APPLICATION OF SELF-ADJOINT EXTENSIONS TO THE RELATIVISTIC AND NON-RELATIVISTIC COULOMB PROBLEM by SCOTT BECK submitted in partial fulfillment for the degree of Doctor of Physics CASE WESTERN RESERVE UNIVERSITY August, 2016 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of Scott Beck candidate for the degree of Physics∗. Committee Chair Harsh Mathur Committee Member Walter Lambrecht Committee Member Mark Meckes Committee Member Andrew Tolley Date of Defense 9 June 2016 ∗ We also certify that written approval has been obtained for any proprietary material contained therein. 1 Contents 1 Introduction 1 2 Self-Adjoint Extensions 7 2.1 Particle in a box . .9 2.2 Delta Function . 10 2.3 Scale Invariance . 13 3 Experimental systems 18 3.1 Liquid Helium . 18 3.2 Rydberg Atoms . 20 3.2.1 First Order Quantum Defect . 21 3.2.2 Higher Order Corrections . 25 3.3 Graphene . 27 4 Non-Relativistic Coulomb Problem 31 4.1 Helium . 38 4.2 Alkali atoms . 41 5 Graphene 46 5.1 Separation of Variables . 47 5.2 Self-Adjoint Extensions . 49 5.3 Solutions and Phase Shift . 52 2 6 Conclusions 55 A Special Functions 58 B Numerical Fitting 62 C Akaike's Information Criterion 67 Bibliography 68 3 List of Figures 3.1 Sample spectra of the transition in the electron bound states made by Grimes and Brown in 1976 [15]. 20 3.2 Observed linearity in the conductivity of the graphene with varying gate voltages. The point at Vg = 0 corresponds to the fermi energy [20]. 28 3.3 Calculated conductivity in graphene based on Random Phase Approxima- tion screening [19]. The conducivity curve exhibits the same linear pattern observed in the experiments of Geim et. al. in figure 3.2 . 29 3.4 Local Density of states (inset) exhibits the same linear behavior as the conductivity curves observed in the experiments, as seen in figure 3.2 [18]. 30 4.1 Graph of λ(κ) described by equation (4.0.14). The x-axis is in units of Hartrees corresponding the κ values. The horizontal line is at λ ≈ :025, the fitted value from the cesium data listed in table (4.2), with the intersections showing the quantized energy levels. 37 4 List of Tables 4.1 Reproduction of the transitions measured in Lithium 7 [25]. All of the states are in the s orbital, with spin angular momentum 1=2. Measure- ments that included orbitals with ` 6= 0 have been excluded. All units are MHz, converted from the original cm−1.................... 42 4.2 Reproduction of the transitions measured in Sodium 23 [25]. All of the states are in the s orbital, with spin angular momentum 1=2. Measure- ments that included orbitals with ` 6= 0 have been excluded. Units are in MHz, converted from the original cm−1 ................... 42 4.3 Reproduction of the transitions measured in Potassium 39 by Lorenzen et. al. [26]. All of the states are in the s orbital, with spin angular momentum 1=2. Measurements have been converted from the original wave numbers reported (cm−1) to frequencies in MHz. The error associated with each measurement is approximately 21 Mhz. 43 4.4 Reproduction of the transitions measured in Cesium 133 by Goy et. al. [24]. All of the states are in the s orbital, with spin angular momentum 1=2. Measurements that included orbitals with ` 6= 0 have been excluded. 43 4.5 Results of the fitting procedure for the first four Alkali atoms. The residual values reported are the average square residual, determined by dividing the sum of the square residuals by the total number of energy levels fitted. All are in units of MHz2 ............................. 44 5 4.6 Comparison of the AIC for the quantum defect model (δ) and the self- adjoint extension model (λ). The difference ∆ = AICλ − AICδ shows that there is almost no evidence to support the self-adjoint as the preferable model. See Appendix C for further discussion on the AIC. 45 6 Acknowledgments I would sincerely like to thank my advisor, Harsh Mathur. His guidance and sugges- tions during the writing and defending of the thesis were invaluable. I would also like to thank David Jacobs, who made significant contributions during the early stages of this project. Finally, I would like to thank my friends and family for their continued support throughout this process. 7 Abstract The Coulomb problem was one of the first successful applications of quantum theory and is a staple topic in textbooks. However there is an ambiguity in the solution to the problem that is seldom discussed in either textbooks or the literature. The ambiguity arises in the boundary conditions that must be applied at the origin where the Coulomb potential is singular. The textbook boundary condition is generally not the only one that is permissible or the one that is most appropriate. Here we revisit the question of boundary conditions using the mathematical method of self-adjoint extensions in context of modern realizations of the Coulomb problem in electrons on helium, Rydberg atoms and graphene. We determine the family of allowed boundary conditions for the non- relativistic Schr¨odingerequation in one and three dimensions and the relativistic Dirac equation in two dimensions. The boundary conditions are found to break the classical SO(4) Runge-Lenz symmetry of the non-relativistic Coulomb problem in three dimensions and to break scale invariance for the two dimensional Dirac problem. The symmetry breaking is analogous to the anomaly phenomenon in quantum field theory. Electrons on helium have been extensively studied for their potential use in quantum computing and as a laboratory for condensed matter physics. The trapped electrons provide a realization of the one dimensional non-relativistic Coulomb problem. Using the method of self-adjoint extensions we are able to reproduce the observed energy levels of electrons on helium which are known to deviate from the textbook Balmer formula. We also study the connection between the method of self-adjoint extensions and an older theoretical model introduced by Cole [1]. Rydberg atoms have potential applications to atomic clocks and precision atomic experiments. They are hydrogen-like in that they have a single highly excited electron that orbits a small positively charged core. We compare the observed spectrum of several species of Rydberg atoms to the predictions of the Coulomb model with self-adjoint extension and to the predictions of the more elaborate quantum defect model which is generally found to be more accurate. The motion of electrons on atomically flat sheets of graphene is governed by the massless Dirac equation. The effect of charged impurities on the electronic states of graphene has been studied using scanning probe microscopy. Here we use the method of self-adjoint extensions to analyze the scattering of electrons from the charged impurities; our results generalize prior theoretical work which considered only one of the family of possible boundary conditions. 2 Chapter 1 Introduction Historically the Coulomb problem was one of the first significant applications of quan- tum mechanics [2, 3]. Pauli solved the Coulomb problem using algebraic methods before the invention of the wave mechanics. Schr¨odingerthen solved his eponymous equation to obtain the same result. The method followed by Schr¨odingeris now a staple of text- books [4]. However, it is seldom mentioned that there is an ambiguity in the boundary conditions that should be applied to the origin in the ` = 0 channel corresponding to zero angular momentum. The conventional treatment assumes that the wave function should be regular at the origin and this can generally be justified on the grounds that quantum mechanical wave functions should be square integrable. In the ` = 0 channel, the irregular solution has a sufficiently weak divergence that it also remains square in- tegrable and cannot be discarded on that ground alone. In fact it is now understood that there is a one parameter family of boundary conditions which are compatible with the principles of quantum mechanics. Physically these boundary conditions represent a contact interaction experienced by the particle at the origin. The textbook boundary condition that the wave function must be regular at the origin corresponds to the case of no contact interaction. The mathematical methods needed to analyze these subtle boundary problems were 1 developed by von Neumann [5] in the course of his wide ranging study of the mathemati- cal foundations of quantum mechanics. The key mathematical idea is that the boundary conditions must be compatible with the requirement that the Hamiltonian operator for the problem is self-adjoint, instead of merely hermitian. This distinction will be elabo- rated in Chapter 2. The method of self-adjoint extensions allows determination of all the boundary conditions that are compatible with the self-adjointness of the Hamiltonian. However it does not determine which of these boundary conditions should actually be used. That is a question of physics and must be determined by either appeal to experi- ment or by appeal to a more complete theory to which the Hamiltonian being analyzed is an effective approximation. Although the method of self-adjoint extensions was available soon after the discovery of quantum mechanics it has found few applications and is not part of the standard toolbox of most physicists. For example, there is no discussion of the method in any of the undergraduate or graduate textbooks that are in common use. A goal of this thesis is to find new applications of the method to problems that are of current interest. The textbook analysis of the Coulomb problem leads to the celebrated Balmer formula 2 for the energy levels, En = 1=(2n ) [4].