Supersymmetric Quantum Mechanics and Reflectionless Potentials by Kahlil Dixon (Howard University)

My research Goals Challenges: 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) ψ 0 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… 𝐻 1 ψ 0 𝑥 =− ħ 2 2𝑚 𝑑 2 𝑑𝑥 2 ψ 0 𝑥 + 𝑉 1 𝑥 ψ 0 𝑥 =0 …where… ħ 2 2𝑚 ψ 0 ′′(𝑥) ψ 0 𝑥 = 𝑉 1 (𝑥) …for now… 𝐻 1 = 𝐴 † 𝐴 Our first Hamiltonian’s super partner 𝐻 2 = 𝐴𝐴 † 𝐴= ħ 2𝑚 𝑑 𝑑𝑥 +𝑊 𝑥 where 𝑊 𝑥 is the Super Potential 𝑉 1 (𝑥)= 𝑊 2 − ħ 2𝑚 𝑊 ′ (𝑥) 𝑉 2 (𝑥)= 𝑊 2 + ħ 2𝑚 𝑊 ′ (𝑥)

The Eigen Relation 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 H1(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

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 𝑉 𝑥 =− ħ 2 2𝑚 𝑎 2 𝑛(𝑛+1) 𝑐𝑜𝑠ℎ 2 ( 𝑥 𝑎 ) 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, nano-science, 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.