eLecture:Quantum Graphs

Taksu Cheon

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Quantum Graphs : Physics of Quantum Singularity (5-1)

Duality and Aholonomy 1

Looking at the boundary conditions for delta and epsilon potentials, we can sense the exsitence of some symmetry. First thing to be noticed is the fact that the delta potential has no effects on antisymmetric wavefunctions, while epsilon has no effect on symmetric ones.

When we consider these effects in the problem of two spinless particles on one dimension, an important fact emerges: If we assume the coordinate x to be a relative coordinate of two particles, delta potential leaves no effects on two fermions whose relative wave function must be antisymmetric and, at the same time, eplsilon potential has no effects on two bosons whose relative wave functions are always symmetric. Anotyher way of expressing this fact is that just as the delta potential being a generic one-parameter family of inter-bosonic point interaction, epsilon potential is a generic one-parameter family of inter-fermionic point interaction.

Now we examine the two-body scattering problem of delta and epsilon potentials. Writing the scattering matrix as S, we have the scattering wave function at positive and negative x regions as



(5.1)

Here, in the combined signs, + stands for the case of symmetrized scattering of bosons and - for antisymmetrized scattering of fermions. The simbol k is the relative momentum. From the connection condition (2.8), we obtain the expression for S. For bosons, it is given by

 ; 
 ; 
(5.2)

and for fermions, it is

 ; 
 ; 
(5.3)

As observed earlier, epsilon has no effects on bosons resulting in S_+=1, and delta has no effects on fermions resulting in S_-=1. A more surprising is the fact that S_+ for delta and S_- for epsilon become identical if we impose the following condition between the delta and epsilon strengths b and d;

(5.4)

In other words, if we consider two identical particles, strongly couped bosonic system and the weekly coupled fermionic system (and vice versa) are mathematically identical. It can be shown that this duality holds for systems with arbitrary number of particles.

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