Abstract The enhancement factor αD ≡ 1+ δαD is defined (in D dimensions) as the ratio of the particle density at the origin to the density far upstream in the incident beam. At high incident momentum p (and for regular potentials) the first Born approximation is known to be adequate in 3D and 1D, and is assumed to be adequate also in 2D; it entails δα1 = ─δα3 if U(r) in 3D equals U(x) in 1D when r = x. For central potentials U(r) with U(0) finite, previous work implies δα3 ~ ─δα1 ~ ─mU(0)/p2, and δα2 = 0 to the same order h°. It is shown that if U(r→0) ~ U(0) + rU'(0) + ..., then δα2 ~ (2mU'(0)h/p3)(π/16), the value of U(0) being irrelevant. IfU(r→0) ~ ─C/rq, with 0 < q < 2, then δα3 ~ ─δα1 ~ (2mC/hqp2-q) π½Γ(1-½ q)/2Γ(½+½q); and, with -1 < q < 2, δα2 ~ (2mC/hqp2-q)π½Γ2(1-½q)/2Γ(½q)Γ(3/2-½q). The only mathematics needed in 3D and 1D is the standard asymptotic estimation of Fourier integrals; but in 2D one needs to develop corresponding methods for integrals where the sine or cosine has been replaced by a product J0Y0 of two Bessel functions.
Abstract The enhancement factor αD ≡ 1+ δαD is defined (in D dimensions) as the ratio of the particle density at the origin to the density far upstream in the incident beam. At high incident momentum p (and for regular potentials) the first Born approximation is known to be adequate in 3D and 1D, and is assumed to be adequate also in 2D; it entails δα1 = ─δα3 if U(r) in 3D equals U(x) in 1D when r = x. For central potentials U(r) with U(0) finite, previous work implies δα3 ~ ─δα1 ~ ─mU(0)/p2, and δα2 = 0 to the same order h°. It is shown that if U(r→0) ~ U(0) + rU'(0) + ..., then δα2 ~ (2mU'(0)h/p3)(π/16), the value of U(0) being irrelevant. IfU(r→0) ~ ─C/rq, with 0 < q < 2, then δα3 ~ ─δα1 ~ (2mC/hqp2-q) π½Γ(1-½ q)/2Γ(½+½q); and, with -1 < q < 2, δα2 ~ (2mC/hqp2-q)π½Γ2(1-½q)/2Γ(½q)Γ(3/2-½q). The only mathematics needed in 3D and 1D is the standard asymptotic estimation of Fourier integrals; but in 2D one needs to develop corresponding methods for integrals where the sine or cosine has been replaced by a product J0Y0 of two Bessel functions.
Abstract In classical mechanics (c.m.), and near the semi-classical limit h→0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ0/ρ∞ are found for scattering by attractive central potentials U(r); here ρ0,∞ (and v0,∞) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials (U(0) > — ∞), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high v∞ ), α1 = v∞/v0, α2 = 1, α3 = v0/v∞ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U(r→0) ~ ─ C/rq, with 0 < q < 1 in ID (where r ≡ |x|), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., α1(c.m.) = 0, α3(c.m.) = ∞, and α2(c.m.) = (1 —½q) N, where N = [integer part of (1 ─½q)-1]is the number of trajectories through any point (r, θ) in the limit r → 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low v∞) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r; for small v∞/u it yields the leading terms α1(s.c.l.) = Λ1(q)v∞/u, α2(s.c.I) = (1 ─½q)-1, α3(s.c.l.)= Λ3(q)u/v∞, where u ≡ (C/hqm1-q)1/(2-q) is a generalized Bohr velocity. Here Λ1,3 are functions of q alone, given in the text; as q →0 the α (s.c.l.) agree with the α quoted above for finite potentials. Even in the limit h = 0, α2 (s.c.l.) and α2 (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential ─ C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter α(e.g. U = ─ Cexp (─r/a)/rq), and if, near the s.c.l., v∞/u is regarded as a function not of h but more realistically of v∞, then the expressions α (s.c.l.) above apply only in an intermediate range 1/a ≪ mv∞/h ≪ (mC/h2)1/(2-q) which exists only if a ≫ (h2/mC)1/(2-q)).
Parabolic coordinates allow Schrödinger’s equation to be solved analytically for a particle confined to the xy plane and incident with velocity v on a potential C/r. In sharp contrast to the three-dimensional case, the cross section assumes its classical value only in the rationally foreseeable classical regime ‖C‖/ℏv≫1, but not otherwise; and in this low-velocity regime the enhancement factor (probability density at the origin) for an attractive potential tends to 2 (independently of the value of ‖C‖), instead of diverging as in three dimensions like some inverse power of v.
We consider the scattering of nonrelativistic particles in three dimensions by a contact potential Ωℏ 2 δ( r )/2µ r α , which is defined as the a →0 limit of Ωℏ 2 δ( r - a )/2µ r α . It is surprising that it gives a nonvanishing cross section when α = 1 and Ω = -1. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter Ω that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.
By solving the Schrodinger equations in three-dimensional (3D) and two-dimensional (2D) finite depth center spherically symmetric square potential well, the eigenequations of bounded energy levels and their distributions are obtained, and the analytical solution for any bound state wave function is given. Both the eigenequations of the energy levels as well as the stationary state wave functions are closely related to the (spherical) Bessel functions. It is found that for the 3D case, there exists a bound solution when 2μa2V0/ħ2>π2/4 is present. While for the 2D case, there exists a bound state if 2μa2V0/ħ2>0.1993. Finally, numerical methods are used to analyze the characteristics of probability density distribution of different quantum states. It is found that in both three-dimensional and two-dimensional situations, the maximum number of probability density distributions inside the well corresponds exactly to the quantum number n. The larger n or the smaller α is, the higher the probability of discovering particles outside the well. In addition, the angular momentum quantum number l or m can characterize the degree to which particles deviate from the center or the uniform distribution.
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.
Analytic expressions are derived for the root-mean-square (rms) radius of a pair of fermions in a BCS many-fermion state in one, two, and three dimensions, in terms of the BCS gap energy and the associated chemical potential. These expressions are valid for any coupling strength of any pair interaction model implying a momentum-independent gap energy. The latter holds, e.g., for an attractive \ensuremath{\delta} pair potential examined in the one-dimensional (1D) case (whose N-fermion ground state can be determined exactly) or for the BCS (electron-phonon) model interaction in any dimension. Weak-coupling and/or high-density limits for the rms radius are identical in 1D, 2D, and 3D, and reduce to the familiar well-known Pippard result to within a factor of order unity. In contrast, strong-coupling and/or low-density limits coincide in 1D and 3D, but differ by a factor of order unity in the 2D limit, and in each case are essentially the size of a single, isolated pair. The 1D \ensuremath{\delta} interaction McGuire-Yang-Gaudin many-fermion model is studied in detail. The interaction renormalization scheme of Miyake and of Randeria, Duan, and Shieh, and the BCS interaction model, both in 2D, are employed to analyze cuprate superconductor empirical results. Reasonable agreement between theoretical rms radii with experimental coherence lengths suggests that cuprates can be described moderately well as weakly coupled superconductors within the BCS-Bose formalism.