2 My research Goals Challenges: To prepare for more competitive research by expanding my knowledge through study of:Basic Quantum Mechanics and SupersymmetryAs well as looking at topological modes in Classical (mass and spring) latticesChallenges:No previous experience with quantum mechanics, supersymmetry, or modern algebra
3 What is Supersymmetry? Math… A principle Very general mathematical symmetryA supersymmetric theory allows for the interchanging of mass and force termsHas several interesting consequences such asEvery fundamental particle has a super particle (matches bosons to fermionic super partners and vice versaIn my studies supersymmetry simply allows for the existence of super partner potential fields
4 Q M Terminology (1) QM= Quantum Mechanics ħ= Max Planck’s constant / 2 πm= massψ(x)= an arbitrary one dimensional wave function (think matter waves)ψ 0 𝑥 = The ground state wave function= the wave function at its lowest possible energy for the corresponding potential well
5 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 systemA= the annihilation operator= a factor of the Hamiltonian HA † = the creation operator= another factor of the HamiltonianSUSY= Supersymmetry or supersymmetricW= the Super Potential function
7 The Eigen RelationThe 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 H1H1ψ(1)n = A†A ψ(1)n= E(1)n ψ(1)nimpliesH2(Aψ(1)n) = AA†Aψ(1)n= E(1)n(A ψ(1)n)Similarly, the Schrodinger equation for H2H2ψ(2)n= AA† ψ(2)n = E(2)n ψ(2)nH1(A†ψ(2)n ) = A†AA†ψ(2)n = E(2)n(A†ψ(2)n)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
8 Reflectionless potentials, Another, consequence of SUSY QMEven constant potential functions can have supersymmetric partner’sIn some cases this leads to potential barriers allowing complete transmission of matter wavesThese potentials are often classified by their super potential function𝑉 𝑥 =− ħ 2 2𝑚 𝑎 2 𝑛(𝑛+1) 𝑐𝑜𝑠ℎ 2 ( 𝑥 𝑎 )Where n is a positive integern=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.
9 More cutting edge research and applications Reflectionless potentials are predicted to speed up optical connectionsSUSY QM can be used in examining modes in isostatic latticesLattices are very important in the fields of condensed matter, nano-science, optics, quantum information, etc.
10 Acknowledgements Helping make this possible my mentor this summer Dr. Victor GalitskiMy mentors during spring semester at Howard University Dr. James Lindesay and Dr. Marcus AlfredDr. Edward (Joe) Reddish
11 ReferencesCooper, 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): Print.Lekner, John. "Reflectionless Eigenstates of the Sech[sup 2] Potential." American Journal of Physics (2007): 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, Print.