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Physical Review

Quantum Mechanics in Fock Space

M. Moshińsky

Published Nov 1, 1951• 14 citations
Publisher

Abstract

In the present paper we propose to develop a quantum-mechanical scheme in Fock space that would describe interactions that take place through the formation of a compound particle. The discussion will be restricted to a dynamical system representing a single-level scattering process. The state of this dynamical system can be found in two stages: initial particles and compound nucleus. The dynamical behavior of this state will be given in terms of a hamiltonian (that has no classical analog) which will be derived from the boundary conditions satisfied by the state. The representation of the dynamical variables, associated with the two stages of the state, will be discussed, and a complete set of constants of motion of our dynamical system will be given. The expansion of an arbitarary Fock state in terms of the simultaneous eigenstates of this complete set of constants of motion, leads to the generalized Hankel transforms, used in a preceding article for the determination of the time dependent states for scattering and disintegration. In a representation in which the constants of motion are diagonal, the operator representing the relative kinetic energy of the two initial particles will be nondiagonal, particularly in the neighborhood of the resonance energy. The unitary matrix connecting any initial Fock state with the corresponding state at time $t$ will be obtained, and with its help the time-dependent behavior of the Heisenberg dynamical variables of our dynamical system can be derived.

DOI
10.1103/physrev.84.533

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Quantum chemistry in Fock space. I. The universal wave and energy operators

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Polaron formation in a medium of damped excitations

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Two schemes of the quantization of dynamical variables of dissipative systems are studied with an example of a two-level atom interacting with phonons scattered from an environment. Transformations of relevant Fock-space operators or rigged-Fock-space operators are proposed in order to diagonalize their equations of motion. It is shown that the damping of dispersionless phonons leads to the formation of a quasi-stationary state of the system—a polaron, which is found in both approaches. A time dependence of the polaron energy and the spectral lineshape of the two-level subsystem are evaluated. A transformation diagonalizing the equations of motion of the relevant classical problem (of a harmonic oscillator coupled to a field of damped waves) is found to be similar to the quantum case.

Journal of Physics A Mathematical and General•May 10, 2005•1 citation
In the present paper we propose to develop a quantum-mechanical scheme in Fock space that would describe interactions that take place through the formation of a compound particle. The discussion will be restricted to a dynamical system representing a single-level scattering process. The state of this dynamical system can be found in two stages: initial particles and compound nucleus. The dynamical behavior of this state will be given in terms of a hamiltonian (that has no classical analog) which will be derived from the boundary conditions satisfied by the state. The representation of the dynamical variables, associated with the two stages of the state, will be discussed, and a complete set of constants of motion of our dynamical system will be given. The expansion of an arbitarary Fock state in terms of the simultaneous eigenstates of this complete set of constants of motion, leads to the generalized Hankel transforms, used in a preceding article for the determination of the time dependent states for scattering and disintegration. In a representation in which the constants of motion are diagonal, the operator representing the relative kinetic energy of the two initial particles will be nondiagonal, particularly in the neighborhood of the resonance energy. The unitary matrix connecting any initial Fock state with the corresponding state at time ttt will be obtained, and with its help the time-dependent behavior of the Heisenberg dynamical variables of our dynamical system can be derived.
The Fock space Hamiltonian H has a simpler structure than its projection Hn to n-particle Hilbert space. It is therefore recommended to diagonalize H—to the extent that this is possible—before one specifies n. Diagonalization of H is possible if one defines the diagonal and nondiagonal parts of an operator appropriately. It is shown that the diagonalized Fock space Hamiltonian L, called an energy operator, contains all information about the eigenvalues of H in a simply coded form. For a spinfree Hamiltonian, spin can be completely eliminated and all interesting quantities are expressible in terms of spinfree excitation operators (generators of the unitary group). The construction of the wave operator W and the energy operator L is formulated in terms of perturbation theory both in the strictly degenerate and the quasidegenerate version. Three variants are discussed that differ in the normalization of W, namely, a intermediate normalization, b and c unitary normalization with two additional conditions, b: WD=WD+, c:σD=(ln W)D=0. The transformation of operator products to normal product form is achieved by means of a generalization of Wick’s theorem to spinfree quantum chemistry. The contributions to the various orders of L, W, and σ are illustrated by means of diagrams. Only variant c guarantees a connected-diagram expansion of the Fock space energy operator L. None of the three variants satisfies a generalization of the Wigner (2n+1) rule, but with a slight change in the normalization from b to b′ or c to c′ (that does, in the case of c′, not affect the connectedness of the diagram expansion) L(2n) and L(2n+1) are expressible in terms of the W(k), k⩽n. A generalization of the variation principle for the Fock space energy operator is given and its consequences are discussed. Generalized Fock space Brillouin conditions are derived and a variational scheme for the construction of the energy operator is proposed. We finally compare the present Fock space approach with the traditional degenerate perturbation theory in n-particle Hilbert space. If the diagonal part ΩD of an operator Ω is defined in the spirit of the theory of the universal wave and energy operators, then the two approaches lead to the same results (for the same normalization), but the Fock space approach looks simpler and more transparent.
Quantum mechanics gives us information about the spectra of dynamical variables and transition rates including scattering cross sections. They can be exhibited as spectral information in analytically continued spaces and their duals. Quantum mechanics formulated in these generalized spaces is used to study scattering and time evolution. It is shown that the usual asymptotic condition is inadequate to deal with the scattering of composite or unstable particles. Scattering theory needs an amendment when the interacting system is not isospectral with the free Hamiltonian; the amendment is formulated. Perturbation theory in generalized spaces is developed and used to study the deletion and augmentation of the spectrum of the Hamiltonian. A complete set of algebraically independent constants for an interacting system is obtained. The question of the breaking of time symmetry is discussed.
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
Abstract The new quantum mechanics consists of a scheme of equations which are very closely analogous to the equations of classical mechanics, with the funda­mental difference that the dynamical variables do not obey the commutative law of multiplication, but satisfy instead the well-known quantum conditions. It follows that one cannot suppose the dynamical variables to be ordinary numbers (c-numbers), but may call them numbers of a special type (q-numbers). The theory shows that these q-numbers can in general be represented by matrices whose elements are c-numbers (functions of a time parameter). When one has performed the calculations with the q-numbers and obtained all the matrices one wants, the question arises how one is to get physical results from the theory, i. e., how can one obtain c-numbers from the theory that one can compare with experimental values ? Hitherto this has been done with the help of a number of special assumptions. In Heisenberg’s original matrix mechanics it was assumed that the elements of the diagonal matrix that repre­sents the energy are the energy levels of the system, and the elements of the matrix that represents the total polarisation, which are periodic functions of the time, determine the frequencies and intensities of the spectral lines in analogy to the classical theory. Schrodinger’s wave representation of the quantum mechanics has provided new ways of obtaining physical results from the theory, based on the assumption that the square of the amplitude of the wave function can in certain cases be interpreted as a probability. From this assumption one can, for instance, work out the probability of a transition being produced in a system (or the number of transitions produced in an assembly of like systems) by an arbitrary external perturbing force, and can thus, by supposing the perturbation to consist of incident radiation, obtain directly Einstein’s B coefficients. Again in Born’s treatment of collision problems it is assumed that the square of the amplitude of the wave function scattered in any direction determines the probability of the colliding electron (or other body) being scattered in that direction.
Two schemes of the quantization of dynamical variables of dissipative systems are studied with an example of a two-level atom interacting with phonons scattered from an environment. Transformations of relevant Fock-space operators or rigged-Fock-space operators are proposed in order to diagonalize their equations of motion. It is shown that the damping of dispersionless phonons leads to the formation of a quasi-stationary state of the system—a polaron, which is found in both approaches. A time dependence of the polaron energy and the spectral lineshape of the two-level subsystem are evaluated. A transformation diagonalizing the equations of motion of the relevant classical problem (of a harmonic oscillator coupled to a field of damped waves) is found to be similar to the quantum case.