Perturbative and nonperturbative studies with the delta function potential

Abstract

Spontaneous symmetry breakdown in non-relativistic quantum mechanics Am. J. Phys. 80, 891 (2012) Understanding the damping of a quantum harmonic oscillator coupled to a two-level system using analogies to classical friction Am. J. Phys. 80, 810 (2012) Relation between Poisson and Schrödinger equations Am. J. Phys. 80, 715 (2012) Comment on "Exactly solvable models to illustrate supersymmetry and test approximation methods in quantum mechanics," Am. J. Phys. 79, 755-761 (2011) Am. J. Phys. 80, 734 (2012) The uncertainty product of position and momentum in classical dynamics Am.We show that the ␦-function potential can be exploited along with perturbation theory to yield the result of certain infinite series. The idea is that any exactly soluble potential, if coupled with a ␦ function potential, remains exactly soluble. We use the strength of the ␦ function as an expansion parameter and express the second-order energy shift as an infinite sum in perturbation theory. The analytical solution is used to determine the second-order energy shift and hence the sum of an infinite series. By an appropriate choice of the unperturbed system, we can show the importance of the continuum in the energy shift of bound states.

Exact solutions of a particle in a box with a delta function potential: The factorization method

“…Equation (20) is similar to Eq. (5). Note that, although we calculated the ground state energy of the Hamiltonian in the first step of the factorization method, we found the full spectrum.…”

“…4 The solution for a particle in a box with a delta function potential has been investigated using a perturbative expansion in the strength of the delta function potential λ. 5 Exact solutions have been obtained for the weak (λ → 0) and the strong (1/λ → 0) coupling limits. 6 In this paper we discuss the solution for a particle in a box with a delta function potential using the factorization method and show that the presence of the delta function simplifies the factorization procedure.…”

Dimensional analysis and the correspondence between classical and quantum uncertainty

“…and the discontinuity of its first derivative The latter is obtained by integrating the Schrödinger equation with Hamiltonian operator specified by (80) over the small interval (0 − , 0 + ) [37,38]. From [32,33], we can see that for a particle confined in a box of length 2L, i.e.…”

“…In addition to the box boundary conditions, the presence of delta-function potential in the relative coordinate Hamiltonian imposes two sets of boundary conditions on the wavefunction at the location of the delta-function potential. These are the continuity of the wavefunction ψ| x1=x2+0 = ψ| x1=x2−0 , (84) and the discontinuity of its first derivative The latter is obtained by integrating the Schrödinger equation with Hamiltonian operator specified by (80) over the small interval (0 − , 0 + ) [37,38]. From [32,33], we can see that for a particle confined in a box of length 2L, i.e.…”

Effects of walls

“…Patil 7 and Atkinson and Crater 8 give analyses for the case that the delta function potential is located precisely at the center of the well. Bera and his co-workers 9 place it anywhere within the well and utilize perturbation theory. Joglekar 10 computes the energy eigenvalues when it is located at irrational multiples of the well's width and considers weak and strong coupling limits.…”