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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) 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 (H 1 ) let… 𝐻 1 ψ 0 ħ 2 𝑑 2 𝑥 = − 2𝑚 𝑑𝑥 2 ψ 0 𝑥 + 𝑉 1 𝑥 ψ 0 𝑥 = 0 𝐻 1 = 𝐴 † 𝐴 …where… ħ 2 ψ 0 ′′(𝑥) 2𝑚 ψ 0 𝑥 = 𝑉 1 (𝑥) 𝐻 2 where 𝐴 = ħ 2𝑚 𝑑 𝑑𝑥 + 𝑊 𝑥 𝑊 𝑥 is the Super Potential = 𝐴𝐴 † 𝑉 1 (𝑥) = 𝑊 2 𝑉 2 (𝑥) = 𝑊 2 − + ħ 2𝑚 𝑊 ′ (𝑥) ħ 2𝑚 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 H 1 ψ (1) n implies = A † A ψ (1) n = E (1) n ψ (1) n H 2 (Aψ (1) n ) = AA † Aψ (1) n = E (1) n (A ψ (1) n ) Similarly, the Schrodinger equation for H2 H 2 ψ (2) n = AA † implies H 1 (A † ψ (2) n ψ (2) n = E (2) n 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 𝑉 𝑥 = − ħ 2 2𝑚𝑎 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."