Dirac Wave Function Research Articles

Elastic electron-nucleus scattering in a relativistic theory of nuclear structure

The transverse elastic form factor of 209Bi is calculated using a Dirac wave function for the valence proton orbiting a relativistic Hartree 208Pb core. Comparison is made with a corresponding nonrelativistic calculation, and relativistic interaction effects in the nuclear current operator are discussed.

Relativistic corrections of three-nucleon wave functions and muon capture inHe3

Wave functions calculated with separable potentials in standard nonrelativistic and covariant Faddeev equations were used to estimate relativistic corrections of the overall muon-capture rate in $^{3}\mathrm{He}$. They seem to cancel other relativistic corrections obtained from considering small Dirac wave function components in the effective Hamiltonian for the weak interaction.NUCLEAR REACTIONS $\ensuremath{\mu}$ capture in $^{3}\mathrm{He}$; Primakoff equation; relativistic corrections from nuclear wave functions.

Operator density current and relativistic localization problem

The well-known difficulties with the relativistic localization in quantum theory of one particle are revisited. Among them the noncausal propagation is shown to be connected with the positivity of the time translations generator. A proposal for solving this problem is presented, namely searching for an operator probability density current being the 4-vector quantity. The operator of multiplication by the argument of the Dirac wavefunction is an example of a causal position operator.

Compton scattering from electrons in bound states: Binding and "off-mass"-shell effects

The Compton scattering of a photon from an unpolarized electron bound in a central field is treated relativistically. The distribution of the emitted electrons for a photon scattering in a given direction is obtained as a function of the radial components of the initial Dirac wave function. This expression, in the lowest power of the coupling constant, leads to the Klein-Nishina formula. The off-mass shell and binding corrections to the free Compton scattering formula, which are important near the ionization threshold, are found. Using exact hydrogenlike wave functions, various approximate formulas as functions of the atomic number $Z$ are discussed, and some possible limitations of the usual description of Compton profiles are considered.

Inner shell electron ejection in the α-decay of 210Po

A model for inner shell electron ejection in nuclear α-decay is proposed. The model accounts for the change of the electron Hamiltonian on nuclear decay, treats the α-motion classically as it emerges and also includes the effect on the electron due to nuclear recoil. We have used the Dirac wave function for the electron with and without shielding. We calculate both the K- and L-shell electron ejection probabilities in 210Po α-decay. The ejected electron energy spectra fit measured data reasonably well. The calculated ejection probabilities are P K S = 2.24 × 10 −6 and P L S = 2.74 × 10 −4 with shielding, and the assumption that the α-tunnels through the nuclear barrier in a time short compared to electron orbital times is made. These values compare favourably with measured values.

Symmetrical formulation of electron–photon scattering in relativistic quantum mechanics

In this paper, electron–photon scattering in relativistic quantum mechanics is formulated in a symmetrical form by making use of a modified form of the photon wave equation proposed by the author in a previous article, together with the usual form of the Dirac wave equation. By solving the coupled wave equations two S matrices are deduced, one with the photon wave function and the other with the Dirac wave function. For a given scattering process, either one may be used to give the correct result, if it is obtainable from both. Thus, the symmetry of quantum electrodynamics is explicitly revealed in this formulation.

Translations as gauge transformations

AbstractA local description of space and time in which translations are included in the group of gauge transformations is studied using the formalism of fibre bundles. It is shown that the flat Minkowski space–time may be obtained from a non-flat connection in a de Sitter structured fibre bundle by choosing at least two different cross-sections. The interaction terms in the covariant derivative of a Dirac wave function that correspond to translations may be interpreted as the mass term of the Dirac equation, and then the two cross-sections (gauges) correspond to the description of a fermion and antifermion respectively.

De Sitter Gauge Transformations of Dirac Equation

Dirac equation is generalized in a way that combines the mass term with the term describing interaction with the gravitational field. The new equation is de Sitter invariant and involves covariant derivatives of the Dirac wave function in a fibre bundle associated with a principal fibre bundle which has a five-dimensional base manifold and de Sitter structure group. Such a principal fibre bundle may be reduced to the bundle of linear frames of a four-dimensional manifold when a specific cross-section is chosen. The generalized Dirac equation reduces then to the ordinary one with the mass term generated by the translational part of the covariant derivative. In such a way analogy between charge and mass is obtained. Every cross-section has its conjugate with reversed vertical components of all tangent vectors. It gives a geometrical meaning to charge conjugation.

Quark theory with internal coordinates

The free-particle Dirac wave function $\ensuremath{\psi}(x)$ is generalized to $\ensuremath{\psi}(x){\ensuremath{\xi}}^{a}(z)$. Here, $z$ denotes a set of three complex coordinates, called internal coordinates, in an abstract complex three-dimensional space, called internal space, $a$ runs from 1 to 3, and ${\ensuremath{\xi}}^{a}(z)$ is assumed to contain a representation of the state of a quark triplet. The mass in the free-particle Dirac equation is replaced by a second-order operator $\ensuremath{-}{\ensuremath{\partial}}_{a}^{b}$ operating on ${\ensuremath{\xi}}^{a}(z)$. The so-modified Dirac equation is assumed to include a description of a free-quark triplet. Subsequently, symmetry-preserving interactions in space-time as well as in the internal space between two quark triplets are introduced. Two S${\mathrm{U}}_{3}$-symmetry-breaking interactions, one transforming like the eighth component of an S${\mathrm{U}}_{3}$ octet vector and the other like the S${\mathrm{U}}_{3}$ charge operator, are also introduced. A similarly generalized Bethe-Salpeter equation in the ladder approximation is obtained. This equation is treated in greater detail in the following paper in which the Gell-Mann-Okubo formula for pseudoscalar mesons is derived with the coefficients determined by given relations. Then, spherical coordinates and corresponding spherical harmonics in the internal space are introduced. Finally, the equation for a one-quark system is briefly treated.

The relativistic virial theorem by the elimination method and nonrelativistic approximations to this theorem

The relativistic virial theorem for bound states of a particle is derived within the framework of the elimination method by the use of Fock's variational argument. The variational equation and the virial theorem involving only the large component of the Dirac wavefunction are deduced in closed form. The general formulae are illustrated for the case of the relativistic electron in an electrostatic central field. By use of series expansions, a nonrelativistic approximation to the virial theorem up to second order in the fine-structure constant is obtained.

Neutron Beta Decay in a Strong Magnetic Field

The $\ensuremath{\beta}$ decay of a neutron in a strong quantizing magnetic field is studied by making use of the exact Dirac wave function for the electron in a magnetic field of arbitrary strength. The general conclusion is that the lifetime is unaffected by the magnetic field for $H<{10}^{12}$ G. For $H>{10}^{12}$ G, the lifetime in a vacuum is appreciably decreased. This conclusion is still valid when the decay takes place in a degenerate electron gas at densities lower than ${10}^{6}$ g/${\mathrm{cm}}^{3}$. At higher densities, the lifetime in a magnetic field tends to reach the value in the absence of the field, for any field strength.

Reduction of Relativistic Wavefunctions to the Irreducible Representations of the Inhomogeneous Lorentz Group. I. Nonzero Mass Components

When the space-time coordinates of a relativistic system undergo the transformations of the proper, orthochronous, inhomogeneous Lorentz group, the wavefunction of the system undergoes transformations which may be considered to constitute a representation of the group. We give a simple algorithm for reducing this representation to the irreducible unitary ray representations if we assume that only nonzero mass representations occur. The extension to cases in which zero mass representations occur will be given in a later paper. The form in which the reduction is given is an expansion of the wavefunction as given in configuration space in terms of a basis such that the coefficients transform in accordance with the Foldy-Shirokov realization of the irreducible representations. Any wave equation which the wavefunction satisfies and any auxiliary conditions, such as the Lorentz condition or reality conditions, eliminate or relate in a simple way some of the representations which can appear. As examples, we reduce the scalar wavefunction, the four-vector with and without the Lorentz condition, the Dirac wavefunction, the wavefunction which transforms like the electromagnetic field, and a wavefunction which transforms as a generalization of the Dirac wavefunction. In these examples it is also shown that if one replaces the amplitudes associated with the irreducible representations by annihilation and creation operators in a suitable manner, one obtains the usual canonical formalism for second quantization in configuration space. The reduction technique given herein is a simple application of the results of an earlier paper by the author and J. S. Lomont in which is shown how to reduce any unitary ray representation of the inhomogeneous Lorentz group.

New Class of Classical Uncertainty Relations Giving Uncertainty for Long and Certainty for Short Times

Bohr's concept of complementarity enters any statistical description of physical phenomena. In quantum theory the complementary quantities are dynamical variables. In classical theory complementarity exists between dynamical and statistical variables. Slowing down of a large particle of mass $m$ by multiple collisions with a gas of small molecules leads to certainty for times. For times, the multiple collisions introduce statistics leading to uncertainty. An uncertainty relation has been derived for the coordinate $q$ as the dynamical and the drift momentum ${p}_{d}$ as the statistical complementary variable: $\ensuremath{\Delta}{p}_{d}\ensuremath{\Delta}q\ensuremath{\ge}mD(1\ensuremath{-}{e}^{\ensuremath{-}\frac{t}{\ensuremath{\tau}}})$, where $D$ is a diffusion coefficient and $\ensuremath{\tau}$ a relaxation time. This relation gives uncertainty for long and certainty for short times.The existence of such a classical uncertainty relation poses the question of whether a generalization of (relativistic) quantum theory is possible, such that for high $\ensuremath{\Delta}p\ensuremath{\Delta}q\ensuremath{\rightarrow}0$, while for low energies $\ensuremath{\Delta}p\ensuremath{\Delta}q\ensuremath{\geqq}\frac{\ensuremath{\hbar}}{2}$. This question is not answered in this note; however, it is pointed out that the telegrapher's equation (which classically implies the aforementioned generalized uncertainty principle) with $\ensuremath{\tau}=\frac{i\ensuremath{\hbar}}{2m{c}^{2}}$ is satisfied by the Klein-Gordon (also Dirac) wave function for a free particle if the time dependence of the rest mass is split off from it.

Relativistic Corrections to the Fermi Matrix Element

The uncertainty in the variations of the ft values in the scheme J = 0/ sup +/ -- J = 0/sup +/ produced by the energydependent relativistic terms of the scalar nuclear matrix elements is shown to be insufficient to mask the presence of a Fierz term larger than the limit set by Gerhart. The calculation of the matrix element is made in the shell model with harmonic wave functions. From this, the ratio of ft values for O/sup 14/ and C/sup 34/ positron. decay is determined. To check whether the result is sensitive to the shape of the potential, the calculation is repeated using the Dirac wave function for a particle in a finite square well. The comparison of the ratios shows that the effect of the relativistic corrections to the ratio of the ft values is uncertain in the exact values; the magnitude is a fraction of one per cent. (T.B.A.)

Biquadratic Spinor Identities

The representation-independent biquadratic identities which Pauli has proved to hold between a Dirac wave function and its adjoint are shown to be generalizable in several ways. To obtain these generalizations it is first shown how the symmetrical Kronecker product of two spinor representations of the orthogonal group in $n$ dimensions decomposes. Then a method is given to express the tensors formed in a particular way from two covariant and two contravariant spinors in terms of those formed in any other way. These results are applied to write explicitly all biquadratic scalar and pseudoscalar identities in $2\ensuremath{\nu}$ dimensions and all scalar identities in $2\ensuremath{\nu}+1$ dimensions. The way to obtain more general tensor identities is indicated.

On the Structure of Dirac Wave Function

On the Structure of Dirac Wave Function

The Neutron in Quantum Mechanics

IT can be shown with the help of Dirac's relativity equation that the neutron can be properly placed in the scheme of the wave-mechanical theory. Dirac's Hamiltonian for a hydrogen-like atom in polar coordinates is in which and 3 are to be taken as the matrices respectively, leading to the equations where the wave-functions are and is small1. f and g must be finite series, if a is real, or H<mc2. The radius of the smallest orbit will be the smallest value of r that makes () a maximum, since Dirac's wave-function is r times the Schrödinger function.2