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 • = Max Planck’s constant / 2 π• m= mass• ψ(x)= an arbitrary one dimensional wave

function (think matter waves)• 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• = the creation operator= another factor of the

Hamiltonian • SUSY= Supersymmetry or supersymmetric• W= the Super Potential function

Hamiltonian Formalism

• …for some Hamiltonian (H1) let…

Our first Hamiltonian’s super partner 𝐻1 ¿ 𝐴† 𝐴

…where… …for now…

where

The Eigen Relation • So why does it matter that

one can create or even find a potential function that can be constructed from ?– 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

impliesH2(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

impliesH1(A†ψ(2)

n ) = A†AA†ψ(2)n = E(2)

n(A†ψ(2)n)

Where n is a positive integer

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

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.