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
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.
Many-particle systems pose commonly known computational challenges in quantum theory. The obstacles arise from the difficulty in finding sets of eigenvalues and eigenvectors of the underlying Hamiltonian while enforcing fermion or boson statistics, not to mention the prohibitive increase in the computational cost with the system’s size. The first obvious step in this direction is to elaborate the theory for Fermi or Bose gases without inter-particle interactions. The traditional approach to the work is with the ideal gases confined in a cubic container with impenetrable walls (in arbitrary dimensions). This approach allows one to find the particle’s spectra and compute all thermodynamic quantities of the confined gas. In the present work, we consider the gas confined in a spherical container (in other words, an infinitely deep spherical potential well in D dimensions), solving the corresponding Schrödinger equation using zero boundary conditions. We address the case of a finite number of particles N, either bosons or fermions, in the spherical potential box, as well as the thermodynamic limit, corresponding to N → ∞. Owing to Weyl’s relations, in the latter limit, the results do not depend on the shape of the box and thus approach the commonly known ones valid in the infinite space. Owing to the underlying SO(D) symmetry, we are dealing with particles carrying a well-defined angular momentum that, together with sorted energy eigenvalues, imparts a shell structure to the system.
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.
Abstract In this work we solve the time-independent Schrödinger equation of a particle restricted to move on the surface of a circular cone of finite height. The energy eigenvalues, as well as the corresponding wave functions, are obtained analytically as a function of, r and ϕ , the radial distance to the apex, 0 ≤ r ≤ r 0 , and the angular variable around the axis of the cone. We compute the Shannon entropy of this system in both configuration and momentum space as a function of r 0 and θ 0 , the angular semi-aperture of the cone. In configuration space, the Shannon entropy decreases, signalling a more pronounced localization, as either r 0 or θ 0 diminish; in momentum space, an opposite behaviour happens, i.e., the Shannon entropy increases when either, r 0 or θ 0 , decrease. We also compute the radial standard deviation; we find that the Shannon entropy better describes the localization-delocalization phenomena. The present results agree with those previously published for a particle confined to a circle of radius r 0 , which corresponds to θ 0 = π / 2 in the present case.
The quantum model of finite potential well is a most subject in the book of quantum mechanics. The solutions maybe derive from one dimension Schrödinger’s equation for describe a bound state energy of confined particle in the finite potential well. Sometime, the problem of confined particle in the finite potential well does not have an exact solution yet, but there are in fact exact solution.In this work a numerical technique is adopted to find the exact solution for the quantum model of finite potential well. To achieve this numerical solution, I use Fortran code computer program by the Newton-Raphson method for calculating the Eigen state energy for a particle (electron) confined in the finite potential well. Five values of potential width and depth have studied, the results showed that large potential width and large potential depth led me to more bound states, while smaller potential width and depth led me to fewer bound states.A comparison between infinite potential and finite potential has been presented. The results bound state energy and penetration. The results of comparison showed that the wave function of particle confined in the infinite potential cannot penetrate in the well, while the wave function of particle in the finite potential was penetrate in the well; the energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well. For confined particle in 2 or 3-dimension, we can reduce from result of 1-dimension.