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)
• ψ0 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…
• 𝐻1 ψ0 𝑥 =
𝐻1 =
𝐴†
ħ2 𝑑 2
−
ψ
2𝑚 𝑑𝑥 2 0
𝑥 + 𝑉1 𝑥 ψ0 𝑥 = 0
…where…
ħ2 ψ0 ′′(𝑥)
2𝑚 ψ0 𝑥
𝐴
= 𝑉1 (𝑥)
…for now…
Our first Hamiltonian’s super partner
†
𝐻2 = 𝐴𝐴
ħ
𝑑
𝐴=
+𝑊 𝑥
2𝑚 𝑑𝑥
where 𝑊 𝑥 is the Super Potential
𝑉1 (𝑥) =
𝑊2
−
𝑉2 (𝑥) = 𝑊 2 +
ħ
2𝑚
ħ
2𝑚
𝑊 ′ (𝑥)
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
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
implies
H1(A†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
ħ2 𝑛(𝑛 + 1)
𝑉 𝑥 =−
2𝑚𝑎2 𝑐𝑜𝑠ℎ2 (𝑎𝑥 )
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, nanoscience, 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.