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

[ − ℏ2

2m∇2 + V (r)

] 휓(r) = E휓(r) H휓(r) = E휓(r). Consider the functional E[휓] =

∫ dX휓∗(X)H휓(X) ∫ dX휓∗(X)휓(X) = ⟨휓∣H∣휓⟩ ⟨휓∣휓⟩

Solve the SE by considering the stationary condition of the functional: 훿E = 0. Defining P = ⟨휓∣H∣휓⟩ and Q = ⟨휓∣휓⟩ so that E = P/Q 훿E = ⟨휓+훿휓∣H∣휓+훿휓⟩

⟨휓+훿휓∣휓+훿휓⟩ − ⟨휓∣H∣휓⟩ ⟨휓∣휓⟩ ≈ ⟨훿휓∣H∣휓⟩−(P/Q)⟨훿휓∣휓⟩ Q

+ ⟨휓∣H∣훿휓⟩−(P/Q)⟨휓∣훿휓⟩

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 ∣휒p⟩, p = 1, 2, ⋅ ⋅ ⋅ , N. The stationary states is then given by the expansion in terms

• f the basis vector, ∣휓⟩ = ∑

p Cp∣휒p⟩.

E =

∑N

p,q=1 C ∗ p CqHpq

∑N

p,q=1 C ∗ p CqSpq .

Hpq = ⟨휒p∣H∣휒q⟩. Spq = ⟨휒p∣휒q⟩. ∑N

q=1(Hpq − ESpq)Cq = 0 for p = 1, 2, ⋅ ⋅ ⋅ , N.

Generalised Eigen value problem HC = ESC. 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(N3). Choice of basis set is crucial. Only upper limit of the true ground state can be found, E ≥ E0. 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

∂휌(x,휏) ∂휏

= 1

2 ∂2휌(x,휏) ∂x2

− [V (x) − ET]휌(x, 휏) ≡ ℒ휌(x, 휏). 휏 → it → 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, t → 휏 = it): − 1

2 ∂2휌(x,휏) ∂x2

+ V (x)휌(x, 휏) = ET휌(x, 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

Green’s funcition

Normalised Green’s function of the modified diffusion equation is G(x, y; Δt) = ⟨x∣etℒ∣y⟩. Note that ℒ comprised of two part: the kinetic part, K = p2/2 = 1

2(−i ∂ ∂x )2 and the potential part, V − ET.

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 TΔt(x → y) = G(x, y; Δt). It turns out that G(x, y; Δt) = GKin(x, y; Δt)e−Δ휏[V (y)−ET ] + 풪(Δt)2 GKin(x, y; Δt) =

1 √ 2휋Δ휏 exp

(

−(x−y)2 2Δ휏

) .

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 ′): PN(X1, X2, ⋅ ⋅ ⋅ , XN) = P1(X1)T(X1 → X2)T(X2 → X3) ⋅ ⋅ ⋅ T(XN−1 → XN). ∑

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), TΔt(x → y) = G(x, y; Δt). We can generate the distribution 휌(x, t) given the knowledge

• f G(x, y; Δt) = GKin(x, y; Δt)e−Δ휏(V (y)−ET ),

GKin(x, y; Δt) =

1 √ 2휋Δ휏 exp

(

−(x−x′)2 2Δ휏

) . x(t + Δt) = x(x) + 휂 √ Δt, P(휂) =

1 √4휋훾 e−휂2/4훾, a gaussian with width √2훾Δt; 훾 = 1/2

is the coefficient to the p2 = −∂2/∂x2 term in the diffusion equation. The shifting of a walker’s position x takes care of the GKin term. The e−Δ휏(V (y)−ET ) terms is taken care of by accepting the trial shifting with a probability with weight exp(−Δt[V (x′) − ET]). ET 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 + Δt) = x(t) + 휂 √ Δt, and the trial move is accepted with probability exp(−Δt[V (x′) − ET]). 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 + Δt) = x(t) + 휂 √ Δt, and the trial move is accepted with probability exp(−Δt[V (x′) − ET]). 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 to the Gaussian transition probability P(휂) =

1 √4휋훾 exp

(

−휂2 4훾

) . Evaluate q = − exp{Δt[V (R′) − ET]}; 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 ET 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

More details

A code is developed to impliment the MC time-stepping. Initially spawn ˜ M (e.g. 400) walkers are randomly sprinkle in a range of choice (e.g (0,5)) in the configuration space (i.e. the x−axis.) Then for each walkers shifted the ‘merit function’ q is calculated at the shifted position q(x′) for that walker. Generate r to be added to q to give s = q + r. [s] is then

• evaluated. If [s] = 0, the walker is ‘killed’; If [s] = 1, the

walker are accpeted at the new position x′. If [s] ≥ 2, [s] − 1 walkers at the new position x′ are created.

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

How ET is found

ET is found by adjusting it such that the Green’s function (overall transition rate) is normalise to maintain the stability

• f the population (not over grow nor vanish):

ET = E0 + 훼 ln( ˜

M M ).

As MC time-stepping proceeds, the number of walker M in the configuration space changes, hence the energy ET = E0 + 훼 ln ( ˜

M M

) will be adjusted from MC step to

• MCstep. M changes in the way to that is ‘steered’ by the

‘target number’ ˜ M. ET will fluctuate in each MC step. Accumulate the values of ET at each MC step to calculate ⟨ET⟩ (and ΔE = √ ⟨E 2⟩ − (⟨E⟩)2) after the MC steps has run for some long enough ‘InitStep’, i.e. when equilibrium has achieved. The MC time stepping shall stop after running for some predefined times, MaxStep. E.g. MaxStep = 4000, InitStep = 1000.

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

Pros and cons

Pro: no systematic error from guessing for trial functions (the basis set) Pro: Fast (as compared to other non-QMC methods, esp. for system with large DOF) Pro: In principle statistical error could be made vanishingly small with increasing MC steps. Con: error due to the approximation e−Δ휏(− 1

2 ∂2 ∂x2 +V ) ≈ eΔ휏

(

1 2 ∂2 ∂x2

)

e−Δ휏V + 풪(Δ휏 2) while splitting the Green’s functon into kinetic and potential part in G(x, y; Δt) = GKin(x, y; Δt)e−Δ휏(V (y)−ET ). Con: Only for ground state energy

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

Application to 1D QHM

H = p2/(2m) + 1

2kx2; p = −i d dx ; m = k = 1 for simplicity

sake. V (x) = x2/2 Eigen function for ground state 휓0(x) = A exp (

−x2 2

) . GS energy E0 = 1

2.

Eigen function for ground state can be obtained from the DMC code by histrogramming the walkers. For an initial ˜ M = 300 walkers, and total MC step = 4000, the DMC code obtains E0 = 0.498 ± 0.011 as compared to the exact value of 0.5 Accuracy can be improved by running more MC step.

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

Generalisation to higher dimension and non unit masses

For SHO with N DOF x1, x2, ⋅ ⋅ ⋅ , xN, H = ∑i=N

i=1 p2 i /(2mi) + 1 2kx2 i ; pi = −i ∂ ∂xi ;

V (x) = ∑

i x2 i /2

GS energy E0 = 0.5 ∑

i 1 √mi .

The DMC can be modified to take into account of the general case where not all mi has unit mass. A transformation has to be carried out on H to take it into a basis where the kinetic energy terms are diagonal and having unit mass, i.e. e.g ∑i=N

i=1 p2 i /(2mi) → ∑i=N i=1 1 2p′2 i .

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

Summary

Give me a Hamiltonian in the form of ∑i=N

i=1 1 2mi ∂2 ∂x2

i + V (r), I

will tell you what its ground state energy and eigen function are.

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