arXiv subjects

Scott E. Hoffmann

Publications and source records attributed to Scott E. Hoffmann.

At least 19 recordsLinked to original sources

arXiv2023

Incorporating the Coulomb potential into a finite, unitary perturbation theory

We have constructed a perturbation theory to treat interactions that can include the Coulomb interaction, describing a physical problem that is often encountered in nuclear physics. The Coulomb part is not treated perturbatively; the exact solutions are employed. The method is an extension of the results presented in Hoffmann (2021 J. Math. Phys. 62 032105). It is designed to calculate phase shifts directly rather than the full form of the wavefunctions in position space. We present formulas that allow calculation of the phase shifts to second order in the perturbation. The phase shift results to second order, for a short-range potential, were compared with the exact solution, where we found an error of third order in the coupling strength. A different model, meant as a simple approximation of nuclear scattering of a proton on Helium-4 and including a Coulomb potential and a spherical well, was constructed to test the theory. The wavepacket scattering formalism of Hoffmann (2017 J. Phy. B: At. Mol. Opt. Phys 50 215302), known to give everywhere finite results, was employed. We found physically acceptable results and a cross section of the correct order of magnitude.

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arXiv2021

An observable for the separation distance of two photons

Motivated by an intention to apply partial wave analysis to systems containing photons, we construct eigenvectors of the distance of separation between two photons. A choice is made that makes the case of two photons most closely resemble the case of two massive particles. The treatment is relativistic. An Hermitian separation observable can then be defined, unlike a position operator for a single photon, which cannot be defined. We test these results on a model system of two photons meant to describe the initial or final state of a scattering experiment. We find that the separation probability density gives a meaningful picture of localization.

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arXiv2021

Finite perturbation theory for the relativistic Coulomb problem

We present a novel form of relativistic quantum mechanics and demonstrate how to solve it using a recently derived unitary perturbation theory, within partial wave analysis. The theory is tested on a relativistic problem, with two spinless, equal mass particles, in which the interaction is entirely given by a Coulomb potential. As such, it is not meant to reproduce experimental results for the scattering of two electrons, but is intended as a test of our calculation methods. We find that this perturbation theory gives finite results at second order. This is unlike other versions of perturbation theory, which find divergent results at second and all higher orders. We calculate differential cross sections in the nonrelativistic regime, where we find excellent agreement with the Rutherford formula. Then, well into the relativistic regime, we find differential cross sections with similar shapes to the Møller formula and differing from that formula by less than an order of magnitude.

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arXiv2020

Unitary, continuum, stationary perturbation theory for the radial Schrödinger equation

The commutators of the Poincaré group generators will be unchanged in form if a unitary transformation relates the free generators to the generators of an interacting relativistic theory. We test the concept of unitary transformations of generators in the nonrelativistic case, requiring that the free and interacting Hamiltonians be related by a unitary transformation. Other authors have applied this concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In our case, a stationary perturbation theory can be constructed to find approximate solutions of the radial Schrödinger equation for scattering from a spherically symmetric potential. General formulae are obtained for the phase shifts at first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to second order of the exact solution.

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arXiv2020

No point-localized photon states

The aim of this paper is to critically examine claims that it is possible to construct point-localized state vectors for the photon. We supply a brief proof of the impossibility of this. Then it is found that the authors making these claims use a non-standard scalar product, not equal to the quantum-mechanical one. This alternative scalar product is found to be proportional to a Dirac delta function in position for the state vectors they use, but the remaining elements of the proof, namely satisfying all three Newton-Wigner criteria, are lacking.

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arXiv2020

Electromagnetic field expectations as measures of photon localization

The questions of whether a photon can be localized in an arbitrarily small volume and what is the allowable strength of that localization (the decrease with distance of the functional form) are questions of current interest. We propose a measure of localization for the single photon that is the expectation values of the electromagnetic field strength components in a coherent wavepacket state of mean photon number unity. As such, we deal with real quantities that have a physical meaning rather than complex amplitudes. It is seen that the real parts of complex amplitudes proposed previously as measures of localization are equal to our field expectations. With this measure, we examine two test states. The first has a well-resolved momentum. The field expectations show Gaussian (quadratic exponential) localization in all directions, although the localization length scale is much larger than the mean wavelength. For the other test state, with a spherically symmetric momentum distribution, we find almost exponential localization in all directions at t = 0. The profiles are scale invariant, so choosing the momentum width very large would make the localization length arbitrarily small. We conclude that there is no lower bound on the localization length scale of a photon as determined by this measure.

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arXiv2020

Measurability of Coulomb wavepacket scattering effects

A previous paper [J. Phys. B: At. Mol. Opt. Phys. 50, 215302 (2017)] showed that partial wave analysis becomes applicable to nonrelativistic Coulomb scattering if wavepackets are used. The scattering geometry considered was special: that of a head-on collision between the wavepacket and the centre of the potential. Our results predicted, in this case, a shadow zone of low probability for small angles around the forward direction for the description of alpha scattering from a gold foil. In this paper we generalize the results to the case of a nonzero impact parameter, a displacement of the wavepacket centre perpendicular to the average momentum direction. We predict a large flux in the forward direction from events with large impact parameters. We find a significant probability of scattering into the deviation region for impact parameters of order the spatial width of the wavepacket. Averaging over impact parameters produces predictions in excellent agreement with the Rutherford formula down to lower angles than for the zero impact parameter prediction. We consider issues that would arise in a real experiment and discuss the possibility of measuring a deviation from the Rutherford formula.

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arXiv2019

Finite second-order Born term for Coulomb wavepacket scattering

It has been known for some time that, for nonrelativistic Coulomb scattering, the terms in the Born series of second and higher order diverge when using the standard method of calculation. In this paper we take the matrix elements between square-integrable wavepacket state vectors. We reproduce the Rutherford cross section from the first-order contribution. We find that the second-order contribution is finite and negligible compared to the first-order contribution, away from the forward direction. At first order, the contribution to the amplitude in the forward direction is found to be finite and physically reasonable. We comment on how a similar procedure applied to the divergences of quantum field theories might render them finite.

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arXiv2018

Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator

We construct ladder operators, $\tilde{C}$ and $\tilde{C^\dagger}$, for a multi-step rational extension of the harmonic oscillator on the half plane, $x\ge0$. These ladder operators connect all states of the spectrum in only infinite-dimensional representations of their polynomial Heisenberg algebra. For comparison, we also construct two different classes of ladder operator acting on this system that form finite-dimensional as well as infinite-dimensional representations of their respective polynomial Heisenberg algebras. For the rational extension, we construct the position wavefunctions in terms of exceptional orthogonal polynomials. For a particular choice of parameters, we construct the coherent states, eigenvectors of $\tilde{C}$ with generally complex eigenvalues, $z$, as superpositions of a subset of the energy eigenvectors. Then we calculate the properties of these coherent states, looking for classical or non-classical behaviour. We calculate the energy expectation as a function of $|z|$. We plot position probability densities for the coherent states and for the even and odd cat states formed from these coherent states. We plot the Wigner function for a particular choice of $z$. For these coherent states on one arm of a beamsplitter, we calculate the two excitation number distribution and the linear entropy of the output state. We plot the standard deviations in $x$ and $p$ and find no squeezing in the regime considered. By plotting the Mandel $Q$ parameter for the coherent states as a function of $|z|$, we find that the number statistics is sub-Poissonian.

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arXiv2018

No relativistic probability current for any spin

We investigate whether the Newton-Wigner position probability density, extended from spinless particles to electrons/positrons and particles of higher spin, can be incorporated as the zero component of a four-component probability current density that transforms locally as a four-vector function of the spacetime coordinates. We find that this is not possible, in all cases.

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arXiv2018

The minimum width in relativistic quantum mechanics

We challenge the widespread belief, originated by Newton and Wigner (Rev. Mod. Phys, 21, 400 (1949)) that the incorporation of special relativity into quantum mechanics implies that a massive particle cannot be localized within an arbitrarily small spatial extent, that there is a minimum width approximately equal to the Compton wavelength. Our argument is in four parts. First, the scalar function used by Newton and Wigner as a measure of localization is not a position probability amplitude. The correct relativistic position probability amplitude becomes a delta function for a state vector localized according to the criteria of Newton and Wigner. Second, the possibility of Lorentz contraction as observed from a boosted frame means that the wavepacket width in the boost direction can take arbitrarily small values. Third, we refer to the work of Almeida and Jabs (Am. J. Phys. 52, 921 (1984)) who show that the long time wavepacket spreading rate for relativistic position probability amplitudes is always less than the speed of light no matter how small the initial width of the wavepacket. Lastly, we show that it is a simple matter to construct scalar amplitudes with spatial widths smaller than the supposed minimum.

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arXiv2018

Relativistic probability amplitudes II. The photon

We identify momentum/helicity probability amplitudes for the photon and find their relativistic transformation properties. We also find their behaviour under space inversion and time reversal. The discussion begins with a review of the unitary, irreducible representations of the Poincare group for massless particles. The little group, the set of Lorentz transformations that leave the momentum unchanged, is distinctly different for massless particles compared to the little group of rest frame rotations for a massive particle. We give a physical interpretation of the little group for a massless particle. The explicit forms of the Wigner rotations for general rotations and boosts are given. In normalized superpositions of the basis vectors, we identify the momentum/helicity probability amplitudes, show their probability interpretation and find their transformation properties. We see that position eigenvectors for the photon are not possible, not because of their masslessness but because of their limited helicity spectrum. Instead, to have a measure of localization for the photon, we use the expectations of the electromagnetic field strength operators in a coherent state with a mean photon number of unity. In the construction of the polarization vectors appearing in the field strengths, we see the connection between gauge invariance and Lorentz covariance.

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arXiv2018

Spreading of relativistic probability densities and Lorentz contraction

We find the laws for the spreading of the spatial widths (parallel and transverse to the direction of average motion) of the relativistic position probability density for a massive, spinless particle. We find that when the momentum width of the wavepacket is small compared to the average momentum, there is a long time over which spreading is minimal. This result may be useful in particle accelerator design. We also demonstrate the Lorentz contraction of a wavepacket using relativistic probability amplitudes.

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arXiv2018

Relativistic probability amplitudes I. Massive particles of any spin

We consider a massive particle of arbitrary spin and the basis vectors that carry the unitary, irreducible representations of the Poincaré group. From the complex coefficients in normalizable superpositions of these basis vectors, we identify momentum/spin-component probability amplitudes with the same interpretation as in the nonrelativistic theory. We find the relativistic transformations of these amplitudes, which are unitary in that they preserve the modulus-squared of scalar products from frame to frame. Space inversion and time reversal are also treated. We reconsider the Newton- Wigner construction of eigenvectors of position and the position operator. Position/spin-component probability amplitudes are also identified and their relativistic, unitary, transformations derived. Again, space inversion and time reversal are considered. For reference, we show how to construct positive energy solutions of the Klein-Gordon and Dirac equations in terms of probability amplitudes. We find the boost transformation of the position operator in the spinless case and present some results on the relativity of position measurements. We consider issues surrounding the classical concept of causality as it applies in quantum mechanics. We briefly examine the relevance of the results presented here for theories of interaction.

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arXiv2018

Coherent states for ladder operators of general order related to exceptional orthogonal polynomials

We construct the coherent states of general order, $m$ for the ladder operators, $c(m)$ and $c^\dagger(m)$, which act on rational deformations of the harmonic oscillator. The position wavefunctions of the eigenvectors involve type III Hermite exceptional orthogonal polynomials. We plot energy expectations, time-dependent position probability densities for the coherent states and for the even and odd cat states, Wigner functions, and Heisenberg uncertainty relations. We find generally non-classical behaviour, with one exception: there is a regime of large magnitude of the coherent state parameter, $z$, where the otherwise indistinct position probability density separates into $m+1$ distinct wavepackets oscillating and colliding in the potential, forming interference fringes when they collide. The Mandel $Q$ parameter is calculated to find sub-Poissonian statistics, another indicator of non-classical behaviour. We plot the position standard deviation and find squeezing in many of the cases. We calculate the two-photon-number probability density for the output state when the $m=4$, $μ=-5$ coherent states (where $μ$ labels the lowest weight in the superposition) are placed on one arm of a beamsplitter. We find that it does not factorize, again indicating non-classical behaviour. Calculation of the linear entropy for this beamsplitter output state shows significant entanglement, another non-classical feature. We also construct linearized versions, $\tilde c(m)$, of the annihilation operators and their coherent states and calculate the same properties that we investigate for the coherent states. For these we find similar behaviour to the $c(m)$ coherent states, at much smaller magnitudes of $z$, but comparable average energies.

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arXiv2017

Uniform analytic approximation of Wigner rotation matrices

We derive the leading asymptotic approximation, for low angle θ, of the Wigner rotation matrix elements $d^j_{m_1m_2}(θ)$, uniform in $j,m_1$ and $m_2$. The result is in terms of a Bessel function of integer order. We numerically investigate the error for a variety of cases and find that the approximation can be useful over a significant range of angles. This approximation has application in the partial wave analysis of wavepacket scattering.

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arXiv2017

Prediction of deviations from the Rutherford formula for low-energy Coulomb scattering of wavepackets

We calculate the nonrelativistic scattering of a wavepacket from a Coulomb potential and find deviations from the Rutherford formula in all cases. These generally occur only at low scattering angles, where they would be obscured by the part of the incident beam that emerges essentially unscattered. For a model experiment, the scattering of helium nuclei from a thin gold foil, we find the deviation region is magnified for low incident energies (in the keV range), so that a large shadow zone of low probability around the forward direction is expected to be measurable. From a theoretical perspective, the use of wavepackets makes partial wave analysis applicable to this infinite-range potential. It allows us to calculate the everywhere finite probability for a wavepacket to wavepacket transition and to relate this to the differential cross section. Time delays and advancements in the detection probabilities can be calculated. We investigate the optical theorem as applied to this special case.

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arXiv2017

Non-classical behaviour of coherent states for systems constructed using exceptional orthogonal polynomials

We construct the coherent states and Schrödinger cat states associated with new types of ladder operators for a particular case of a rationally extended harmonic oscillator involving type III Hermite exceptional orthogonal polynomials. In addition to the coherent states of the annihilation operator, $c$, we form the linearised version, \tilde{c}, and obtain its coherent states. We find that while the coherent states defined as eigenvectors of the annihilation operator $c$ display only quantum behaviour, those of the linearised version, \tilde{c}, have position probability densities displaying distinct wavepackets oscillating and colliding in the potential. The collisions are certainly quantum, as interference fringes are produced, but the remaining evolution indicates a classical analogue.

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