Eigenstates of a particle confined to a cone of finite length capped by a spherical surface element are derived. A countable infinite set of solutions is obtained corresponding to integer azimuthal and orbital quantum numbers ( m , l ) \left ( m, l \right ) . These solutions apply to a discrete subset of the domain of half vertex angles, 0 ≤ θ 0 ≤ π / 2 0 \le {\theta _0} \le \pi /2 . For arbitrary real orbital quantum numbers, l → ν l \to \nu , solutions are given in terms of the hypergeometric function, with ν = ν ( θ 0 ) \nu = \nu \left ( {\theta _0} \right ) , and are valid in the θ 0 {\theta _0} domain, 0 ≤ θ 0 > π / 2 0 \le {\theta _0} > \pi /2 . Eigenstates are either nondegenerate or two-fold degenerate. Numerical examples of both classes of solutions are included. For the case μ = cos π / 4 \mu = \cos \pi /4 , the ground-state wavefunction and eigenenergy are [ φ G
Abstract We consider the quantum problem of a particle in either a spherical box or a finite spherical well confined by a circular cone with an apex angle 2θ 0 emanating from the center of the sphere, with 0 < θ 0 < π. This non-central potential can be solved by an extension of techniques used in spherically-symmetric problems. The angular parts of the eigenstates depend on azimuthal angle φ and polar angle θ as P λ m ( cos θ ) e im φ where P λ m is the associated Legendre function of integer order m and (usually noninteger) degree λ. There is an infinite discrete set of values λ = λ i m (i = 0, 1, 3, ...) that depend on m and θ 0. Each λ i m has an infinite sequence of eigenenergies E n ( λ i m ) , with corresponding radial parts of eigenfunctions. In a spherical box the discrete energy spectrum is determined by the zeros of the spherical Bessel functions. For several θ 0 we demonstrate the validity of Weyl’s continuous estimate W for the exact number of states up to energy E, and evaluate the fluctuations of around W . We examine the behavior of bound states in a well of finite depth U 0, and find the critical value U c (θ 0) when all bound states disappear. The radial part of the zero energy eigenstate outside the well is 1/r λ+1, which is not square-integrable for λ ≤ 1/2. (0 < λ ≤ 1/2) can appear for θ
We solve the stationary Schrödinger equation for a particle confined to a 3D spherical wedge -- the region
{(r,θ,φ):0≤r≤R,0≤θ≤π,0≤φ≤Φ} with Dirichlet BCs on all surfaces. This exactly solvable constrained-domain model exhibits spectral reorganisation under symmetry-breaking BCs and provides an operator-domain viewpoint on angular momentum quantisation. We obtain three main results. First, the stationary states are standing waves in the azimuthal coordinate and consequently are \emph{not} eigenstates of
L^z; we prove
⟨Lz⟩=0 with
ΔLz=ℏnφπ/Φ=0, demonstrating that angular momentum projection becomes an observable with genuine quantum uncertainty rather than a good quantum number. Second, the effective azimuthal quantum number
μ=nφπ/Φ is generically non-integer, and square-integrability of the polar wavefunctions at both poles requires the angular eigenvalue parameter
ν to satisfy
ν−μ∈Z≥0. This regularity constraint yields a hierarchy: sectoral solutions (
ν=μ, satisfying the first-order highest-weight condition) exist for any real
μ>0, while tesseral and zonal solutions require integer steps, appearing only when
μ itself is integer. Third, application to a Coulomb potential shows that the familiar integer angular momentum spectrum of hydrogen arises from the periodic identification
φ∼φ+2π that defines the full-sphere Hilbert space domain; modified boundary conditions yield a reorganised spectrum with non-integer effective angular momentum. The model clarifies the distinct roles of single-valuedness (selecting integer
m via azimuthal topology) and polar regularity (selecting integer
ℓ≥∣m∣ via analytic constraints) in the standard quantisation of orbital angular momentum.
We examine the quantum mechanical eigensolutions of the two-dimensional infinite-well or quantum billiard system consisting of a circular boundary with an infinite barrier or baffle along a radius. Because of the change in boundary conditions, this system includes quantized angular momentum values corresponding to half-integral multiples of ℏ/2. We discuss the resulting energy eigenvalue spectrum and visualize some of the novel energy eigenstates found in this system. We also discuss the density of energy eigenvalues, N ( E ), comparing this system to the standard circular well. These two billiard geometries have the same area ( A = π R 2 ), but different perimeters ( P = 2π R versus (2π + 2) R ), and we compare both cases to fits of N ( E ) which make use of purely geometric arguments involving only A and P . We also point out connections between the angular solutions of this system and the familiar pedagogical example of the one-dimensional infinite-well plus δ-function potential.
We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and
Ntoinfty regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
The present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates. The ‘‘polygon ground-state theorem’’ is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the ‘‘elemental polygons’’ (defined in the text). The ‘‘polygon first excited-state theorem’’ is established which states that for every N-sided regular polygon, N equivalent first excited states exist, each of which contains a nodal curve that is a line of mirror symmetry of the related polygon. A vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the N-sided regular polygon is spanned by any two of these N nodal-diagonal eigenstates (i.e., the first excited state is twofold degenerate). At various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations. Thus, for example, correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive, nonoverlapping classes of polygons. Stemming from the preceding conclusions, a discussion is included on the possibility of employing transverse magnetic (TM) modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic.
We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.