Solving the ground state energy of the quantum harmonic oscillator - PowerPoint PPT Presentation

Solving the ground state energy of the quantum harmonic oscillator with Diffusion Monte Carlo method Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Wednesday, 20 Dec 2010 Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Abstract Non-relatisitic quantum mechanical systems are discribed by the Shroedinger Equation. To obtain the ground state solution for quantum system with large degree of freedom, variantional method could be numerically inefficient and time consuming. A far more superior numerical method to solve the Shroedinger equation with large degree of freedom is the diffusion Monte Carlo (DMC) method. In this talk I will briefly explain some technical detail of the DMC. I will illustrate the application using the example of a 1-D quantum harmonic oscillator. Generalisation to higher dimensional case is straight forward. Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Variation method for the Schroedinger Equation [ 2 m ∇ 2 + V ( r ) ] − ℏ 2 휓 ( r ) = E 휓 ( r ) H 휓 ( r ) = E 휓 ( r ). ∫ dX 휓 ∗ ( X ) H 휓 ( X ) dX 휓 ∗ ( X ) 휓 ( X ) = ⟨ 휓 ∣ H ∣ 휓 ⟩ Consider the functional E [ 휓 ] = ∫ ⟨ 휓 ∣ 휓 ⟩ Solve the SE by considering the stationary condition of the functional: 훿 E = 0 . Defining P = ⟨ 휓 ∣ H ∣ 휓 ⟩ and Q = ⟨ 휓 ∣ 휓 ⟩ so that E = P / Q 훿 E = ⟨ 휓 + 훿휓 ∣ H ∣ 휓 + 훿휓 ⟩ ⟨ 휓 + 훿휓 ∣ 휓 + 훿휓 ⟩ − ⟨ 휓 ∣ H ∣ 휓 ⟩ ⟨ 휓 ∣ 휓 ⟩ ≈ ⟨ 훿휓 ∣ H ∣ 휓 ⟩− ( P / Q ) ⟨ 훿휓 ∣ 휓 ⟩ + ⟨ 휓 ∣ H ∣ 훿휓 ⟩− ( P / Q ) ⟨ 휓 ∣ 훿휓 ⟩ Q Q H ∣ 휓 ⟩ = E ∣ 휓 ⟩ . Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Variation method for the Schroedinger Equation (cont.) In variational calculus, stationary states of the energy-functional are spanned by a set of basis vector ∣ 휒 p ⟩ , p = 1 , 2 , ⋅ ⋅ ⋅ , N . The stationary states is then given by the expansion in terms of the basis vector, ∣ 휓 ⟩ = ∑ p C p ∣ 휒 p ⟩ . ∑ N p , q =1 C ∗ p C q H pq E = p C q S pq . ∑ N p , q =1 C ∗ H pq = ⟨ 휒 p ∣ H ∣ 휒 q ⟩ . S pq = ⟨ 휒 p ∣ 휒 q ⟩ . ∑ N q =1 ( H pq − ES pq ) C q = 0 for p = 1 , 2 , ⋅ ⋅ ⋅ , N . Generalised Eigen value problem HC = E SC . Solve using numerical routines, e.g. LAPACK’s DSYGV subroutine. Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Weakness of variation method ∼ 풪 ( N 3 ). Choice of basis set is crucial. Only upper limit of the true ground state can be found, E ≥ E 0 . For system with complicated potential, N required maybe to large to be handled by the numerical routines. Monte Carlo Method come to the rescue. Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

(Modified) Diffusion equation ∂ 2 휌 ( x ,휏 ) ∂휌 ( x ,휏 ) = 1 − [ V ( x ) − E T ] 휌 ( x , 휏 ) ≡ ℒ 휌 ( x , 휏 ). ∂휏 2 ∂ x 2 휏 → i t → 1D time dependent Schrodinger Equation. The diffusion equation describes how the probability distribution 휌 evolve in time. 휌 ( x , t ) is the density distribution for a large collection of independent walkers. For stationary distribution, 휌 ( x , 휏 → 0) = 휌 ( x ), we recover the stationary Schroedinger equation (with imaginary time, ∂ 2 휌 ( x ,휏 ) t → 휏 = it ): − 1 + V ( x ) 휌 ( x , 휏 ) = E T 휌 ( x , 휏 ). ∂ x 2 2 Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Green’s funcition Normalised Green’s function of the modified diffusion equation is G ( x , y ; Δ t ) = ⟨ x ∣ e t ℒ ∣ y ⟩ . Note that ℒ comprised of two part: the kinetic part, ∂ x ) 2 and the potential part, V − E T . K = p 2 / 2 = 1 2 ( − i ∂ ∫ 휌 ( y , t ) = dxG ( x , y ; t ) 휌 ( x , 0). ∫ dyG ( x , y ; t ) = 1; G ( x , y ; t ) → 훿 ( x − y ) for t → 0 . GF is the probability distribution of a single walker which starts of at position x at t = 0. Use the GF to construct a Markov process with transition probability T Δ t ( x → y ) = G ( x , y ; Δ t ). It turns out that G ( x , y ; Δ t ) = G Kin ( x , y ; Δ t ) e − Δ 휏 [ V ( y ) − E T ] + 풪 (Δ t ) 2 ( − ( x − y ) 2 ) 1 G Kin ( x , y ; Δ t ) = 2 휋 Δ 휏 exp . √ 2Δ 휏 Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Markov chain A Markov chain is defined in terms of the transition probability T ( X → X ′ ): P N ( X 1 , X 2 , ⋅ ⋅ ⋅ , X N ) = P 1 ( X 1 ) T ( X 1 → X 2 ) T ( X 2 → X 3 ) ⋅ ⋅ ⋅ T ( X N − 1 → X N ). X ′ T ( X → X ′ ) = 1. ∑ X denote the configurations of a given system. Given the transitional probability T ( X → X ′ ) Markov chain generates a sequence of configurations that depend only on the last configuration. After a long time, the Markov chain will forget its initial configuration, and the configurations it generates will sample a distribution 휌 . In other words, if we want to sample a desired distribution 휌 , we need to know what is the T ( X → X ′ ) that leads to 휌 . Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Generating random walk in a Markov process In the case of the diffusion equation with distribution 휌 ( x , t ), T Δ t ( x → y ) = G ( x , y ; Δ t ). We can generate the distribution 휌 ( x , t ) given the knowledge of G ( x , y ; Δ t ) = G Kin ( x , y ; Δ t ) e − Δ 휏 ( V ( y ) − E T ) , ( − ( x − x ′ ) 2 ) 1 G Kin ( x , y ; Δ t ) = 2 휋 Δ 휏 exp . √ 2Δ 휏 √ x ( t + Δ t ) = x ( x ) + 휂 Δ t , √ 4 휋훾 e − 휂 2 / 4 훾 , a gaussian with width √ 2 훾 Δ t ; 훾 = 1 / 2 1 P ( 휂 ) = is the coefficient to the p 2 = − ∂ 2 /∂ x 2 term in the diffusion equation. The shifting of a walker’s position x takes care of the G Kin term. The e − Δ 휏 ( V ( y ) − E T ) terms is taken care of by accepting the trial shifting with a probability with weight exp( − Δ t [ V ( x ′ ) − E T ]). E T is the ground state energy we wish to find. Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Birth and Dealth In the MC stepping, ˜ M walkers are spawn randomly over an initial range, e.g. ( − 10 . 0 , 10 . 0). √ Then the walkers are shifted via x ( t + Δ t ) = x ( t ) + 휂 Δ t , and the trial move is accepted with probability exp( − Δ t [ V ( x ′ ) − E T ]). After some moves, the population of the walker will change. Use ‘birth and dealth’ method to improve the efficiency of the computation Kill walker with small acceptance rate, duplicate those favarable one. This will avoid spending too much computational effort on those poor walker visiting unfavarouble region. Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

Birth and Death In the MC stepping, ˜ M walkers are spawn randomly over an initial range, e.g. ( − 10 . 0 , 10 . 0). √ Then the walkers are shifted via x ( t + Δ t ) = x ( t ) + 휂 Δ t , and the trial move is accepted with probability exp( − Δ t [ V ( x ′ ) − E T ]). After some moves, the population of the walker will change. Use ‘birth and death’ method to improve the efficiency of the computation Kill walker with small acceptance rate, duplicate those favarable one. This will avoid spending too much computational effort on those poor walker visiting unfavarouble region. Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil

The algorithm Put the walkers at random positions in configuration space; REPEAT FOR all walkers DO Shift walker from its position R to a new position R ′ according ( ) − 휂 2 1 to the Gaussian transition probability P ( 휂 ) = √ 4 휋훾 exp . 4 훾 Evaluate q = − exp { Δ t [ V ( R ′ ) − E T ] } ; Eliminate the walker or create new ones at R ′ , depending on s = q + r , r is random, uniform between 0 and 1; (effectively, at [ s ] walkers are created at R ′ ([ s ] = 0 , 1 , 2 , ⋅ ⋅ ⋅ ); if [ s ] = 0 then the walker is deleted.) END FOR; Update E T Until Finished Yoon Tiem Leong Talk given at theory group weekly seminar, School of Physics, Universiti Sains Malaysia Solving the ground state energy of the quantum harmonic oscil