Abstract Biological cells, aerosols and dust grains are corrugated rather than spherical, yet quantum scattering theory is developed almost entirely for smooth spherical targets. We lift that restriction for a controlled family of corrugated quasi-spheres and identify the matter-wave observables that carry a readable signature of the surface structure. We study the non-relativistic scattering of a plane wave by a three-dimensional Chebyshev particle, a sphere whose radius is modulated by a Chebyshev polynomial. The surface is modeled as a boundary-wall δ -potential and the Lippmann–Schwinger equation is solved through position-dependent reflection and transmission coefficients, yielding the probability density, the differential and total cross-sections, and the quantum refractive index. The β -fold azimuthal symmetry imposes a selection rule m = m ′ + q β on the coupling between partial waves; in the ε → 0 limit the formalism recovers the analytical spherical δ -shell result, which we use as a benchmark. The total cross-section shows transparency windows at the zeros of j ℓ ( k R ) for either sign of γ , which we interpret as a three-dimensional Ramsauer–Townsend effect; the attractive barrier adds sharper resonances associated with quasi-bound states. Switching on the polar undulation α breaks separability in the polar angle and couples partial waves, producing narrow resonances whose positions shift with small changes in the deformation. For attractive particles the refractive index has a negative real part and a positive imaginary part tied to the total cross-section; for repulsive particles it is smooth and monotonic. In both regimes the index approaches its vacuum value n → 1 at high energy, where the particle becomes transparent.
Abstract Biological cells, aerosols and dust grains are corrugated rather than spherical, yet quantum scattering theory is developed almost entirely for smooth spherical targets. We lift that restriction for a controlled family of corrugated quasi-spheres and identify the matter-wave observables that carry a readable signature of the surface structure. We study the non-relativistic scattering of a plane wave by a three-dimensional Chebyshev particle, a sphere whose radius is modulated by a Chebyshev polynomial. The surface is modeled as a boundary-wall δ -potential and the Lippmann–Schwinger equation is solved through position-dependent reflection and transmission coefficients, yielding the probability density, the differential and total cross-sections, and the quantum refractive index. The β -fold azimuthal symmetry imposes a selection rule m = m ′ + q β on the coupling between partial waves; in the ε → 0 limit the formalism recovers the analytical spherical δ -shell result, which we use as a benchmark. The total cross-section shows transparency windows at the zeros of j ℓ ( k R ) for either sign of γ , which we interpret as a three-dimensional Ramsauer–Townsend effect; the attractive barrier adds sharper resonances associated with quasi-bound states. Switching on the polar undulation α breaks separability in the polar angle and couples partial waves, producing narrow resonances whose positions shift with small changes in the deformation. For attractive particles the refractive index has a negative real part and a positive imaginary part tied to the total cross-section; for repulsive particles it is smooth and monotonic. In both regimes the index approaches its vacuum value n → 1 at high energy, where the particle becomes transparent.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. The integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. The relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. The chapter concludes with a consideration of identical particle scattering and an application to the proton–proton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. e integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. e relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. e chapter concludes with a consideration of identical particle scattering and an application to the protonproton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
We investigate the non-relativistic scattering of a plane wave by a vertical segment formulating the problem in terms of the Lippmann–Schwinger equation in two spatial dimensions. Adjusting the coupling strength function we show how to implement the scattering by a system of multiple slits and by a Cantor set. We present detailed calculations of the scattered wave function for the line segment, as well as for the single, double, and multiple slits. We define reflection and transmission functions that are position-dependent in a defined region. From these results, we obtain the probability densities and differential and total cross-sections for these problems.
We are concerned with the justification of the statement, commonly (explicitly or implicitly) used in quantum scattering theory, that for a free non-relativistic quantum particle with initial wave function Ψ0(x), surrounded by detectors along a sphere of large radius R, the probability distribution of the detection time and place has asymptotic density (i.e., scattering cross section) σ(x,t)=m3ℏ−3Rt−4∣Ψ0(mx/ℏt)∣2 with Ψ0 the Fourier transform of Ψ0. We give two derivations of this formula, based on different macroscopic models of the detection process. The first one consists of a negative imaginary potential of strength λ>0 in the detector volume (i.e., outside the sphere of radius R) in the limit R→∞,λ→0,Rλ→∞. The second one consists of repeated nearly-projective measurements of (approximately) the observable 1∣x∣>R at times T,2T,3T,… in the limit R→∞,T→∞,T/R→0; this setup is similar to that of the quantum Zeno effect, except that there one considers T→0 instead of T→∞. We also provide a comparison to Bohmian mechanics: while in the absence of detectors, the arrival times and places of the Bohmian trajectories on the sphere of radius R have asymptotic distribution density given by the same formula as σ, their deviation from the detection times and places is not necessarily small, although it is small compared to R, so the effect of the presence of detectors on the particle can be neglected in the far-field regime. We also cover the generalization to surfaces with non-spherical shape, to the case of N non-interacting particles, to time-dependent surfaces, and to the Dirac equation.
We quantize the electromagnetic field in the presence of a nonmoving dielectric sphere in vacuum. The sphere is assumed to be lossless, dispersionless, isotropic, and homogeneous. The quantization is performed using normalized eigenmodes as well as plane-wave modes. We specify two useful alternative bases of normalized eigenmodes: spherical eigenmodes and scattering eigenmodes. A canonical transformation between plane-wave modes and normalized eigenmodes is derived. This formalism is employed to study the scattering of a single photon, coherent squeezed light, and two-photon states off a dielectric sphere. In the latter case, we calculate the second-order correlation function of the scattered field, thereby unveiling the angular distribution of the Hong–Ou–Mandel interference for a dielectric sphere acting as a three-dimensional beam splitter. Our results are analytically derived for a dielectric sphere of arbitrary refractive index and size with a particular emphasis on the small-particle limit. As shown in Phys. Rev. A 108 , 033714 ( 2023 ) PLRAAN 1050-2947 10.1103/PhysRevA.108.033714 , this work sets the theoretical foundation for describing the quantum interaction between light and the motional, rotational, and vibrational degrees of freedom of a dielectric sphere.