Department of Physics, Chemistry and Biology

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1 Department of Physics, Chemistry and Biology Master s Thesis Quantum Chaos On A Curved Surface John Wärnå LiTH-IFM-A-EX-8/7-SE Department of Physics, Chemistry and Biology Linköpings universitet, SE Linköping, Sweden

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3 Master s Thesis LiTH-IFM-A-EX-8/7-SE Quantum Chaos On A Curved Surface John Wärnå Adviser: Examiner: Irina Yakimenko Theoretical Physics Karl-Fredrik Berggren Theoretical Physics Irina Yakimenko Theoretical Physics Linköping, November, 8

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5 Avdelning, Institution Division, Department Theoretical Physics Department of Physics, Chemistry and Biology Linköpings universitet, SE Linköping, Sweden Datum Date 8-- Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN LiTH-IFM-A-EX-8/7-SE Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version Titel Title Kvantkaos På Krökt Yta Quantum Chaos On A Curved Surface Författare Author John Wärnå 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. Nyckelord Keywords Put your keywords here

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7 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. v

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9 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 Björn Wahlstrand for his patient help, good advice and for his role as the opponent. vii

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11 Contents Introduction. The aim of the thesis Theory 3. Quantum mechanics Regular systems become chaotic Quantum chaos Method 7 3. Finite Difference Method The kinetic energy part of the Hamiltonian matrix Building the box Test of the method 3 5 Statistical Properties 7 5. Distribution of the probability function Results Pictures The Porter-Thomas distribution for different eigenfunctions Results Pictures Amplitude distribution of the eigenfunction Results Pictures Show the curvature 3 7 Limits and tips for the reader Resolution of the FDM Better resolution of the box The smallest sphere posible ix

12 x Contents 8 Discussion and conclusion The eigenfunctions ψ and the probability density ψ Porter-Thomas Amplitude distribution and Gaussian clock curve Conclusion Bibliography 37

13 Chapter Introduction The Schrödinger equation and the particle in a box-solution [] are often used because of their simplicity and because real systems, for example microwave billiards, can be modelled using this approximation. Two-dimensional quantum billiards and microwave billiards 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 []. 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 radius is small enough. The studies discussed are purely theoretical, however, they could be of interest for manufacturing of chaotic quantum structures.. 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.

14 Introduction

15 Chapter Theory Here follows a short introduction of quantum mechanics and quantum chaos.. Quantum mechanics There is no rigorous presentation of quantum mechanics needed in this thesis. The only thing needed is the time-independent Schrödinger equation: m ψ + V ψ = Eψ (.) In the system particle in a box V is zero inside the box and goes to infinity outside. Therefore Eq. (.) can be written as, m ψ = Eψ (.) where E is the total energy of the particle: E n,m = π (n m e a + m ), (.3) b 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: ψ n,m = a sin( nπx) a b sin( mπy) b and ψ is the probability density of the function []. (.4). Regular systems become chaotic In a rectangular system (discussed in section.) the Schrödinger equation is regular or integrable, which means that it is possible to separate the equation into 3

16 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. and.) 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] Figure.. The system goes from a rectangular grid to a grid shaped like a hockey arena Figure.. The system goes from a flat rectangular grid to a grid on a curved surface.

17 .3 Quantum chaos 5.3 Quantum chaos Quantum chaos can be shown through the statistical properties of the eigenfunction, 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 wavevector magnitude with random amplitude, phase and direction. The amplitude distribution of a chaotic wave follows a Gaussian curve [4], P(ψ) = σ π exp( ψ σ ) (.5) where σ is the standard deviation,σ = ψ ψ P(ψ). The intensity distribution follows the Porter-Thomas distribution function [4]: P( ψ ) = exp( ψ ) π ψ (.6)

18 6 Theory

19 Chapter 3 Method To compute the Schrödinger equation Matlab uses matrix multiplication with a vector, f: Af + V f = E i f (3.) E i is the eigenvalue. For a rectangle E i is E n,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 [, x end ] and [, y end ] and θ φ plane defined by [θ start, θ end ] and [φ start, φ end ] are divided in grid points, therefore the vector f i,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 (A l,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.) [5]. 3. 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 = f i+,j f i,j a (3.) and in y-direction: δ f(x, y) δx δf(x, y) δy = f i+,j + f i,j f i,j a (3.3) = f i,j+ f i,j b (3.4) 7

20 8 Method and δ f(x, y) δy = f i,j+ + f i,j f i,j b (3.5) f(x, y) = a f i,j+ + a f i,j + b f i+,j + b f i,j a f i,j b f i,j a b (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 f(x, y) is written in spherical coordinates [6]: f(θ, φ, R) = δ ( R R δf ) + δr δr sin(θ) δ ( δθ sin(θ) δf δθ ) + With the constant radius and by performing differentiation if gives: f(θ, φ) = ( R sin sin(θ)cos(θ) δf (θ) δθ + sin (θ) δ f ) δθ + δ f R sin (θ) δφ (3.7) δ f R sin (θ) δφ (3.8) In order to discretize this expression 3.8 we use the Eqs. (3.) (3.3) (3.4) (3.5) for the derivatives δf δ f δθ δθ and δ f δφ, which results in: f(θ i, φ j ) = ( R sin sin(θ i )cos(θ i ) f i+,j f i,j (θ i ) a + sin (θ i ) f i+,j + f i,j f i,j a + f i,j+ + f i,j f i,j b ) (3.9) 3. The kinetic energy part of the Hamiltonian matrix Eq. (3.9) can be written in a better and more useful way, as ( ( sin(θi )cos(θ i ) f(θ i, φ j ) = R sin f i+,j + sin (θ i ) ) (θ i ) a a + ( sin (θ i ) f i,j a sin(θ i)cos(θ i ) a ( sin (θ i ) f i,j a + ) b + f i,j+ b + f i,j ) ) b (3.) The kinetic part of the Hamiltonian matrix is shown in figure 3., where different shades represent different values for the grid point. Every value for the grid points is put into the matrix using Eq. (3.). For example, the first row of the matrix is the one referring to the first element of the vector f, :

21 3.3 Building the box 9 ( ( ) ( f(θ, φ ) = R sin (θ ) f sin(θ)cos(θ ), a + sin (θ ) a f sin (θ ), a + b )+ ) f, b All the other elements of the first row of the matrix are zero, so directly f, only depends on its two neighboring points f,, f,. This is of course different for the points somewhere in the middle of the grid, which depend on four neighboring points Figure 3.. 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 ([θ start, θ end ] and [φ start, φ end ]), the box would be wider near the equator (θ = π ) (see figure 3.). Some grid points have to be taken away. To solve the problem the potential energy part of the Hamiltonian matrix (V i,j ) is included to the Schrödinger equation (Eq. (3.)). 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.), (3.), (3.3) [6]. x = Rsin(θ)cos(φ) (3.)

22 Method Figure 3.. The distance between the meridians is larger near the equator Figure 3.3. The potential matrix for the radius.5 times the sidelength of the box.

23 3.3 Building the box y = Rsin(θ)sin(φ) (3.) z = Rcos(θ) (3.3) For example, if the box lays around the y-axis (around θ = π and φ = π on the sphere), the boundaries would be restricted by Eqs. (3.4), (3.5) (B x and B z are the box s dimensions in x and z direction.) B x Rsin(θ)cos(φ) B x (3.4) Eq. (3.5) gives arccos ( B z R B z Rcos(θ) B z (3.5) ) ( B z ) θ arccos R Eq. (3.6) determines [θ start, θ end ]. From Eqs. (3.6), (3.4) we get arccos ( B x ) ( B x ) φ arccos Rsin(θ start ) Rsin(θ end ) (3.6) (3.7) Eq. (3.7) determines [φ start, φ end ]. If φ was restricted only by Eq. (3.7), the box would still be wider near the equator. Therefore [φ start, φ end ] depends on the latitude (the angle θ) according to, arccos ( B x ) ( B x ) φ arccos Rsin(θ i ) Rsin(θ i ) (3.8)

24 Method

25 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 m on the surface of earth. So a test would be to compare the numerical eigenvalues from the the system with radius, R = 6 B (B = B x = B z ), with the theoretical eigenvalues for a two-dimensional box in the x-y-plane in Eq. (4.). The results are shown in figure 4.. E n,m = π m e ( n (B x ) + m (B z ) ),where m e is the mass of the particle and n and m are quantum numbers []. (4.) The graph (figure 4.) show a good agreement between the theoretical and nummerical eigenvalues up to the th 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. (.4). (see figure 4.) 3

26 4 Test of the method Figure 4.. 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.

27 Figure 4.. The first, second, fourth and the th eigenfunction for the radius R = 6 B.

28 6 Test of the method

29 Chapter 5 Statistical Properties 5. Distribution of the probability function Chaotic behaviour can be shown by the plots of the probability density, ψ. 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 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.3) [3]. P(ρ) = Where ρ = A ψ and A is the total area of the system. πρ exp( ρ ) (5.) 5.. 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 =. 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 B and.5 B. 7

30 8 Statistical Properties 5.. Pictures The pictures (figure 5., 5., 5.3, 5.4, 5.5) show the probability density, ψ, for the th eigenfunction and the Porter-Thomas distribution for the th eigenfunction. There is a statistical fluctuation in the Porter-Thomas distribution and to get rid this the average of the Porter-Thomas distribution for the 9th to the th eigenfunction is made. Pictures from different systems: spheres with radii R = B, R = 5 B, R = 3 B, R = B and R =.5 B Figure 5.. The Porter-Thomas distribution and probability density for the th eigenfunction and an average of the Porter-Thomas distribution for the 9th to the th eigenfunction for the sphere with radius R = B.

31 5. Distribution of the probability function Figure 5.. The Porter-Thomas distribution and probability density for the th eigenfunction and an average of the Porter-Thomas distribution for the 9th to the th eigenfunction for the sphere with radius R = 5 B.

32 Statistical Properties Figure 5.3. The Porter-Thomas distribution and probability density for the th eigenfunction and an average of the Porter-Thomas distribution for the 9th to the th eigenfunction for the sphere with radius R = 3 B.

33 5. Distribution of the probability function Figure 5.4. The Porter-Thomas distribution and probability density for the th eigenfunction and an average of the Porter-Thomas distribution for the 9th to the th eigenfunction for the sphere with radius R = B.

34 Statistical Properties Figure 5.5. The Porter-Thomas distribution and probability density for the th eigenfunction and an average of the Porter-Thomas distribution for the 9th to the th eigenfunction for the sphere with radius R =.5 B.

35 5. The Porter-Thomas distribution for different eigenfunctions 3 5. The Porter-Thomas distribution for different eigenfunctions Chaotic behaviour is shown better for eigenfunctions with higher energy [3]. 5.. Results We can see a difference in the Porter-Thomas distribution for different eigenfunctions. 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.. Pictures The pictures (figure 5.6, 5.7, 5.8) show the Porter-Thomas distribution for the same system, but for different eigenfunctions. The 5th, th, 5th and the th eigenfunction for the systems with the radii R = B, R = 5 B and R = B Figure 5.6. The Porter-Thomas distribution of the 5th, th, 5th and th eigenfunction for the radius R = B.

36 4 Statistical Properties Figure 5.7. The Porter-Thomas distribution of the 5th, th, 5th and th eigenfunction for the radius R = 5 B.

37 5. The Porter-Thomas distribution for different eigenfunctions Figure 5.8. The Porter-Thomas distribution of the 5th, th, 5th and th eigenfunction for the radius R = B.

38 6 Statistical Properties 5.3 Amplitude distribution of the eigenfunction Chaotic behaviour can be shown by the distribution of amplitudes of ψ (see section.3). The amplitudes of the eigenfunction is spread equally on both sides of zero. Amplitudes 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 ψ exp( π σ ) [3]. In this thesis the the Gaussian curve has been approximated the fit the amplitude distribution Results The pictures show that the amplitude distribution follows the Gaussian curve better and better with decreased radius and for increased energy Pictures The numerical results show (figure 5.9, 5., 5.) the amplitude distribution and a Gaussian function to match the distribution for the radii R = B, R = 4 B and R = B and the eigenfunction, for which the amplitude distribution is made. Chaotic behavior is shown better for eigenfunctions with higher energy. Therefore, a comparison of the amplitude distribution for different eigenfunctions for the same system is shown in figure 5..The picture shows the amplitude distributions for the 5th, th, 5th and the th eigenfunction in the system with the radius R = B.

39 5.3 Amplitude distribution of the eigenfunction Figure 5.9. The distribution of the amplitudes and th eigenfunction for the radius R = B. Steps=5R=4 Nr Grid 6* Figure 5.. The distribution of the amplitudes and the th eigenfunction for the radius R = 4 B.

40 8 Statistical Properties Figure 5.. The distribution of the amplitudes and the th eigenfunction for the radius R = B.

41 5.3 Amplitude distribution of the eigenfunction Figure 5.. The amplitude distribution of the 5th, th, 5th and th eigenfunction for the radius R = B.

42 3 Statistical Properties

43 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 =.5 B in figure 6.. It shows that the box really is square Figure 6.. The first eigenfunction in [x,y,z]-space 3

44 3 Show the curvature

45 Chapter 7 Limits and tips for the reader 7. Resolution of the FDM The accuracy of the simulation depends on the resolution of the FDM in Eq. (3.) (the size of a and b compared to B). With a higher number of grid points (higher values of rows and columns in the Schrödinger 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. 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 B x B z, 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 ( B z ) ( B x ) the theoretical limit would be if the box covered the whole half sphere, R = B R = B.4 B. But because of the singularity for the angles θ = and θ = π in Eq. (3.) the program only works down to R =.5 B z. But with a higher number of grid points it is possible to come closer to the edges, θ = and θ = π. 33

46 34 Limits and tips for the reader

47 Chapter 8 Discussion and conclusion First the results from the different chapters are shown, then there is a general conclusion. 8. The eigenfunctions ψ and the probability density ψ It looks like there are less and higher amplitude peaks for the th eigenfuntion for smaller radii. The probability density functions show less order for smaller radii. 8. Porter-Thomas Because of the poor resolution of the grid (discussed in section 7.), 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, ρ. 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.). The total area of the box is not exactly as big as it should be (discussed in section 7.) 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 35

48 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.

49 Bibliography [] P. A. Tipler, R. A. Llewellyn, Modern Physics Fourth edition (W. H. Freeman and Company, New York, U.S.A., 3) [] K-F. Berggren, Kvantkaos, KOSMOS : 43-64, Svenska Fysikersamfundet () [3] H. J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, U.K.,999), 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. 46 (7) [5] Arieh Iserles, A First Course in the Numerical Analysis of the Differential Equations (University Press, Cambridge, U.K., 3) [6] C. Nordling, J. Österman, Physics Handbook for Science and Engineering Sixth edition (Studentlitteratur, Lund, Sweden, 999) 37

50 38 Bibliography