supersymmetric quantum mechanics and reflectionless potentials
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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 - PowerPoint PPT PresentationTRANSCRIPT
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.