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Meshing Strategies of Spectral Method for<br />
Schrödinger equation<br />
Qian Lilong<br />
Department of Mathematics, NUS<br />
March 30, 2016<br />
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Outline<br />
Schrödinger equation<br />
General Formulation<br />
History<br />
Application<br />
Spectral Method<br />
Preliminary<br />
Construction<br />
Meshing Strategy<br />
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Schrödinger equation<br />
∂Ψ(x, t)<br />
i = − 1 ∂t 2 ∇2 Ψ(x, t) + V (x)Ψ(x, t) + β|Ψ(x, t)| 2 Ψ(x, t), x ∈ R d<br />
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Schrödinger equation<br />
∂Ψ(x, t)<br />
i = − 1 ∂t 2 ∇2 Ψ(x, t) + V (x)Ψ(x, t) + β|Ψ(x, t)| 2 Ψ(x, t), x ∈ R d<br />
▶ where i is the imaginary unit, β is a constant, V is a given<br />
function.<br />
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Schrödinger equation<br />
∂Ψ(x, t)<br />
i = − 1 ∂t 2 ∇2 Ψ(x, t) + V (x)Ψ(x, t) + β|Ψ(x, t)| 2 Ψ(x, t), x ∈ R d<br />
▶ where i is the imaginary unit, β is a constant, V is a given<br />
function.<br />
▶ The Schrödinger equation is one of the five basic hypotheses<br />
in quantum mechanics.<br />
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History of Schrödinger equation<br />
▶ 1905: Albert Einstein invents the concept of a photon<br />
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History of Schrödinger equation<br />
▶ 1905: Albert Einstein invents the concept of a photon<br />
▶ 1922: Louis de Broglie (French) proposes wave-particle duality<br />
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History of Schrödinger equation<br />
▶ 1905: Albert Einstein invents the concept of a photon<br />
▶ 1922: Louis de Broglie (French) proposes wave-particle duality<br />
▶ 1927: Erwin Schrdinger (Austrian) constructs a wave equation<br />
for de Broglie’s matter waves<br />
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Application of Schrödinger equation<br />
Table: The ten most important equations<br />
1 Maxwell’s Equations<br />
2 Euler’s Identity<br />
3 Newton’s Second Law of Motion<br />
4 Pythagorean Theorem<br />
5 Massenergy Equivalence<br />
6 Schrödinger Equation<br />
7 1+1=2<br />
8 De Broglie Relations<br />
9 Fourier Transform<br />
10 Length of the Circumference of a Circle<br />
voted by the readers of Physical World<br />
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Application of Schrödinger equation<br />
▶ Scanning Tunneling Microscope (STM)<br />
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Application of Schrödinger equation<br />
▶ Scanning Tunneling Microscope (STM)<br />
▶ Atomic-force microscopy (AFM)<br />
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Application of Schrödinger equation<br />
▶ Scanning Tunneling Microscope (STM)<br />
▶ Atomic-force microscopy (AFM)<br />
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Application of Schrödinger equation<br />
▶ Scanning Tunneling Microscope (STM)<br />
▶ Atomic-force microscopy (AFM)<br />
▶ Quantum Computer<br />
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Preliminary<br />
1. Derivatives<br />
▶<br />
▶<br />
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Preliminary<br />
1. Derivatives<br />
▶<br />
▶<br />
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Preliminary<br />
1. Derivatives<br />
2. Partial Derivatives<br />
▶<br />
▶<br />
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Preliminary<br />
1. Derivatives<br />
2. Partial Derivatives<br />
▶<br />
▶<br />
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Preliminary<br />
1. Derivatives<br />
2. Partial Derivatives<br />
3. Partial Derivative Equation<br />
▶<br />
▶<br />
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Preliminary<br />
1. Derivatives<br />
2. Partial Derivatives<br />
3. Partial Derivative Equation<br />
▶ Heat equation<br />
▶<br />
∂u(x, t)<br />
− = ∇ 2 u(x, y, t) = ∂2 u(x, y, t)<br />
∂t<br />
∂x 2 + ∂2 u(x, y, t)<br />
∂y 2<br />
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Preliminary<br />
1. Derivatives<br />
2. Partial Derivatives<br />
3. Partial Derivative Equation<br />
▶ Heat equation<br />
∂u(x, t)<br />
− = ∇ 2 u(x, y, t) = ∂2 u(x, y, t)<br />
∂t<br />
∂x 2 + ∂2 u(x, y, t)<br />
∂y 2<br />
▶<br />
Schrödinger equation<br />
∂Ψ(x, t)<br />
i = − 1 ∂t 2 ∇2 Ψ(x, t) + V (x)Ψ(x, t) + β|Ψ(x, t)| 2 Ψ(x, t)<br />
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Construction of Spectral Method<br />
1. Orthonormal basis<br />
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Construction of Spectral Method<br />
1. Orthonormal basis<br />
2. Discretization<br />
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Construction of Spectral Method<br />
1. Orthonormal basis<br />
2. Discretization<br />
3. Solving the coefficients<br />
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Orthonormal Basis<br />
▶ basis in R 3 8 / 12
Orthonormal Basis<br />
▶ basis in R 3 8 / 12
Orthonormal Basis<br />
▶ basis in R 3<br />
▶ Legendre orthogonal polynomials<br />
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Orthonormal Basis<br />
▶ basis in R 3<br />
▶ Legendre orthogonal polynomials<br />
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Orthonormal Basis<br />
▶ basis in R 3<br />
▶ Legendre orthogonal polynomials<br />
▶ Sine<br />
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Orthonormal Basis<br />
▶ basis in R 3<br />
▶ Legendre orthogonal polynomials<br />
▶ Sine<br />
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Solving the coefficient<br />
▶ Approximation with combination from the basis<br />
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Solving the coefficient<br />
▶ Approximation with combination from the basis<br />
u(x) =<br />
{ϕ(x)} is a basis<br />
N∑<br />
û n ϕ n (x)<br />
n=1<br />
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Solving the coefficient<br />
▶ Approximation with combination from the basis<br />
▶ Choose the coefficients to make it closest<br />
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Solving the coefficient<br />
▶ Approximation with combination from the basis<br />
▶ Choose the coefficients to make it closest<br />
▶ Make the difference from the target function as small as<br />
possible<br />
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Solving the coefficient<br />
▶ Approximation with combination from the basis<br />
▶ Choose the coefficients to make it closest<br />
▶ Make the difference from the target function as small as<br />
possible<br />
▶ Satisfy the PDE as closely as possible<br />
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Discretization<br />
▶ Discretization on 1 dimension<br />
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Discretization<br />
▶ Discretization on 1 dimension<br />
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Discretization<br />
▶ Discretization on 1 dimension<br />
▶ Discretization on 2 dimension<br />
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Discretization<br />
▶ Discretization on 1 dimension<br />
▶ Discretization on 2 dimension<br />
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Discretization<br />
▶ Discretization on 1 dimension<br />
▶ Discretization on 2 dimension<br />
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Meshing Strategies<br />
▶ The mesh size in time axis t<br />
▶ The mesh size in spatial axis h<br />
Meshing Strategy : the relationship between t and k<br />
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Meshing Strategies<br />
▶ The mesh size in time axis t<br />
▶ The mesh size in spatial axis h<br />
Meshing Strategy : the relationship between t and k<br />
▶ What effects does it take?<br />
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Meshing Strategies<br />
▶ The mesh size in time axis t<br />
▶ The mesh size in spatial axis h<br />
Meshing Strategy : the relationship between t and k<br />
▶ What effects does it take?<br />
▶ How does it affect?<br />
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Meshing Strategies<br />
▶ The mesh size in time axis t<br />
▶ The mesh size in spatial axis h<br />
Meshing Strategy : the relationship between t and k<br />
▶ What effects does it take?<br />
▶ How does it affect?<br />
▶ How to choose the optimum one?<br />
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Q&A<br />
τħαηκ ⋎∅∪<br />
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