The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a na... more The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a natural complex structure on a four-dimensional space. If paravectors, which are formed sums of scalars and vectors, are taken as the “real” elements of the space, then the space can be shown to have a Minkowski spacetime metric, and the paravectors may be identified with spacetime vectors. Physical Lorentz transformations of spacetime vectors are described by spin transformations of the paravectors. The transformation elements are unimodular elements of the algebra, and they form the six-parameter group SL(2,C), the two-fold covering group of restricted Lorentz transformations, SO+(l,3). Its elements are also reducible spinors, whose elements carry a reducible spin representation of SL(2,C). The spin representation is reduced by splitting elements into complementary minimal left ideals. The complex spin space that results has a symplectic structure and its elements belong to Sp(2).
The multiconfiguration Dirac-Fock model is employed to evaluate the effects of relativity and ele... more The multiconfiguration Dirac-Fock model is employed to evaluate the effects of relativity and electron correlation on the hyperfine interaction constants of the 4d5s 2 2 D 3/2 and 2 D 5/2 levels of yttrium. The wavefunction expansions are obtained with the active space method, where configuration-state functions of a specific parity and J value are generated by substitutions from the reference configurations to an active set of orbitals. The active set is then increased in a systematic way, allowing the convergence of the hyperfine interaction constants to be studied. The calculated electric quadrupole hyperfine constants are used to evaluate the nuclear electric quadrupole moment of isotope 90 of yttrium, for which the value QϭϪ0.125(11) b is found. ͓S1050-2947͑98͒06509-3͔
The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a na... more The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a natural complex structure on a four-dimensional space. If paravectors, which are formed sums of scalars and vectors, are taken as the “real” elements of the space, then the space can be shown to have a Minkowski spacetime metric, and the paravectors may be identified with spacetime vectors. Physical Lorentz transformations of spacetime vectors are described by spin transformations of the paravectors. The transformation elements are unimodular elements of the algebra, and they form the six-parameter group SL(2,C), the two-fold covering group of restricted Lorentz transformations, SO+(l,3). Its elements are also reducible spinors, whose elements carry a reducible spin representation of SL(2,C). The spin representation is reduced by splitting elements into complementary minimal left ideals. The complex spin space that results has a symplectic structure and its elements belong to Sp(2).
Theoretical Methods in the Physical Sciences, 1994
Often it’s convenient or even necessary to approximate the value of a function at a point or over... more Often it’s convenient or even necessary to approximate the value of a function at a point or over a small domain of points. Approximations can, of course, be very precise at times, and we often use them as though they were exact. Whenever you compute sin(0.1) on a calculator or in Maple, for example, the value must be approximated. A pocket calculator approximates the value to a limited accuracy of usually less than 12 decimal places, but Maple can approximate such numbers to what is for most practical purposes essentially arbitrary accuracy.
A quasiclassical correspondent for the fermion degrees of freedom is obtained by using a timedepe... more A quasiclassical correspondent for the fermion degrees of freedom is obtained by using a timedependent variational principle with Grassmann coherent states as trial functions. In the real parametrization provided by the canonical coordinates, these states satisfy a closure relationship, and this is used to calculate the partition function at finite temperature. The particular example considered here consists of a many-fermion system in a quantum double-well. Entanglement between the one-particle orbital states leads to deviations from the normal Fermi-Dirac distribution. This deviation is reflected in the dependence of the chemical potential on concentration. In the physical example of two adjacent CuO planes in a high-T c superconductor, the dependence is suggestive of the pseudogap observed at temperatures T > T c .
Potential energies for the ground and several excited states of the HgCd excimer molecule have be... more Potential energies for the ground and several excited states of the HgCd excimer molecule have been calculated. The CI singles procedure, followed by a second-order Meller-Plesset perturbation theory correction, has been applied for the valence electrons. The core electrons have been replaced by an effective core potential. The effect of spin-orbit interaction has not been investigated. Spectroscopic constants of the ground states of all other group IIB metal dimers have been also calculated with the help of the RHF method, followed by fourth-order M¢ller-Plesset perturbation theory and a quadratic CI approach. Our computed constants are compared with known experimental and other theoretical data.
. Clifford’s geometric algebra, in particular the algebra of physical space (APS), provides a ne... more . Clifford’s geometric algebra, in particular the algebra of physical space (APS), provides a new relativistic approach to the Quantum/Classical interface. It describes classical relativistic dynamics in quantum terms: spinor amplitudes with projectors describe classical motion and satisfy the Dirac equation of relativistic quantum theory. Some basic properties such as the spin-1/2 nature of elementary systems are seen to be a simple result of the geometry of physical space. The nature of “quantum reality” is constrained: the pure state of any single fermion is fully polarized and determines an exact spin direction, but an entangled pair can be unpolarized. Measurements can form or break entanglement, or they can transfer it between particle pairs. Quantum “weirdness” can arise from the need to use amplitudes of entangled systems.
De Broglie waves are a simple consequence of special relativity applied to the complex-phase osci... more De Broglie waves are a simple consequence of special relativity applied to the complex-phase oscillations of stationary states. As de Broglie showed in his doctoral thesis, the synchronized oscillations of an extended system at rest, even a classical one, become de Broglie-like waves when boosted to finite velocity. The waves illustrate the well-known but seldom demonstrated relativistic effect of clock desynchroniation (or dephasing) in moving frames. Although common manifestations of stationary-state oscillations in interference experiments are sensitive only to energy differences, de Broglie wavelengths are inversely proportional to rest-frame oscillation frequency, and their observed values require that the oscillation frequencies are proportional to the the total absolute energy, including the rest component mc2. PACS Nos.: 03.65.Ta, 03.30.+p, 01.65.+g
A century after its formulation by Einstein, it is time to incorporate special relativity early i... more A century after its formulation by Einstein, it is time to incorporate special relativity early in the physics curriculum. The approach advocated here employs a simple algebraic extension of vector formalism that generates Minkowski spacetime, displays covariant symmetries, and enables calculations of boosts and spatial rotations without matrices or tensors. The approach is part of a comprehensive geometric algebra with applications in many areas of physics, but only an intuitive subset is needed at the introductory level. The approach and some of its extensions are given here and illustrated with insights into the geometry of spacetime. PACS Nos.: 03.30.+p, 01.40.Gm, 03.50.De, 02.10.Hh
Theoretical Methods in the Physical Sciences, 1994
With complex numbers, mathematics becomes richer, and its applications to the physical sciences, ... more With complex numbers, mathematics becomes richer, and its applications to the physical sciences, more marvelous. The numbers with an imaginary term, proportional to \\( i = \\sqrt { - 1,} \\) have practical applications far beyond what the famous mathematician Descartes (15961650) could have imagined when he rejected them.1
The eigenspinor approach uses the classical amplitude of the algebraic Lorentz rotation connectin... more The eigenspinor approach uses the classical amplitude of the algebraic Lorentz rotation connecting the lab and rest frames to study the relativistic motion of particles. It suggests a simple covariant extension of the common definition of the electric field: the electromagnetic field can be defined as the proper spacetime rotation rate it induces in the particle frame times its mass-to-charge ratio. When applied to the dynamics of a point charge in an external electromagnetic field, the eigenspinor approach reveals surprising symmetries, particularly the invariance of some field properties in the rest frame of the accelerating charge. The symmetries facilitate the discovery of analytic solutions of the charge motion and are simply explained in terms of the geometry of spacetime. Symmetries of the uniformly accelerated charge and electric dipole are also briefly discussed.
The intensity distribution in the core region of the Lyman-α line of hydrogen perturbed by neutra... more The intensity distribution in the core region of the Lyman-α line of hydrogen perturbed by neutral hydrogen atoms in the presence of an external cone-rotating electric field has been evaluated using an impact approximation. It was found that the line shape in this region can be described as the sum of four Lorentzian profiles representing contributions coming from transitions between various Stark and Berry sublevels of the resonance and ground states of the hydrogen atom. The effects due to Berry's phase are shown to produce a Lorentzian-shaped feature on each side of the Lyman-α line, with maxima located at frequencies linearly dependent on the angular frequency with which the electric field rotates about the laboratory fixed z axis, as well as on the cosine of the polar angle that the field makes with respect to this axis. The remaining two Lorentzian profiles are due to the Stark effect, and their maxima are located on both sides of the Lyman-α line at frequencies corresponding to the Stark shift.
The importance of quasibound resonances is emphasized, with special reference to alkalinoble-gas ... more The importance of quasibound resonances is emphasized, with special reference to alkalinoble-gas systems. A modified WKB method for calculating the distribution and properties of quasibound resonances as well as of bound states is reviewed and applied to calculations of KAr and RbAr ground states. The calculated distribution of states in a Lennard-Jones 6-10 potential agrees precisely with the exact results of Mahan and Lapp. Inclusion of quasibound resonances is found to increase the total number of states by about 3. Tunneling lifetimes much larger than typical collisional destruction times are found for most resonances.
The choice of y which eliminates the first-order contribution will also eliminate the first contr... more The choice of y which eliminates the first-order contribution will also eliminate the first contribution. The second-order potentials will enter identically in the 3-3 and 4-4 equations by virtue of condition (16) and so the process of forming (15) from (14) will cancel this contribution. Therefore the expansion in (15) will be accurate up to third-order terms. Research supported by the Office of Naval Research under Contract No. N00014-72-A-0406-0004 and by CUNY Faculty Research Award No. 1657.
Journal of Physics B: Atomic and Molecular Physics
Atomic and molecular properties dependent on the valence electrons are sensitive to intra-atomic ... more Atomic and molecular properties dependent on the valence electrons are sensitive to intra-atomic correlation effects, and these tend to be largest in atoms of high atomic number where accurate ab initio computations are difficult. A simple model potential method is proposed for including correlation effects in Hartree-Fock calculations. Application is made to an electron-gas computation of the Hg2(X1 Sigma g+) potential.
The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a na... more The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a natural complex structure on a four-dimensional space. If paravectors, which are formed sums of scalars and vectors, are taken as the “real” elements of the space, then the space can be shown to have a Minkowski spacetime metric, and the paravectors may be identified with spacetime vectors. Physical Lorentz transformations of spacetime vectors are described by spin transformations of the paravectors. The transformation elements are unimodular elements of the algebra, and they form the six-parameter group SL(2,C), the two-fold covering group of restricted Lorentz transformations, SO+(l,3). Its elements are also reducible spinors, whose elements carry a reducible spin representation of SL(2,C). The spin representation is reduced by splitting elements into complementary minimal left ideals. The complex spin space that results has a symplectic structure and its elements belong to Sp(2).
The multiconfiguration Dirac-Fock model is employed to evaluate the effects of relativity and ele... more The multiconfiguration Dirac-Fock model is employed to evaluate the effects of relativity and electron correlation on the hyperfine interaction constants of the 4d5s 2 2 D 3/2 and 2 D 5/2 levels of yttrium. The wavefunction expansions are obtained with the active space method, where configuration-state functions of a specific parity and J value are generated by substitutions from the reference configurations to an active set of orbitals. The active set is then increased in a systematic way, allowing the convergence of the hyperfine interaction constants to be studied. The calculated electric quadrupole hyperfine constants are used to evaluate the nuclear electric quadrupole moment of isotope 90 of yttrium, for which the value QϭϪ0.125(11) b is found. ͓S1050-2947͑98͒06509-3͔
The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a na... more The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a natural complex structure on a four-dimensional space. If paravectors, which are formed sums of scalars and vectors, are taken as the “real” elements of the space, then the space can be shown to have a Minkowski spacetime metric, and the paravectors may be identified with spacetime vectors. Physical Lorentz transformations of spacetime vectors are described by spin transformations of the paravectors. The transformation elements are unimodular elements of the algebra, and they form the six-parameter group SL(2,C), the two-fold covering group of restricted Lorentz transformations, SO+(l,3). Its elements are also reducible spinors, whose elements carry a reducible spin representation of SL(2,C). The spin representation is reduced by splitting elements into complementary minimal left ideals. The complex spin space that results has a symplectic structure and its elements belong to Sp(2).
Theoretical Methods in the Physical Sciences, 1994
Often it’s convenient or even necessary to approximate the value of a function at a point or over... more Often it’s convenient or even necessary to approximate the value of a function at a point or over a small domain of points. Approximations can, of course, be very precise at times, and we often use them as though they were exact. Whenever you compute sin(0.1) on a calculator or in Maple, for example, the value must be approximated. A pocket calculator approximates the value to a limited accuracy of usually less than 12 decimal places, but Maple can approximate such numbers to what is for most practical purposes essentially arbitrary accuracy.
A quasiclassical correspondent for the fermion degrees of freedom is obtained by using a timedepe... more A quasiclassical correspondent for the fermion degrees of freedom is obtained by using a timedependent variational principle with Grassmann coherent states as trial functions. In the real parametrization provided by the canonical coordinates, these states satisfy a closure relationship, and this is used to calculate the partition function at finite temperature. The particular example considered here consists of a many-fermion system in a quantum double-well. Entanglement between the one-particle orbital states leads to deviations from the normal Fermi-Dirac distribution. This deviation is reflected in the dependence of the chemical potential on concentration. In the physical example of two adjacent CuO planes in a high-T c superconductor, the dependence is suggestive of the pseudogap observed at temperatures T > T c .
Potential energies for the ground and several excited states of the HgCd excimer molecule have be... more Potential energies for the ground and several excited states of the HgCd excimer molecule have been calculated. The CI singles procedure, followed by a second-order Meller-Plesset perturbation theory correction, has been applied for the valence electrons. The core electrons have been replaced by an effective core potential. The effect of spin-orbit interaction has not been investigated. Spectroscopic constants of the ground states of all other group IIB metal dimers have been also calculated with the help of the RHF method, followed by fourth-order M¢ller-Plesset perturbation theory and a quadratic CI approach. Our computed constants are compared with known experimental and other theoretical data.
. Clifford’s geometric algebra, in particular the algebra of physical space (APS), provides a ne... more . Clifford’s geometric algebra, in particular the algebra of physical space (APS), provides a new relativistic approach to the Quantum/Classical interface. It describes classical relativistic dynamics in quantum terms: spinor amplitudes with projectors describe classical motion and satisfy the Dirac equation of relativistic quantum theory. Some basic properties such as the spin-1/2 nature of elementary systems are seen to be a simple result of the geometry of physical space. The nature of “quantum reality” is constrained: the pure state of any single fermion is fully polarized and determines an exact spin direction, but an entangled pair can be unpolarized. Measurements can form or break entanglement, or they can transfer it between particle pairs. Quantum “weirdness” can arise from the need to use amplitudes of entangled systems.
De Broglie waves are a simple consequence of special relativity applied to the complex-phase osci... more De Broglie waves are a simple consequence of special relativity applied to the complex-phase oscillations of stationary states. As de Broglie showed in his doctoral thesis, the synchronized oscillations of an extended system at rest, even a classical one, become de Broglie-like waves when boosted to finite velocity. The waves illustrate the well-known but seldom demonstrated relativistic effect of clock desynchroniation (or dephasing) in moving frames. Although common manifestations of stationary-state oscillations in interference experiments are sensitive only to energy differences, de Broglie wavelengths are inversely proportional to rest-frame oscillation frequency, and their observed values require that the oscillation frequencies are proportional to the the total absolute energy, including the rest component mc2. PACS Nos.: 03.65.Ta, 03.30.+p, 01.65.+g
A century after its formulation by Einstein, it is time to incorporate special relativity early i... more A century after its formulation by Einstein, it is time to incorporate special relativity early in the physics curriculum. The approach advocated here employs a simple algebraic extension of vector formalism that generates Minkowski spacetime, displays covariant symmetries, and enables calculations of boosts and spatial rotations without matrices or tensors. The approach is part of a comprehensive geometric algebra with applications in many areas of physics, but only an intuitive subset is needed at the introductory level. The approach and some of its extensions are given here and illustrated with insights into the geometry of spacetime. PACS Nos.: 03.30.+p, 01.40.Gm, 03.50.De, 02.10.Hh
Theoretical Methods in the Physical Sciences, 1994
With complex numbers, mathematics becomes richer, and its applications to the physical sciences, ... more With complex numbers, mathematics becomes richer, and its applications to the physical sciences, more marvelous. The numbers with an imaginary term, proportional to \\( i = \\sqrt { - 1,} \\) have practical applications far beyond what the famous mathematician Descartes (15961650) could have imagined when he rejected them.1
The eigenspinor approach uses the classical amplitude of the algebraic Lorentz rotation connectin... more The eigenspinor approach uses the classical amplitude of the algebraic Lorentz rotation connecting the lab and rest frames to study the relativistic motion of particles. It suggests a simple covariant extension of the common definition of the electric field: the electromagnetic field can be defined as the proper spacetime rotation rate it induces in the particle frame times its mass-to-charge ratio. When applied to the dynamics of a point charge in an external electromagnetic field, the eigenspinor approach reveals surprising symmetries, particularly the invariance of some field properties in the rest frame of the accelerating charge. The symmetries facilitate the discovery of analytic solutions of the charge motion and are simply explained in terms of the geometry of spacetime. Symmetries of the uniformly accelerated charge and electric dipole are also briefly discussed.
The intensity distribution in the core region of the Lyman-α line of hydrogen perturbed by neutra... more The intensity distribution in the core region of the Lyman-α line of hydrogen perturbed by neutral hydrogen atoms in the presence of an external cone-rotating electric field has been evaluated using an impact approximation. It was found that the line shape in this region can be described as the sum of four Lorentzian profiles representing contributions coming from transitions between various Stark and Berry sublevels of the resonance and ground states of the hydrogen atom. The effects due to Berry's phase are shown to produce a Lorentzian-shaped feature on each side of the Lyman-α line, with maxima located at frequencies linearly dependent on the angular frequency with which the electric field rotates about the laboratory fixed z axis, as well as on the cosine of the polar angle that the field makes with respect to this axis. The remaining two Lorentzian profiles are due to the Stark effect, and their maxima are located on both sides of the Lyman-α line at frequencies corresponding to the Stark shift.
The importance of quasibound resonances is emphasized, with special reference to alkalinoble-gas ... more The importance of quasibound resonances is emphasized, with special reference to alkalinoble-gas systems. A modified WKB method for calculating the distribution and properties of quasibound resonances as well as of bound states is reviewed and applied to calculations of KAr and RbAr ground states. The calculated distribution of states in a Lennard-Jones 6-10 potential agrees precisely with the exact results of Mahan and Lapp. Inclusion of quasibound resonances is found to increase the total number of states by about 3. Tunneling lifetimes much larger than typical collisional destruction times are found for most resonances.
The choice of y which eliminates the first-order contribution will also eliminate the first contr... more The choice of y which eliminates the first-order contribution will also eliminate the first contribution. The second-order potentials will enter identically in the 3-3 and 4-4 equations by virtue of condition (16) and so the process of forming (15) from (14) will cancel this contribution. Therefore the expansion in (15) will be accurate up to third-order terms. Research supported by the Office of Naval Research under Contract No. N00014-72-A-0406-0004 and by CUNY Faculty Research Award No. 1657.
Journal of Physics B: Atomic and Molecular Physics
Atomic and molecular properties dependent on the valence electrons are sensitive to intra-atomic ... more Atomic and molecular properties dependent on the valence electrons are sensitive to intra-atomic correlation effects, and these tend to be largest in atoms of high atomic number where accurate ab initio computations are difficult. A simple model potential method is proposed for including correlation effects in Hartree-Fock calculations. Application is made to an electron-gas computation of the Hg2(X1 Sigma g+) potential.
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Papers by William Baylis