Quantum Chaos On A Curved Surface

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Department of Physics, Chemistry and Biology

Master’s Thesis

Quantum Chaos On A Curved Surface

John W¨

arn˚

a

LiTH-IFM-A-EX-08/2027-SE

Department of Physics, Chemistry and Biology

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Master’s Thesis LiTH-IFM-A-EX-08/2027-SE

Quantum Chaos On A Curved Surface

John W¨

arn˚

a

Adviser: Irina Yakimenko

Theoretical Physics

Karl-Fredrik Berggren

Theoretical Physics

Examiner: Irina Yakimenko

Theoretical Physics

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Avdelning, Institution Division, Department Theoretical Physics

Department of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden

Datum Date 2008-11-20 Spr˚ak Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats ¨Ovrig rapport ISBN ISRN

Serietitel och serienummer Title of series, numbering

ISSN

URL f¨or elektronisk version

Titel Title

Kvantkaos P˚a Kr¨okt Yta

Quantum Chaos On A Curved Surface

F¨orfattare Author

John W¨arn˚a

Sammanfattning Abstract

The system studied in the thesis is a particle in a two-dimensional box on the surface of a sphere with constant radius. The different systems have different radii while the box dimension is kept the same, so the curvature of the surface of the box is different for the different systems. In a system with a sphere of a large radius the surface of the box is almost flat. What happens if the radius is decreased and the symmetry is broken? Will the system become chaotic if the radius is small enough? There are some properties of the eigenfunctions, that show different things depending on whether the system is chaotic or regular. The amplitude distribution of the probability density, the amplitude distribution of the eigenfunction and the probability density look different for chaotic and regular systems. The main subject of this thesis is to study these distributions.

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Abstract

The system studied in the thesis is a particle in a two-dimensional box on the surface of a sphere with constant radius. The different systems have different radii while the box dimension is kept the same, so the curvature of the surface of the box is different for the different systems. In a system with a sphere of a large radius the surface of the box is almost flat. What happens if the radius is decreased and the symmetry is broken? Will the system become chaotic if the radius is small enough? There are some properties of the eigenfunctions, that show different things depending on whether the system is chaotic or regular. The amplitude distribution of the probability density, the amplitude distribution of the eigenfunction and the probability density look different for chaotic and regular systems. The main subject of this thesis is to study these distributions.

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Acknowledgements

I would like to thank my supervisors Irina Yakimenko and Karl-Fredrik Berggren for the subject of the diploma work and for all help and support during the work. I also would like to thank Bj¨orn Wahlstrand for his patient help, good advice and for his role as the opponent.

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Introduction

The Schr¨odinger equation and the particle in a box-solution [1] are often used because of their simplicity and because real systems, for example microwave liards, can be modelled using this approximation. Two-dimensional quantum bil-liards and microwave bilbil-liards are well known systems in the quantum physics. The experiments are of importance and of both theoretical and parctical use in the quantum chaos field [2]. In this thesis we go from the two-dimensional billiards into a new field of quasi-three-dimensional structures. The system studied in here is a particle in a two-dimensional box on the surface of a sphere. The sphere can have different radii. The hypothesis is that the system becomes chaotic, if the ra-dius is small enough. The studies discussed are purely theoretical, however, they could be of interest for manufacturing of chaotic quantum structures.

1.1 The aim of the thesis

A large part of this thesis focuses on the method and how to model the system, which is presented in chapter 3 and 4. In chapter 5, 6 and 7 the results of the calculations are presented. Chapter 8 contains practical advice for further work. And finally in chapter 9 there are discussions and conclusions.

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Chapter 2

Theory

Here follows a short introduction of quantum mechanics and quantum chaos.

2.1 Quantum mechanics

There is no rigorous presentation of quantum mechanics needed in this thesis. The only thing needed is the time-independent Schr¨odinger equation:

−~

2

2m∇

2_ψ_{+ V ψ = Eψ} _(2.1)

In the system ”particle in a box” V is zero inside the box and goes to infinity outside. Therefore Eq. (2.1) can be written as,

−~

2

2m∇

2_ψ_{= Eψ} _(2.2)

where E is the total energy of the particle:

En,m= ~2_π2 2me n2 a2 + m2 b2 , (2.3)

where a and b are the sidelengths of the two-dimensional box and n and m are the two quantum numbers of the system in the x-y-direction respectively. The eigenfunctions corresponding to eigenvalues are of the form:

ψn,m=r 1 asin nπx a r 1 bsin mπy b (2.4)

and |ψ|2_{is the probability density of the function [1].}

2.2 Regular systems become chaotic

In a rectangular system (discussed in section 2.1) the Schr¨odinger equation is

regular or integrable, which means that it is possible to separate the equation into 3

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4 Theory

independent equations one for each degree of freedom. It is easy to calculate the corresponding classical path of the particle in the box. If the system is shaped like a hockey arena or the symmetry of the system is broken in an another way, it is no longer possible to separate the equation into independent equations. The system is irregular or non-integrable. (see figure 2.1 and 2.2) If the movement of the particle is calculated, the time evolution of the particle’s path will change rapidly with small changes in the initial conditions. The system is now chaotic [3].

0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 30 35 40 45 50

Figure 2.1. The system goes from a rectangular grid to a grid shaped like a hockey arena. 0 20 40 60 80 100 0 20 40 60 80 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 0 20 40 60 80 100 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

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2.3 Quantum chaos 5

2.3 Quantum chaos

Quantum chaos can be shown through the statistical properties of the eigenfunc-tion, amplitude and intensity distributions [4]. A non chaotic wave function shows its nodal pattern as a regular net of intersecting circles and straight lines. In a chaotic wave function the nodal lines never intersect [3] and the wave function is effectively indistinguishable from a superposition of plane waves of fixed wave-vector magnitude with random amplitude, phase and direction. The amplitude distribution of a chaotic wave follows a Gaussian curve [4],

P(ψ) = 1

σ√2πexp(

−ψ2

2σ2) (2.5)

where σ is the standard deviation,σ =qP

ψψ2P(ψ). The intensity distribution

follows the Porter-Thomas distribution function [4]:

P_(|ψ|2) = 1 p2π|ψ|2exp −|ψ|2 2 (2.6)

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Chapter 3

Method

To compute the Schr¨odinger equation Matlab uses matrix multiplication with a vector, f:

Af+ V f = Eif (3.1)

Ei is the eigenvalue. For a rectangle Ei is En,m, where n and m are quantum

numbers. A and V are the kinetic energy part and the potential energy part

of the Hamiltonian matrix. The x-y-plane defined by [0, xend] and [0, yend] and

θ− φ − plane defined by [θstart, θend] and [φstart, φend] are divided in grid points,

therefore the vector fi,j have indices i and j. The matricies A and V have indices

l and k, where l goes from zero to i × j and k goes from i × j. The maximum of i represents the points on the x or θ axis and j represents points on y or φ axis. The maximum of i determines the number of rows and j determines the number of columns in the grid.

The kinetic energy part of the Hamiltonian matrix (Al,k) is built using the Finite

Difference Method (FDM). Then Matlab solves the matrix eigenvalue problem, which is a system of equations with one equation for each grid point Eq. (3.1) [5].

3.1 Finite Difference Method

In this method the derivaties each represented at each point as the difference of the value of the function in the two (or four in the two-dimensional case) neighbouring grid points divided by the distance between them [5].

For the x-y-grid we have, in x-direction: δf(x, y) δx = fi+1,j− fi−1,j 2a (3.2) δ2_f_{(x, y)} δx2 =

fi+1,j+ f_i−1,j− 2fi,j

a2 (3.3) and in y-direction: δf(x, y) δy = fi,j+1− fi,j−1 2b (3.4) 7

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8 Method

δ2_f_{(x, y)}

δy2 =

fi,j+1+ f_i,j−1− 2fi,j

b2 (3.5)

and

∇2f(x, y) = a

2_f

i,j+1+ a2fi,j−1+ b2fi+1,j+ b2fi−1,j− 2a2fi,j− 2b2fi,j

a2_b2 (3.6)

where a and b are the distances between two grid points in x and y directions respectively or in θ and φ directions.

In the case of a surface on a sphere ∇2_f_{(x, y) is written in spherical coordinates}

[6]: ∇2f(θ, φ, R) = 1 R2 δ δR R2δf δR + 1 sin(θ) δ δθ sin(θ)δf δθ + 1 R2_sin2_(θ) δ2f δφ2 (3.7)

With the constant radius and by performing differentiation if gives:

∇2f(θ, φ) = 1 R2_sin2_(θ) sin(θ)cos(θ)δf δθ + sin 2_(θ)δ2f δθ2 + 1 R2_sin2_(θ) δ2_f δφ2 (3.8)

In order to discretize this expression 3.8 we use the Eqs. (3.2) (3.3) (3.4) (3.5) for

the derivatives δf_δθ δ_δθ2f2 and

δ2_f

δφ2, which results in:

∇2f(θi, φj) = 1 R2_sin2_(θ i) sin(θi)cos(θi) fi+1,j− f_i−1,j 2a + sin2_(θ i)

fi+1,j+ fi−1,j− 2fi,j

a2 +

fi,j+1+ fi,j−1− 2fi,j

b2

(3.9)

3.2 The kinetic energy part of the Hamiltonian

matrix

Eq. (3.9) can be written in a better and more useful way, as

∇2f(θi, φj) = 1 R2_sin2_(θ i) fi+1,j sin(θ_i)cos(θ_i) 2a + sin2(θi) a2 + f_i−1,jsin 2_(θ i) a2 − sin(θi)cos(θi) 2a − 2fi,j sin2(θ_i) a2 + 1 b2 + fi,j+1 1 b2 + fi,j−1 1 b2 (3.10) The kinetic part of the Hamiltonian matrix is shown in figure 3.1, where different shades represent different values for the grid point. Every value for the grid points is put into the matrix using Eq. (3.10). For example, the first row of the matrix

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3.3 Building the box 9 ∇2_f_(θ 1, φ1) = _R2_sin12(θ₁) f2,1 _sin(θ 1)cos(θ1) 2a + sin2_(θ 1) a2 − 2f1,1 _sin2_(θ 1) a2 + 1 b2 + f1,2_b12

All the other elements of the first row of the matrix are zero, so directly f1,1 only

depends on its two neighboring points f2,1, f1,2. This is of course different for

the points somewhere in the middle of the grid, which depend on four neighboring points. 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25

Figure 3.1. The Hamiltonian matrix for the θ − φ-plane in a 5 × 5-grid: Non-periodic (to the left) and θ-periodic (to the right)

3.3 Building the box

If the calculations were made on all the grid points in a grid restricted only by

the angles θ and φ ([θstart, θend] and [φstart, φend]), the box would be wider near

the ”equator” (θ = π

2) (see figure 3.2). Some grid points have to be taken away.

To solve the problem the potential energy part of the Hamiltonian matrix (Vi,j)

is included to the Schr¨odinger equation (Eq. (3.1)). In the grid points, which lay outside the box, the function is multiplied with a large constant, the value of which is at least larger than the largest eigenvalue. (For the ideal theoretical case this value goes to infinity.) The white part corresponds the grid points with no potential added and the black parts correspond the forbidden parts with a large potential in figure 3.3. [x,y,z] written in spherical coordinates are used to restrict the boundaries of the box as in Eqs. (3.11), (3.12), (3.13) [6].

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10 Method −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 3.2. The distance between the meridians is larger near the equator.

10 20 30 40 50 60 10 20 30 40 50 60

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3.3 Building the box 11

y= Rsin(θ)sin(φ) (3.12)

z= Rcos(θ) (3.13)

For example, if the box lays around the y-axis (around θ = π

2 and φ = π2 on the

sphere), the boundaries would be restricted by Eqs. (3.14), (3.15) (2Bxand 2Bz

are the box’s dimensions in x and z direction.)

−Bx≤ Rsin(θ)cos(φ) ≤ Bx (3.14) −Bz≤ Rcos(θ) ≤ Bz (3.15) Eq. (3.15) gives arccos Bz R ≤ θ ≤ arccos −Bz R (3.16)

Eq. (3.16) determines [θstart, θend]. From Eqs. (3.16), (3.14) we get

arccos Bx

Rsin(θstart) ≤ φ ≤ arccos

−Bx

Rsin(θend)

(3.17)

Eq. (3.17) determines [φstart, φend]. If φ was restricted only by Eq. (3.17), the

box would still be wider near the ”equator”. Therefore [φstart, φend] depends on

the latitude (the angle θ) according to,

arccos Bx Rsin(θi) ≤ φ ≤ arccos −Bx Rsin(θi) (3.18)

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Chapter 4

Test of the method

A square on the surface of a sphere with large radius compared to the sidelength

of the square is almost flat, like a square with a area of a couple m2_{on the surface}

of earth. So a test would be to compare the numerical eigenvalues from the the

system with radius, R = 106_{× B (B = B}

x= Bz), with the theoretical eigenvalues

for a two-dimensional box in the x-y-plane in Eq. (4.1). The results are shown in figure 4.1. En,m= π2_~2 2me n2 (2Bx)2 + m 2 (2Bz)2 (4.1)

,where me is the mass of the particle and n and m are quantum numbers [1].

The graph (figure 4.1) show a good agreement between the theoretical and nummerical eigenvalues up to the 200th eigenvalue.

The eigenfunctions should look the same as the ones for the particle in a flat two-dimensional box, ordinary sinus or cosinus waves in two dimensions as in Eq. (2.4). (see figure 4.2)

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14 Test of the method 0 200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 2500 3000 3500 4000 4500 50 100 150 200 250 300 350 400 0 500 1000 1500

Figure 4.1.The dashed curve refers to the exact eigenvalues in a two-dimensional box, the solid line curve refers to the numerical eigenvalues. The picture to the left shows all the eigenvalues and the picture to the right shows an enlargement of the eigenvalues of interest.

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15 0 10 20 30 40 50 60 0 10 20 30 40 50 60 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 0 10 20 30 40 50 60 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 0 10 20 30 40 50 60 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04

Figure 4.2. The first, second, fourth and the 200th eigenfunction for the radius R = 106

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Chapter 5

Statistical Properties

5.1 Distribution of the probability function

Chaotic behaviour can be shown by the plots of the probability density, |ψ|2_.

For a non-chaotic system the probability density function is a periodic function, where the amplitudes go up and down like a simple sinus curve. The distribution of the amplitudes has largest rate around zero, then the rate goes down until the

rate for the maximum of the probability density, |ψ|2 _{and then down to zero. In}

the chaotic case the probability density function has some small island, where the amplitude goes up, and around this islands the function is almost zero and it is a small possibility for really large amplitude. This leads to the Porter-Thomas distribution (see section 2.3) [3].

P(ρ) = r 1 2πρexp(− ρ 2) (5.1)

Where ρ = A|ψ|2 _{and A is the total area of the system.}

5.1.1 Results

The plots of the probability density function and the Porter-Thomas distribution look almost the same for all systems with larger radii than 5 × B. The plots have been cut to be able to compare the Porter-Thomas distribution for the different systems. The results of this action is that the plots only show the distribution up to ρ = 2. This makes all the pictures look alike. The grid points, which lay outside the box, are included as zero elements in the distribution, therefore the pile around zero is too high and the other are too small. For systems with smaller radii than 5×B the probability density functions follow the Porter-Thomas distribution better and better, except the last two, for the systems with radii 2×B and 1.5×B.

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18 Statistical Properties

5.1.2 Pictures

The pictures (figure 5.1, 5.2, 5.3, 5.4, 5.5) show the probability density, |ψ|2_{, for}

the 200th eigenfunction and the Porter-Thomas distribution for the 200th eigen-function. There is a statistical fluctuation in the Porter-Thomas distribution and to get rid this the average of the Porter-Thomas distribution for the 190th to the 210th eigenfunction is made. Pictures from different systems: spheres with radii

R= 1000 × B, R = 50 × B, R = 3 × B, R = 2 × B and R = 1.5 × B 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 10 20 30 40 50 60 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 5.1. The Porter-Thomas distribution and probability density for the 200th eigenfunction and an average of the Porter-Thomas distribution for the 190th to the 210th eigenfunction for the sphere with radius R = 1000 × B.

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5.1 Distribution of the probability function 19 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 10 20 30 40 50 60 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4

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20 Statistical Properties 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 10 20 30 40 50 60 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4

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5.1 Distribution of the probability function 21 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 10 20 30 40 50 60 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4

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22 Statistical Properties 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 10 20 30 40 50 60 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 5.5. The Porter-Thomas distribution and probability density for the 200th eigenfunction and an average of the Porter-Thomas distribution for the 190th to the 210th eigenfunction for the sphere with radius R = 1.5 × B.

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5.2 The Porter-Thomas distribution for different eigenfunctions 23

5.2 The Porter-Thomas distribution for different

eigenfunctions

Chaotic behaviour is shown better for eigenfunctions with higher energy [3].

5.2.1 Results

We can see a difference in the Porter-Thomas distribution for different eigenfunc-tions. The higher number eigenfunctions follow the Porter-Thomas distribution better than the lower ones. The lower energy eigenfunctions show a more smeard out distribution of the probability density.

5.2.2 Pictures

The pictures (figure 5.6, 5.7, 5.8) show the Porter-Thomas distribution for the same system, but for different eigenfunctions. The 50th, 100th, 150th and the 200th eigenfunction for the systems with the radii R = 1000 × B, R = 50 × B and

R_{= 2 × B.} 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 5.6. The Porter-Thomas distribution of the 50th, 100th, 150th and 200th eigen-function for the radius R = 1000 × B.

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24 Statistical Properties 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4

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5.2 The Porter-Thomas distribution for different eigenfunctions 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4

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26 Statistical Properties

5.3 Amplitude distribution of the eigenfunction

Chaotic behaviour can be shown by the distribution of amplitudes of ψ (see section 2.3).

The amplitudes of the eigenfunction is spread equally on both sides of zero. Am-plitudes with the value zero have the highest rate. For a non-chaotic system the rate goes down as the absolute value of the amplitude goes up until it reaches a maximum value, then the rate goes down to zero. For a chaotic system there is a small possibility for extremely high values of amplitudes. The distribution of

the amplitudes follow the Gaussian clock function, P (ψ) =_σ√1

2πexp( −ψ2

2σ2) [3]. In

this thesis the the Gaussian curve has been approximated the fit the amplitude distribution.

5.3.1 Results

The pictures show that the amplitude distribution follows the Gaussian curve better and better with decreased radius and for increased energy.

5.3.2 Pictures

The numerical results show (figure 5.9, 5.10, 5.11) the amplitude distribution and a Gaussian function to match the distribution for the radii R = 1000×B, R = 40×B and R = 2×B and the eigenfunction, for which the amplitude distribution is made. Chaotic behavior is shown better for eigenfunctions with higher energy. There-fore, a comparison of the amplitude distribution for different eigenfunctions for the same system is shown in figure 5.12.The picture shows the amplitude distributions for the 50th, 100th, 150th and the 200th eigenfunction in the system with the radius R = 2 × B.

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5.3 Amplitude distribution of the eigenfunction 27 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 60 0 10 20 30 40 50 60 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04

Figure 5.9. The distribution of the amplitudes and 200th eigenfunction for the radius R= 1000 × B. −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 Steps=25R=40 Nr 200 Grid 60*60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04

Figure 5.10. The distribution of the amplitudes and the 200th eigenfunction for the radius R = 40 × B.

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28 Statistical Properties −8 −6 −4 −2 0 2 4 6 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 10 20 30 40 50 60 0 10 20 30 40 50 60 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08

Figure 5.11. The distribution of the amplitudes and the 200th eigenfunction for the radius R = 2 × B.

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5.3 Amplitude distribution of the eigenfunction 29 −6 −4 −2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 −6 −4 −2 0 2 4 6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −6 −4 −2 0 2 4 6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −8 −6 −4 −2 0 2 4 6 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 5.12. The amplitude distribution of the 50th, 100th, 150th and 200th eigenfunc-tion for the radius R = 2 × B.

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Chapter 6

Show the curvature

Finally a picture below shows that the system really is a curved surface. It is shown best for the eigenfunction for a sphere with small radius. Here is the first eigenfunction for a system of the box on a sphere with radius R = 1.5 ×B in figure 6.1. It shows that the box really is square.

−2 −1 0 1 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 1.2 1.4 1.6

Figure 6.1. The first eigenfunction in [x,y,z]-space

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Chapter 7

Limits and tips for the

reader

7.1 Resolution of the FDM

The accuracy of the simulation depends on the resolution of the FDM in Eq. (3.10) (the size of a and b compared to B). With a higher number of grid points (higher values of rows and columns in the Schr¨odinger matrix) the space between them (a and b) decreases (compared within the same system). A higher number of grid points increases the scale of calculations and the time for each simulation increases.

7.2 Better resolution of the box

More grid points make a better definition of the square on the surface, because the grid points get closer to the edge of the square. The total area of the box is calculated as the sum of the area element of the grid points restriced by potetial energy part of the Hamiltonian matrix (see section 3.3). The total area of the box is used in some of the calculations, for example, the Porter-Thomas distribution.

The area is supposed to be 2Bx× 2Bz, but the numerical one is less than that.

7.3 The smallest sphere posible

Another limit is the smallest radius possible. If the dimensions of the box are

(2 × Bz) × (2 × Bx) the theoretical limit would be if the box covered the whole

half sphere, 2R = 2√2B ↔ R =√2B ≈ 1.41 × B. But because of the singularity

for the angles θ = 0 and θ = 2 × π in Eq. (3.10) the program only works down

to R = 1.5 × Bz. But with a higher number of grid points it is possible to come

closer to the edges, θ = 0 and θ = 2 × π.

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Chapter 8

Discussion and conclusion

First the results from the different chapters are shown, then there is a general conclusion.

8.1 The eigenfunctions ψ and the probability

den-sity

_|ψ|

2

It looks like there are less and higher amplitude peaks for the 200th eigenfuntion for smaller radii. The probability density functions show less order for smaller radii.

8.2 Porter-Thomas

Because of the poor resolution of the grid (discussed in section 7.1), the θ−φ−grid looks almost square for systems with larger radii than 5 × B, as it is not for the systems with smaller radii (see figure 3.3). Therefore the pictures showing the Porter-Thomas distribution look almost the same for the systems with larger radius than 5 × B. To get the same scale on the pictures showing the Porter-Thomas distribution the pictures only show the distribution for the probability

density, ρR 2. Therefore more chaotic systems, systems with larger maximum of

ρ do not show their actual distribution. If they did, they would probably show

a better correspondence to the curve all the way up to the maximum of ρ, for which the distribution is zero. The more regular systems only follow the curve up to their maximum of ρ, for which the distribution is larger than zero. Compare with the distribution of the amplitudes (see figures in section 5.3.2). The total area of the box is not exactly as big as it should be (discussed in section 7.2) and it is different for the different systems. Therefore, the calculations of the Porter-Thomas distribution do not only depend on how regular the systems are, but also on the error in the summation of the total area. The statistics is made on all grid points. The grid points, which lay outside the box are multiplied with zero (see

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36 Discussion and conclusion figure 3.3). So the distribution of the value around zero is too large and therefore the distribution of the other values is too small.

8.3 Amplitude distribution and Gaussian clock

curve

The large pile around zero is an effect of how the systems were modelled. In the grid points, where the potential matrix elements are not zero (see figure 3.3), the function is zero. These elements do not belong to the system and therefore the rate distribution of the amplitudes equal zero is larger than it should be. It shows that the distribution follows the Gaussian clock curve better and better with a decreased radius and for functions with larger energy.

8.4 Conclusion

All the pictures indicate that the distributions follow the theoretical curves better for a smaller radius. The distributions do not follow the curves perfectly. But if it was possible to decrease the radius of the sphere even more, it could be an almost perfect match. But it is not possible to decrease the radius of the sphere more, so the hypothesis is not true.

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Bibliography

[1] P. A. Tipler, R. A. Llewellyn, Modern Physics Fourth edition (W. H. Freeman and Company, New York, U.S.A., 2003)

[2] K-F. Berggren, Kvantkaos, KOSMOS 2001: 43-64, Svenska Fysikersamfundet (2001)

[3] H. J. St¨ockmann, Quantum Chaos: An Introduction (Cambridge University

Press, Cambridge, U.K.,1999), and references cited therein.

[4] C. C. Chen, C. C. Liu, K. W. Su, T.H. Lu, Y.F. Chen, and K. F. Huang, Phys Rev, E vol. 75, p. 046202 (2007)

[5] Arieh Iserles, A First Course in the Numerical Analysis of the Differential Equations (University Press, Cambridge, U.K., 2003)

[6] C. Nordling, J. ¨Osterman, Physics Handbook for Science and Engineering

Sixth edition (Studentlitteratur, Lund, Sweden, 1999)

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Quantum Chaos On A Curved Surface

Department of Physics, Chemistry and Biology

Master’s Thesis

Quantum Chaos On A Curved Surface

John W¨

arn˚

a

Quantum Chaos On A Curved Surface

John W¨

arn˚

a

Abstract

Acknowledgements

Contents

Chapter 1

Introduction

1.1

The aim of the thesis

Chapter 2

Theory

2.1

Quantum mechanics

2.2

Regular systems become chaotic

2.3

Quantum chaos

Chapter 3

Method

3.1

Finite Difference Method

3.2

The kinetic energy part of the Hamiltonian

matrix

3.3

Building the box

Chapter 4

Test of the method

Chapter 5

Statistical Properties

5.1

Distribution of the probability function

5.1.1

Results

5.1.2

Pictures

5.2

The Porter-Thomas distribution for different

eigenfunctions

5.2.1

Results

5.2.2

Pictures

5.3

Amplitude distribution of the eigenfunction

5.3.1

Results

5.3.2

Pictures

Chapter 6

Show the curvature

Chapter 7

Limits and tips for the

reader

7.1

Resolution of the FDM

7.2

Better resolution of the box

7.3

The smallest sphere posible

Chapter 8

Discussion and conclusion

8.1

The eigenfunctions ψ and the probability

den-sity

|ψ|

8.2

Porter-Thomas

8.3

Amplitude distribution and Gaussian clock

curve

_|ψ|