http://cds.cern.ch/record/262851/files/9405029.pdf

Supersymmetry and quantum mechanics

Advanced Mathematical Methods for Scientists and Engineers

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Introduction To Electrodynamics

We formulate and discuss one-particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E>0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low-energy behaviour of one theory gives the high-energy behaviour of the transformed theory. Finally, we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs use only known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitian symplectic forms.

https://iopscience.iop.org/article/10.1088/0305-4470/32/4/006/pdf

Kirchhoff's rule for quantum wires

An elementary introduction is given to the subject of supersymmetry in quantum mechanics which can be understood and appreciated by any one who has taken a first course in quantum mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct n new exactly solvable Hamiltonians having n − 1, n − 2,…, 0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one‐dimensional harmonic oscillator problem to a class of potentials called shape‐invariant potentials. It is worth emphasizing that this class includes almost all the solvable problems that are found in the standard text books on quantum mechanics. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s‐matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape‐invariant potentials and further, this approximation is not only exact for large quantum numbers but by construction, it is also exact for the ground state. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.

Supersymmetry in quantum mechanics

We discuss in some detail the self-similar potentials of Shabat [Inverse Prob. 8, 303 (1992)] and Spiridonov [Phys. Rev. Lett. 69, 298 (1992)] which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape-invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling Ansatz for the change of parameters, we obtain a large class of new, reflectionless, shape-invariant potentials of which the Shabat-Spiridonov ones are a special case. These new potentials can be viewed as q deformations of the single-soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for the energy eigenvalues, eigenfunctions, and transmission coefficients for these potentials are obtained. We show that these potentials can also be obtained numerically. Included as an intriguing case is a shape-invariant double-well potential whose supersymmetric partner potential is only a single well. Our class of exactly solvable Hamiltonians is further enlarged by examining two new directions: (i) changes of parameters which are different from the previously studied cases of translation and scaling and (ii) extending the usual concept of shape invariance in one step to a multistep situation. These extensions can be viewed as q deformations of the harmonic oscillator or multisoliton solutions corresponding to the Rosen-Morse potential.

/pdf/new-exactly-solvable-hamiltonians-shape-invariance-and-self-59bh51cvkm.pdf

New exactly solvable Hamiltonians: Shape invariance and self-similarity.

Traditional Quantum Mechanics - The Hard Way Algebraic Solution for the Harmonic Oscillator Supersymmetric Quantum Mechanics (SUSYQM) Shape Invariance The Generators of Supersymmetry Angular Momentum Dirac Theory and SUSYQM WKB and SWKB Scattering in Supersymmetric Quantum Mechanics Isospectral Deformations SUSYQM and Quantum Hamilton-Jacobi Theory Generating Shape Invariant Potentials Singular Superpotentials.

Supersymmetric Quantum Mechanics: An Introduction

It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be applied to several problems with noncentral vector and scalar potentials. As examples, we analyze the bound state spectra of an electron in a Coulomb plus an Aharonov–Bohm field and/or in the magnetic field of a Dirac monopole.

/pdf/noncentral-potentials-and-spherical-harmonics-using-2fs2yta03q.pdf

Noncentral potentials and spherical harmonics using supersymmetry and shape invariance

In supersymmetric quantum mechanics, shape invariance is a sufficient condition for solvability. We show that all conventional additive shape-invariant superpotentials that are independent of $\ensuremath{\hbar}$ can be generated from two partial differential equations. One of these is equivalent to the one-dimensional Euler equation expressing momentum conservation for inviscid fluid flow, and it is closed by the other. We solve these equations, generate the set of all conventional shape-invariant superpotentials, and show that there are no others in this category. We then develop an algorithm for generating all additive shape-invariant superpotentials including those that depend on $\ensuremath{\hbar}$ explicitly.

/pdf/generation-of-a-complete-set-of-additive-shape-invariant-3ul2lfljqc.pdf

Generation of a complete set of additive shape-invariant potentials from an Euler equation.

It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be applied to several problems with non-central vector and scalar potentials. As examples, we analyze the bound state spectra of an electron in a Coulomb plus an Aharonov-Bohm field and/or in the magnetic field of a Dirac monopole.

/pdf/non-central-potentials-and-spherical-harmonics-using-waphy1o60k.pdf

Asim Gangopadhyaya

Papers

New exactly solvable Hamiltonians: Shape invariance and self-similarity.

Supersymmetric Quantum Mechanics: An Introduction

Noncentral potentials and spherical harmonics using supersymmetry and shape invariance

Generation of a complete set of additive shape-invariant potentials from an Euler equation.

Non-Central Potentials and Spherical Harmonics Using Supersymmetry and Shape Invariance