QianLilongPresentatiion

Meshing Strategies of Spectral Method for
Schrödinger equation
Qian Lilong
Department of Mathematics, NUS
March 30, 2016
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Outline
Schrödinger equation
General Formulation
History
Application
Spectral Method
Preliminary
Construction
Meshing Strategy
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Schrödinger equation
∂Ψ(x, t)
i = − 1 ∂t 2 ∇2 Ψ(x, t) + V (x)Ψ(x, t) + β|Ψ(x, t)| 2 Ψ(x, t), x ∈ R d
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Schrödinger equation
∂Ψ(x, t)
i = − 1 ∂t 2 ∇2 Ψ(x, t) + V (x)Ψ(x, t) + β|Ψ(x, t)| 2 Ψ(x, t), x ∈ R d
▶ where i is the imaginary unit, β is a constant, V is a given
function.
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Schrödinger equation
∂Ψ(x, t)
i = − 1 ∂t 2 ∇2 Ψ(x, t) + V (x)Ψ(x, t) + β|Ψ(x, t)| 2 Ψ(x, t), x ∈ R d
▶ where i is the imaginary unit, β is a constant, V is a given
function.
▶ The Schrödinger equation is one of the five basic hypotheses
in quantum mechanics.
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History of Schrödinger equation
▶ 1905: Albert Einstein invents the concept of a photon
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History of Schrödinger equation
▶ 1905: Albert Einstein invents the concept of a photon
▶ 1922: Louis de Broglie (French) proposes wave-particle duality
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History of Schrödinger equation
▶ 1905: Albert Einstein invents the concept of a photon
▶ 1922: Louis de Broglie (French) proposes wave-particle duality
▶ 1927: Erwin Schrdinger (Austrian) constructs a wave equation
for de Broglie’s matter waves
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Application of Schrödinger equation
Table: The ten most important equations
1 Maxwell’s Equations
2 Euler’s Identity
3 Newton’s Second Law of Motion
4 Pythagorean Theorem
5 Massenergy Equivalence
6 Schrödinger Equation
7 1+1=2
8 De Broglie Relations
9 Fourier Transform
10 Length of the Circumference of a Circle
voted by the readers of Physical World
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Application of Schrödinger equation
▶ Scanning Tunneling Microscope (STM)
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Application of Schrödinger equation
▶ Scanning Tunneling Microscope (STM)
▶ Atomic-force microscopy (AFM)
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Application of Schrödinger equation
▶ Scanning Tunneling Microscope (STM)
▶ Atomic-force microscopy (AFM)
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Application of Schrödinger equation
▶ Scanning Tunneling Microscope (STM)
▶ Atomic-force microscopy (AFM)
▶ Quantum Computer
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Preliminary
1. Derivatives
▶
▶
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Preliminary
1. Derivatives
▶
▶
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Preliminary
1. Derivatives
2. Partial Derivatives
▶
▶
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Preliminary
1. Derivatives
2. Partial Derivatives
▶
▶
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Preliminary
1. Derivatives
2. Partial Derivatives
3. Partial Derivative Equation
▶
▶
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Preliminary
1. Derivatives
2. Partial Derivatives
3. Partial Derivative Equation
▶ Heat equation
▶
∂u(x, t)
− = ∇ 2 u(x, y, t) = ∂2 u(x, y, t)
∂t
∂x 2 + ∂2 u(x, y, t)
∂y 2
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Preliminary
1. Derivatives
2. Partial Derivatives
3. Partial Derivative Equation
▶ Heat equation
∂u(x, t)
− = ∇ 2 u(x, y, t) = ∂2 u(x, y, t)
∂t
∂x 2 + ∂2 u(x, y, t)
∂y 2
▶
Schrödinger equation
∂Ψ(x, t)
i = − 1 ∂t 2 ∇2 Ψ(x, t) + V (x)Ψ(x, t) + β|Ψ(x, t)| 2 Ψ(x, t)
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Construction of Spectral Method
1. Orthonormal basis
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Construction of Spectral Method
1. Orthonormal basis
2. Discretization
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Construction of Spectral Method
1. Orthonormal basis
2. Discretization
3. Solving the coefficients
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Orthonormal Basis
▶ basis in R 3 8 / 12

Orthonormal Basis
▶ basis in R 3 8 / 12

Orthonormal Basis
▶ basis in R 3
▶ Legendre orthogonal polynomials
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Orthonormal Basis
▶ basis in R 3
▶ Legendre orthogonal polynomials
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Orthonormal Basis
▶ basis in R 3
▶ Legendre orthogonal polynomials
▶ Sine
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Orthonormal Basis
▶ basis in R 3
▶ Legendre orthogonal polynomials
▶ Sine
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Solving the coefficient
▶ Approximation with combination from the basis
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Solving the coefficient
▶ Approximation with combination from the basis
u(x) =
{ϕ(x)} is a basis
N∑
û n ϕ n (x)
n=1
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Solving the coefficient
▶ Approximation with combination from the basis
▶ Choose the coefficients to make it closest
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Solving the coefficient
▶ Approximation with combination from the basis
▶ Choose the coefficients to make it closest
▶ Make the difference from the target function as small as
possible
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Solving the coefficient
▶ Approximation with combination from the basis
▶ Choose the coefficients to make it closest
▶ Make the difference from the target function as small as
possible
▶ Satisfy the PDE as closely as possible
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Discretization
▶ Discretization on 1 dimension
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Discretization
▶ Discretization on 1 dimension
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Discretization
▶ Discretization on 1 dimension
▶ Discretization on 2 dimension
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Discretization
▶ Discretization on 1 dimension
▶ Discretization on 2 dimension
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Discretization
▶ Discretization on 1 dimension
▶ Discretization on 2 dimension
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Meshing Strategies
▶ The mesh size in time axis t
▶ The mesh size in spatial axis h
Meshing Strategy : the relationship between t and k
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Meshing Strategies
▶ The mesh size in time axis t
▶ The mesh size in spatial axis h
Meshing Strategy : the relationship between t and k
▶ What effects does it take?
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Meshing Strategies
▶ The mesh size in time axis t
▶ The mesh size in spatial axis h
Meshing Strategy : the relationship between t and k
▶ What effects does it take?
▶ How does it affect?
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Meshing Strategies
▶ The mesh size in time axis t
▶ The mesh size in spatial axis h
Meshing Strategy : the relationship between t and k
▶ What effects does it take?
▶ How does it affect?
▶ How to choose the optimum one?
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Q&A
τħαηκ ⋎∅∪
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