Exactly solvable problems in the momentum space with a minimum uncertainty in position

Abstract

A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schroëdinger equation in momentum representation on the case of deformed Heisenberg algebra with minimal length. Assuming that the kernel of potential energy operator do not change in the case of deformation, we obtain exact solution of eigenproblem of a particle in delta potential as well as double delta potential. Particle in Coulomb like potential is revisited and the problem of inversibility and hermicity of inverse coordinate operator is solved.

Cosmological horizons, uncertainty principle, and maximum length quantum mechanics

“…It is easy to show that both representations (2.7)-(2.8) and (2.9)-(2.10) satisfy the generalized commutation relation (2.1) to O(β).. Both operator representations may be used to construct and solve a generalized Schrodinger equation for simple quantum mechanical systems [18,[39][40][41][42][43][44] in one space dimension leading to generalized spectra that are consistent with the existence of a fundamental minimum lengthscale [17]. They may also be used to derive thermodynamics properties of gravity and black holes [26,[45][46][47][48][49][50][51] The operator representation (2.7)-(2.8) is more suitable for perturbative analysis of quantum systems while in the representation (2.9)-(2.10) the Hamiltonian eigenvalue problems may usually be expressed as a relatively simpler second order ODE in momentum space which may lead to exact generalized solutions [17].…”

Scattering solution of Schrödinger equation with $δ$-potential in deformed space with minimal length

“…The influence of the minimum length has been studied in the context of the following problems with singularity in potential energy: hydrogen atom [9][10][11][12][13][14][15], gravitational quantum well [16][17][18], a particle in delta potential and double delta potential [19,20], one-dimensional Coulomb-like problem [20][21][22], particle in the singular inverse square potential [23][24][25][26].…”

Regularization of 1/X2 potential in general case of deformed space with minimal length

“…The study of the effect of the minimal length on systems with singular potentials or point interactions is of particular interest, since such systems are expected to have a nontrivial sensitivity to minimal length. The impact of the minimum length has been studied in the context of the following problems with singularity in potential energy: hydrogen atom [9][10][11][12][13][14][15][16], gravitational quantum well [17][18][19], a particle in delta potential and double delta potential [20,21], one-dimensional Coulomb-like problem [21][22][23], particle in the singular inverse square potential [24][25][26][27], two-body problems with delta and Coulomb-like interactions [28].…”