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Quantum Mechanics for Graphs and CW complexes
This project develops a version of the Schr¨
- dinger model of
Quantum Mechanics for Graphs and CW-Complexes Michael Toriyama, Zhe - - PowerPoint PPT Presentation
Quantum Mechanics for Graphs and CW-Complexes Michael Toriyama, Zhe - - PowerPoint PPT Presentation
Quantum Mechanics for Graphs and CW-Complexes Michael Toriyama, Zhe Hu, Boyan Xu, Chengzheng Yu Sarah Loeb Ivan Contreras University of Illinois at Urbana-Champaign Illinois Geometry Lab Midterm Presentation October 25, 2016 Quantum
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Schr¨
- dinger equation for graphs
We use a discretization of the Schr¨
- dinger equation from
“usual” quantum mechanics which takes the form i ∂ ∂t Ψt = −∆Ψt and is solved by the exponential exp(i∆t)Ψ0. This partition function is typically easy to compute and encodes certain topological and combinatorial data of the graph.
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Graph Laplacian
Definition The adjacency matrix AΓ of a graph Γ is given by A(i, j) = 1 i ∼ j
- therwise
and graph Laplacian ∆Γ ∆Γ(i, j) = val(i) i = j −1 i ∼ j
- therwise
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Example
Given a graph Γ 1 2 3 4 AΓ = 1 1 1 1 1 1 1 1 and ∆Γ = 1 −1 −1 3 −1 −1 −1 2 −1 −1 −1 2
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Result on “new paths”
Theorem For a graph Γ ∆k
Γ(i, j) =
- |γi→j|=k
sgn(γ) where the sum is taken over “new paths” γi→j of length k starting from vertex i and ending at j.
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Application to topology
We can use the Laplacian ∆ to calculate Betti numbers of graphs b0 = dim(ker(∆)) b1 = dim(coker(∆)) 1 2 3 1 2 3 4 1 2 3 4
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Evolution of quantum states
The Schr¨
- dinger operator in QM is -∆ + u(x) We use
u(x)=0. Solve the ODE:
∂φ ∂t = −∆φ
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