Supersymmetric Quantum Mechanics and Reflectionless Potentials by Kahlil Dixon (Howard University) My research • Goals – To prepare for more competitive research by expanding my knowledge through study of: • Basic Quantum Mechanics and Supersymmetry • As well as looking at topological modes in Classical (mass and spring) lattices • Challenges: – No previous experience with quantum mechanics, supersymmetry, or modern algebra What is Supersymmetry? • Math… • A principle – Very general mathematical symmetry • A supersymmetric theory allows for the interchanging of mass and force terms – Has several interesting consequences such as • Every fundamental particle has a super particle (matches bosons to fermionic super partners and vice versa – In my studies supersymmetry simply allows for the existence of super partner potential fields Q M Terminology (1) • • • • QM= Quantum Mechanics ħ= Max Planck’s constant / 2 π m= mass ψ(x)= an arbitrary one dimensional wave function (think matter waves) • ψ0 𝑥 = The ground state wave function= the wave function at its lowest possible energy for the corresponding potential well Q M Terminology (2) • H= usually corresponds to the Hamiltonian… – The Hamiltonian is the sum of the Kinetic (T)and Potential (V) energy of the system • A= the annihilation operator= a factor of the Hamiltonian H • A† = the creation operator= another factor of the Hamiltonian • SUSY= Supersymmetry or supersymmetric • W= the Super Potential function Hamiltonian Formalism • …for some Hamiltonian (H1) let… • 𝐻1 ψ0 𝑥 = ħ2 𝑑 2 − ψ 2𝑚 𝑑𝑥 2 0 𝑥 + 𝑉1 𝑥 ψ0 𝑥 = 0 …where… 𝐻1 = 𝐴† ħ2 ψ0 ′′(𝑥) 2𝑚 ψ0 𝑥 𝐴 = 𝑉1 (𝑥) …for now… Our first Hamiltonian’s super partner † 𝐻2 = 𝐴𝐴 ħ 𝑑 𝐴= +𝑊 𝑥 2𝑚 𝑑𝑥 where 𝑊 𝑥 is the Super Potential 𝑉1 (𝑥) = 𝑊2 − 𝑉2 (𝑥) = 𝑊 2 + ħ 2𝑚 ħ 2𝑚 𝑊 ′ (𝑥) 𝑊 ′ (𝑥) The Eigen Relation • So why does it matter that one can create or even find a potential function that can be constructed from 𝐴𝐴† ? – Because the two potentials share energy spectra The potentials V1(x) and V2(x) are known as supersymmetric partner potentials. As we shall see, the energy eigenvalues, the wave functions and the S-matrices of H1 and H2 are related. To that end notice that the energy eigenvalues of both H1 and H2 are positive semi-definite (E(1,2) n ≥ 0) . For n > 0, the Schrodinger equation for H1 H1ψ(1)n = A†A ψ(1)n= E(1)n ψ(1)n implies H2(Aψ(1)n) = AA†Aψ(1)n= E(1)n(A ψ(1)n) Similarly, the Schrodinger equation for H2 H2ψ(2)n= AA† ψ(2)n = E(2)n ψ(2)n implies H1(A†ψ(2)n ) = A†AA†ψ(2)n = E(2)n(A†ψ(2)n) Reflectionless potentials, • Another, consequence of SUSY QM • Even constant potential functions can have supersymmetric partner’s • In some cases this leads to potential barriers allowing complete transmission of matter waves • These potentials are often classified by their super potential function ħ2 𝑛(𝑛 + 1) 𝑉 𝑥 =− 2𝑚𝑎2 𝑐𝑜𝑠ℎ2 (𝑎𝑥 ) Where n is a positive integer n=1. The wave functions are raised from the x axis to separate them from 2ma2 /2 times the =1 potential, namely −2 sech2x/a filled shape. More cutting edge research and applications • Reflectionless potentials are predicted to speed up optical connections • SUSY QM can be used in examining modes in isostatic lattices • Lattices are very important in the fields of condensed matter, nanoscience, optics, quantum information, etc. Acknowledgements • Helping make this possible – my mentor this summer Dr. Victor Galitski – My mentors during spring semester at Howard University Dr. James Lindesay and Dr. Marcus Alfred – Dr. Edward (Joe) Reddish References Cooper, Fred, Avinash Khare, Uday Sukhatme, and Richard W. Haymaker. "Supersymmetry in Quantum Mechanics." American Journal of Physics 71.4 (2003): 409. Web. Kane, C. L., and T. C. Lubensky. "Topological Boundary Modes in Isostatic Lattices." Nature Physics 10.1 (2013): 39-45. Print. Lekner, John. "Reflectionless Eigenstates of the Sech[sup 2] Potential." American Journal of Physics 75.12 (2007): 1151. Web. Maluck, Jens, and Sebastian De Haro. "An Introduction to Supersymmetric Quantum Mechanics and Shape Invariant Potentials." Thesis. Ed. Jan Pieter Van Der Schaar. Amsterdam University College, 2013. Print.