 ˆ GGG= ) ) (35.3) where only acts on the coordinates of the kth particle, of mass The one-particle potential energy is due to external fields where this particle moves and k∇ .km( trU kk ,G( )jkkj rrU GG , gives the mutual interaction between the particles k and j. It is assumed that the position and momentum operators of different particles commute, and so do all the operators describing single-particle observables if they refer to different particles. The evolution of a many-particle system will be then formulated as a generalization of Eq.(30.6) in terms of given by Eqs.(35.1) and (35.3) HˆandΨ Ψ=∂ Ψ∂ H t i ˆ= (35.4) If the Hamiltonian (35.3) does not depend explicitly on the time, it is usual to express the time dependence of the many-particle state function by means of the evolution operator defined as ( )tUˆ ( ) ( ) ( )NkNk rrrtUtrrr G…G…GG…G…G ,,,,ˆ,,,,, 11 αΨ=Ψ (35.5) which is formally derived from Eq.(35.4) as ( ) ( ) ( )ˆˆ ˆU ti U t i HU t t t ˆα α α ∂∂ ⎡ ⎤Ψ = Ψ =⎣ ⎦∂ ∂= = Ψ It follows that (35.6) ( ) =/ˆˆ tHietU −= which is the analogue of Eq.(30.12). The evolution operator is unitary, Eq.(II.22), since ( ) ( )†ˆ ˆU t U t Iˆ= (35.7) Substituting Eq.(35.5), one obtains from Eq.(35.4) that the stationary states of the Nparticle systems are given by the energy eigenvalue equation ( ) ( )1 1ˆ , , , , , , , ,k N k NH r r r E r r rα α αΨ = ΨG G G G G G… … … … (35.8) The solution to the N-particle problem is facilitated if we take into account the symmetries of many systems of physical interest, which allow us to define constants of motion and criteria for simplifying the form (35.3) of the Hamiltonian operator. An alternative to complex explicit calculations is to associate with specific symmetries time independent unitary operators which transform the state functions into so that Eq.(35.4) gives ˆ ,U Ψ ˆ ,U Ψ ( ) ( )†ˆ ˆ ˆ ˆ ˆ ˆ ˆi U i U UH UHU Ut t∂ ∂ΨΨ = = Ψ = Ψ∂ ∂= = ˆ This is again a Schrödinger equation for the state Uˆ Ψ with the new Hamiltonian †ˆ ˆ ˆ ˆH UHU′ = If the Hamiltonian remains unchanged by the transformation, we have (35.9) †ˆ ˆ ˆ ˆ ˆ ˆor , 0H UHU U H⎡ ⎤= ⎣ ⎦ = According to the evolution equation (29.43), this implies that so that U is a constant of motion. If the unitary transformation which leaves the Hamiltonian invariant, is a function of an observable, its constancy implies that the given observable is a constant in any state of the system.

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The external electromagnetic interactions of hydrogenic atoms, which exhibit global Galilean invariance but preserve the relativistic description of the electron, are investigated. The Hamiltonian for the free atom consists of the Breit Hamiltonian for a relativistic electron interacting with a nonrelativistic spinless proton. Approximate generators of the Galilean group are constructed in terms of individual-particle position, momentum, and spin operators in several unitarily equivalent representations. In one of these the Hamiltonian is separable in conventional relative and center-of-mass coordinates. This separability facilitates the treatment of a number of external-field problems. An effective Hamiltonian for the interaction of the atom with a constant external electric field is given and compared to that obtained by other authors. The low-energy theorem for Compton scattering from the composite system is explicitly verified including corrections due to atomic binding. This also gives an independent verification of recently obtained bound-state corrections to the atomic magnetic moment. Finally, an effective interaction of a charged Dirac particle with a hydrogenic atom is given in the representation in which the atomic Hamiltonian is separable.

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A previously developed representation for many-atom problems is extended by incorporation of all effects of interatomic exchange of electrons and nuclei into a ``projected Hamiltonian.'' An expression for the projected Hamiltonian is derived which is exact on the two-atom subspace, and hence incorporates those exchange effects which are important at low densities. By the use of the center-of-mass separation of atomic wavefunctions, this expression for the projected Hamiltonian is transformed into a representation in terms of atomic field operators ψν(r) and ψν†(r), where r is the center-of-mass position of an atom and v stands for all of its internal quantum numbers.
The many-body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second-quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.
Presented herein is a second-quantized formulation for a system of interacting nuclei and electrons expressed in terms of the creation and annihilation operators of various species of atoms and ions which can be formed by combining various numbers of the "elementary" particles. Processes such as ionization and recombination appear naturally. Explicit expressions for interatomic forces including exchange effects are generated from a "cluster expansion" reminiscent of the Ursell expansion.
A method of constructing ’’atomic second quantization’’ representations is described, based on the introduction of redundant modes (’’ideal atom variables’’) which are then given physical content by carrying out a suitable unitary transformation, the ’’generalized Tani transformation.’’ In such a representation bound atoms or molecules are described by elementary Bose or Fermi operators, the field operators for nuclei and electrons referring only to unbound particles. The Hamiltonian thus obtained contains not only nucleus–nucleus, electron–electron, and nucleus–electron Coulomb interactions, but also atom–atom, atom–nucleus, and atom–electron Coulomb and exchange interactions, including breakup and recombination terms. All possible scattering and reaction channels are exhibited simultaneously in this transformed Hamiltonian. The method is applicable to any species or mixture of species of composite particles, but for simplicity the derivation is restricted here to the case of atomic hydrogen.
If we consider ( ) 2,kr tΨ G to be the probability density for finding, at time t, particle 1 at particle k at and particle N at ,,1 … Gr …G ,kr ,Nr G a normalization condition analogous to Eq.(29.2) holds as (35.2) 1V* =ΨΨ∫ d The integral is defined in a space with 3N dimensions called the configuration space of the system, where a configuration point is given by 3N position coordinates 1 1 1, , , , , , , , , ,k k k N N N ,x y z x y z x y z… … and the volume element is NNNkkkNk dzdydxdzdydxdzdydxdddd ………… 1111 VVVV == Physical observables of the system will be represented by Hermitian operators, obtained from the classical representation in terms of the position and momentum variables as indicated by Postulate 2. Thus the many-particle Hamiltonian results from Eq.(2.32) as ( ) (∑ ∑ = ≠ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ++∇−= k rrUtrU m H ,, 2 ˆ GGG= ) ) (35.3) where only acts on the coordinates of the kth particle, of mass The one-particle potential energy is due to external fields where this particle moves and k∇ .km( trU kk ,G( )jkkj rrU GG , gives the mutual interaction between the particles k and j. It is assumed that the position and momentum operators of different particles commute, and so do all the operators describing single-particle observables if they refer to different particles. The evolution of a many-particle system will be then formulated as a generalization of Eq.(30.6) in terms of given by Eqs.(35.1) and (35.3) HˆandΨ Ψ=∂ Ψ∂ H t i ˆ= (35.4) If the Hamiltonian (35.3) does not depend explicitly on the time, it is usual to express the time dependence of the many-particle state function by means of the evolution operator defined as ( )tUˆ ( ) ( ) ( )NkNk rrrtUtrrr G…G…GG…G…G ,,,,ˆ,,,,, 11 αΨ=Ψ (35.5) which is formally derived from Eq.(35.4) as ( ) ( ) ( )ˆˆ ˆU ti U t i HU t t t ˆα α α ∂∂ ⎡ ⎤Ψ = Ψ =⎣ ⎦∂ ∂= = Ψ It follows that (35.6) ( ) =/ˆˆ tHietU −= which is the analogue of Eq.(30.12). The evolution operator is unitary, Eq.(II.22), since ( ) ( )†ˆ ˆU t U t Iˆ= (35.7) Substituting Eq.(35.5), one obtains from Eq.(35.4) that the stationary states of the Nparticle systems are given by the energy eigenvalue equation ( ) ( )1 1ˆ , , , , , , , ,k N k NH r r r E r r rα α αΨ = ΨG G G G G G… … … … (35.8) The solution to the N-particle problem is facilitated if we take into account the symmetries of many systems of physical interest, which allow us to define constants of motion and criteria for simplifying the form (35.3) of the Hamiltonian operator. An alternative to complex explicit calculations is to associate with specific symmetries time independent unitary operators which transform the state functions into so that Eq.(35.4) gives ˆ ,U Ψ ˆ ,U Ψ ( ) ( )†ˆ ˆ ˆ ˆ ˆ ˆ ˆi U i U UH UHU Ut t∂ ∂ΨΨ = = Ψ = Ψ∂ ∂= = ˆ This is again a Schrödinger equation for the state Uˆ Ψ with the new Hamiltonian †ˆ ˆ ˆ ˆH UHU′ = If the Hamiltonian remains unchanged by the transformation, we have (35.9) †ˆ ˆ ˆ ˆ ˆ ˆor , 0H UHU U H⎡ ⎤= ⎣ ⎦ = According to the evolution equation (29.43), this implies that so that U is a constant of motion. If the unitary transformation which leaves the Hamiltonian invariant, is a function of an observable, its constancy implies that the given observable is a constant in any state of the system.
The external electromagnetic interactions of hydrogenic atoms, which exhibit global Galilean invariance but preserve the relativistic description of the electron, are investigated. The Hamiltonian for the free atom consists of the Breit Hamiltonian for a relativistic electron interacting with a nonrelativistic spinless proton. Approximate generators of the Galilean group are constructed in terms of individual-particle position, momentum, and spin operators in several unitarily equivalent representations. In one of these the Hamiltonian is separable in conventional relative and center-of-mass coordinates. This separability facilitates the treatment of a number of external-field problems. An effective Hamiltonian for the interaction of the atom with a constant external electric field is given and compared to that obtained by other authors. The low-energy theorem for Compton scattering from the composite system is explicitly verified including corrections due to atomic binding. This also gives an independent verification of recently obtained bound-state corrections to the atomic magnetic moment. Finally, an effective interaction of a charged Dirac particle with a hydrogenic atom is given in the representation in which the atomic Hamiltonian is separable.