The Quantum Many-Body Problem

Solutions on a grid?

1 particle in 1D, n-gridpoints matrix: n × n

1 particle in 2D, n-grispoints in each dimension matrix:n2 × n2

2 particles in 2D matrix: n4 × n4

N particles in 3D matrix: n3N × n3N + spin

Grows quickly beyond what can be handled.

Strategies:

mean field methods (each particle in the average field fromthe other particles). Efficient, but limited accuracy.

expand in basis functions (smaller number than needed gridpoints)

solve for clusters of two (and three, and four and...) particles

perturbation theory

Quantum Monte Carlo

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

The Quantum Many-Body Problem

Solutions on a grid?

1 particle in 1D, n-gridpoints matrix: n × n

1 particle in 2D, n-grispoints in each dimension matrix:n2 × n2

2 particles in 2D matrix: n4 × n4

N particles in 3D matrix: n3N × n3N + spin

Grows quickly beyond what can be handled.Strategies:

mean field methods (each particle in the average field fromthe other particles). Efficient, but limited accuracy.

expand in basis functions (smaller number than needed gridpoints)

solve for clusters of two (and three, and four and...) particles

perturbation theory

Quantum Monte CarloComputational Physics Assignment 3, Quantum Monte Carlo Calculations

Quantum Monte Carlo

A stochastic method to solve the Schrodinger equation

several types - we will study Variational Quantum Monte Carlo

Gives the ground state energy

Trial wave function?

Vary the wave function to find the lowest energy (groundstate)

Optimize some parameter through random walks

But you will be able to solve a 2-body 2D problem on yourlaptop!

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

One particle in a 1D harmonic oscillator

Local energy

EL (x) =HΨ

Ψ

Use trial wave function. Example: Harmonic oscillatorΨα(x) = e−αx

2(Comment: the trial wave function does not need

to be normalized - any normalization cancels in the ratio above)

EL,α (x) =

(−1

2∂2

∂x2 + x2

2

)e−αx

2

e−αx2 =

(α + x2

(1

2− 2α2

))Expectation value (assuming a normalized Ψα):∫

EL,α (x) | Ψα(x) |2 dx =

∫ (α + x2

(1

2− 2α2

))| Ψα(x) |2 dx

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

One particle in a 1D harmonic oscillator

∫EL,α (x) | Ψα(x) |2 dx =

∫ (α + x2

(1

2− 2α2

))| Ψα(x) |2 dx

Figure : Optimal α gives ground state energyComputational Physics Assignment 3, Quantum Monte Carlo Calculations

One particle in a 1D harmonic oscillator

Monte Carlo approach for efficient minimization:

skip the integration

sample the probability distribution through random walks

0. Start with choosing arbitrary positions for the particles. Thengo through the following steps many, many times.

1. Choose a new position for particle i

2. Calculatep =| Ψ(x1, x2, . . . , x

newi , . . .) |2 / | Ψ(x1, x2, . . . , x

oldi , . . .) |2.

3. If p ≥ 1 keep the new position. If p < 1, keep the newposition with probability p.

4. Choose a new position for particle j (and so on).

5. Calculate p = . . .

2 particles in 2D → 4× 1 particle in 1D

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

One particle in a 1D harmonic oscillator

0. Start with choosing arbitrary positions for the particles. Thengo through the following steps many, many times.

1. Choose a new position for particle i

2. Calculatep =| Ψ(x1, x2, . . . , x

newi , . . .) |2 / | Ψ(x1, x2, . . . , x

oldi , . . .) |2.

3. If p ≥ 1 keep the new position. If p < 1, keep the newposition with probability p.

4. Choose a new position for particle j (and so on).

5. Calculate p = . . .

. . . Calculate EL, Calculate 〈EL〉 and 〈E 2L 〉 − 〈EL〉2

. . . GO TO 1.

The Metropolis algorithm

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

One particle in a 1D harmonic oscillator

∫EL,α (x) | Ψα(x) |2 dx =

∫ (α + x2

(1

2− 2α2

))| Ψα(x) |2 dx

Figure : Comparison between analytic results for different α and theresults with the Monte Carlo algorithm (here the Golden Searchalgorithm is used to quickly find the α which gives the lowest energy).

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

One particle in a 1D harmonic oscillator

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

QMC: plus and minus

+ We can handle big systems

- We need to guess the trial wave function

- We get only the ground state

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

two electrons in a 2D harmonic oscillator

Now to something unknown

Two interacting particles

Let them sit in a two-dimensional Harmonic oscillator(successful model of a 2D quantum dot)

H =

(2∑

i=1

− ~2

2m

∂2

∂x2i

+mx2

i ω2

2− ~2

2m

∂2

∂y2i

+my2

i ω2

2

)

+e2

4πε0εr

1√(x1 − x2)2 + (y1 − y2)2

.

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

two electrons in a 2D harmonic oscillator

with x = x√

mω/~, y = y√

mω/~, and λ = me2

4πε0εr~2

√~

mω(2∑

i=1

−1

2

∂2

∂x2i

+x2i

2− 1

2

∂2

∂y2i

+y2i

2

)Ψ (x1, y1, x2, y2) +

λ√(x1 − x2)2 + (y1 − y2)2

Ψ (x1, y1, x2, y2) =E

~ωΨ (x1, y1, x2, y2)

Trial wave function

Ψ(x1, y1, x2, y2) = e−(x21 +y2

1 +x22 +y2

2 )/2 e

λ

√(x1−x2)2+(y1−y2)2

1+α

√(x1−x2)2+(y1−y2)2

.

Why?

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

two electrons in a 2D harmonic oscillator

Figure : Results for α when λ = 1. Solid line Energe. Dashed lineVariance. The exact result is E = 3~ω.

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

two electrons in a 2D harmonic oscillator

Figure : The results for some different α when λ = 2. Here theliterature value is E = 3.729 . . . ~ω.

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

two electrons in a 2D harmonic oscillator

Figure : The results for some different α when λ = 8. Here theliterature value is E = 6.618 . . . ~ω.

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

two electrons in a 2D harmonic oscillator

Figure : One particle in r = 0.7. The distribution of particle two ifλ = 0. λ = 1: The distribution of particle two for the optimized α, andfor a too small(solid), respectively too large (dotted) α. The results areshown for an angle close that of the first particle. See comment below onnormalization

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

two electrons in a 2D harmonic oscillator

Figure : One particle in r = 0.7. The distribution of particle two ifλ = 0. λ = 5: The distribution of particle two for the optimized α, andfor a too small(solid), respectively too large (dotted) α. The results areshown for an angle close that of the first particle. See comment below onnormalization

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

QMC

Solve for the Harmonic Oscillator to understand the method.Chapter 12 of Computational Physics, by Jos Thijssens, givesexplicit results for different α. Might be helpful.

Generalize your code to the two-particle, 2 D case.

A working routine needs variations of α to find the energyminimum. See Chapter 10.1 in Numerical Recipes.

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

Comment

Ground state of o two-dimensional harmonic oscillator (oneparticle)Cartesian coordinates:∫ ∞

−∞

∫ ∞−∞

ρ (x , y) dxdy =1

π

∫ ∞−∞

∫ ∞−∞

e−(x2+y2)dxdy = 1

Circular coordinates:∫ ∞0

∫ 2π

0ρ (r , θ) rdrdθ =

1

π

∫ ∞0

∫ 2π

0e−r

2rdrdθ =

∫ ∞0

2e−r2rdr

ρ (r) = 2e−r2r

gives the angular integrated probability to find the particle atradius r . Maximum probability at r =

√2!

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

Probability distribution

Figure : One particle in r = 0.7. The distribution of particle two ifλ = 0, if λ = 1, and if λ = 5. Angular integrated results.

Computational Physics Assignment 3, Quantum Monte Carlo Calculations

The wavefunction

Figure : One particle in r =√

2. The absolute square of the wavefunction for particle two if λ = 0, if λ = 1, and if λ = 5. Angularintegrated results.

Computational Physics Assignment 3, Quantum Monte Carlo Calculations