Application of Self-Adjoint Extensions to The

APPLICATION OF SELF-ADJOINT

EXTENSIONS TO THE RELATIVISTIC AND

NON-RELATIVISTIC COULOMB PROBLEM

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 Schrödingerequation 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 quantum mechanics [2, 3]. Pauli solved the Coulomb problem using algebraic methods before the invention of the wave mechanics. Schrödingerthen solved his eponymous equation to obtain the same result. The method followed by Schrödingeris now a staple of textbooks [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 integrable 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 mathematical 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 experiment 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]. However, even in the 1920’s, it was known that the spectrum of hydrogen had a fine structure that was not captured by Balmer’s formula. This fine structure is due to relativistic effects and was explained by a solution [6] of the Dirac Coulomb problem which does not suffer any ambiguity with regard to boundary conditions, at least for the case of ordinary hydrogen [4] 1. Alternatively, these relativistic effects can be modeled as perturbations to the non-relativistic Coulomb Hamiltonian. Two of these perturbations are fairly straightforward and correspond respectively to the mass increase of moving particles in special relativity and to the spin-orbit coupling of the electron’s magnetic dipole to the Coulomb field when the electron is in motion. The third one is called the Darwin term and corresponds to a contact interaction between the

1Ambiguities do arise if one could create a hydrogen-like ion from an atom with atomic number Z ≥ 118. Needless to day, this remains a distant prospect.

2 electron and the proton that is generally modeled as a delta function potential [7]

Ze2~2 VDarwin(r) = 2 2 δ(r). (1.0.1) 80m c

The standard treatment of fine structure treats the effects of the Darwin term by first order perturbation theory, which gives results consistent with the Dirac equation and the observed fine structure of hydrogen. However, since the Darwin term is a contact interaction, the viewpoint advocated in this thesis is that it would be better to treat it as a modification of the boundary condition that should be applied at the origin. Delta function potentials are inconsistent in more than one dimension: although they produce effects in perturbation theory, if the problem is solved exactly, there are no scattering solutions at all, suggesting that there is no interaction with the delta function potential, a result sometimes called the Beg-Furlong-Huang theorem [8]. The problems associated with delta function potentials can be studied more transparently in the absence of an additional Coulomb potential and it was in this context that these problems were first identified. Pure delta function interactions were studied by Huang in the context of superfluids in the 1960s. Huang wished to consider bosons interaction via a short ranged contact interaction with no additional forces. Later the problem was studied by Beg and Furlong who were motivated by fundamental considerations of quantum field theory. Beg and Furlong showed that two non-relativistic particles that interact via a delta function potential are actually free in two or more dimensions and they argued that this observation supported Wilson’s conjecture regarding the triviality of the φ4 field theory [8]. The consensus that has emerged from these studies is that it is possible to have non-trivial contact interactions in two and higher dimensions and that these interactions should be analyzed by the method of self-adjoint extensions as modified boundary conditions or by renormalization group methods. These studies also established that boundary condi-

3 tions can break symmetries that are present at the classical level in the model and that breaking of these symmetries is analogous to the anomaly phenomenon in quantum field theory. For example, the delta function in two dimensions imposes boundary conditions which anomalously break scale invariance [8]. In Chapter 2, a brief pedagogical review of previous work is provided. This thesis will focus on the study of three new realizations of the Coulomb problem in which potentially contact effects may play a more prominent role than in ordinary hydrogen. The first has to do electrons trapped on the surface of liquid helium by their own image charges [1]. This system is a realization of the one dimensional non- relativistic Coulomb problem. At large distances from the helium one expects the liquid can be treated as a continuous dielectric medium, but at short distances the physics are likely much more complicated. The method of self-adjoint extensions would attempt to condense these complicated short distance interactions into a single boundary contact parameter [9]. The second application covered is to Rydberg atoms [10, 11]. This is a realization of the three dimensional non-relativistic Coulomb problem. Rydberg atoms or molecules are systems where a single electron is highly excited and sees the remaining nuclei and electrons as a small singly charged core. In this case, the goal is to capture the complicated interactions with the core as a single boundary parameter [9]. Finally, the third application considered is the effect of charged impurities in the electronic states of graphene [12]. As graphene is an atomically flat two dimensional crystal, and the electrons are effectively relativistic particles in the system, the motion of the electrons is described by the two dimensional Dirac Coulomb problem. Again, in this case, the charged impurities are clusters of Cobalt or Calcium atom which are far from point- like and hence one might expect significant contact interactions [13]. From a purely theoretical point of view, an interesting find is two examples of anomalous symmetry breaking: the SO(4) Runge-Lenz symmetry for the three dimensional Coulomb problem and scale invariance for the two dimensional Dirac Coulomb problem.

4 In the one dimensional Coulomb problem, application of the self-adjoint extensions led to a modiﬁcation of the Balmer formula,

1 E ∝ , (1.0.2) n (n − 2λ)2 where λ is the self-adjoint parameter, associated with the modified boundary conditions. Using this formula, the results of the liquid helium experiments performed by Grimes and Brown [14, 15] were successfully recreated. Furthermore, the results using the self-adjoint method are shown to be compatible with the generally accepted Cole [1] model for the system. Application of the method to the Rydberg atoms is only somewhat successful. As the extensions are only present in the s-wave channel, scope is much more confined than in the general Rydberg theory. However, the modified Balmer formula (1.0.2) is also derived in the three dimensional case, and is shown to be equivalent to the first order Rydberg Ritz formula. Unfortunately comparison of the two methods by fitting to known spectra of various alkali metals heavily favors the second order Rydberg-Ritz formula [10]. We conclude that a modified boundary condition does not capture the interaction of the excited electron with the core with accuracy comparable to the quantum defect model used to derive the Rydberg Ritz formula [10, 11]. Prior work on charged impurities in graphene did not take into account the ambiguity in the boundary condition that must be applied at the location of the impurity. By treating boundary conditions using the method of self-adjoint extensions we discover that the contact interaction anomalously breaks scale invariance and leads to a scattering phase shift with a non-trivial energy dependence in contrast to prior work which found a featureless energy independent phase shift that might be expected on the basis of scale invariance. The remainder of the thesis is organized as follows: Chapter 2 covers the mathematics

5 behind the self ajoint extensions and reviews past work on contact potentials through the use of self-adjoint extensions and the renormalization group. Chapter 3 provides a brief review of the experimental systems used in the thesis. In chapter 4, the non- relativistic Coulomb problem in one and three dimensions is discussed and compared to the experimental results of electrons bound on liquid helium and Rydberg atoms. Chapter 5 covers the two dimensional Dirac Coulomb problem and compares it to results in graphene systems. The Conclusions are presented in chapter 6. Technical details are relegated to the appendices. Appendix A covers the ﬁner details of the hypergeometric functions; appendix B provides further details into the ﬁtting algorithms used in the Rydberg section and appendix C describes the statistical method used for comparison of the Rydberg model to the self-adjoint model.

6 Chapter 2

Self-Adjoint Extensions

The underlying mathematical principles which this work was based off are a somewhat subtle generalization of concepts surrounding hermiticity. In the most reductive sense, the self-adjoint extensions that are employed are simply changing the boundary conditions in the system to ensure that the operator is still self-adjoint while simultaneously relaxing the requirements for the accepted wave functions. To fully exploit these extensions, they must have a strong set of definitions as well as a clear distinction from the weaker criterion for hermiticity. As the main distinction between a fully self-adjoint operator and one that is hermitian is fully dependent on the domain of the operators, it is best to first define what constitutes the domain for such an operator. An operator T defined on a Hilbert space H is a linear map from some linear subspace of H into H. The domain of T , denoted D(T ) is simply the subspace of H on which it acts [16]. From this definition of the domain, it is now possible to define the extension of an

operator. Consider two operators T1 and T2 deﬁned on some Hilbert space H, T2 is an

extension of T1 if D(T2) ⊃ D(T1). To write this more speciﬁcally, T2 is said to extend T1

if T2φ = T1φ for all φ ∈ D(T1), and is written T2 ⊃ T1 [17]. The final concept to define before differentiating between hermitian and self-adjoint

7 operators is the adjoint of the operator. Let T be an operator on a Hilbert space H with an inner product (·, ·). Let D(T ∗) be the set of φ ∈ H for which there exists a corresponding η ∈ H with

(T ψ, φ) = (ψ, η) for all ψ ∈ D(T ). (2.0.1)

For each such φ ∈ D(T ∗), T ∗ is deﬁned by T ∗φ = η, and T ∗ is called the adjoint of T [17]. For an operator T to be fully self-adjoint it must then meet two criteria. The ﬁrst is that its adjoint acts in same fashion as the operator itself, namely that T φ = T ∗φ, with φ ∈ D(T ). This can also be written in terms of the inner product as

(T ψ, φ) = (ψ, T φ) for all ψ, φ ∈ D (T ) (2.0.2) which is the colloquial definition of a hermitian operator [17]. The second criterion for the self-adjoint definition is the distinction between simply hermitian and fully self-adjoint. For the operator to be self-adjoint, the domain of the operator and its adjoint must coincide, D(T ) = D(T ∗). If the domains are not equal, D(T ) ⊂ D(T ∗), the operator is then simply hermitian [17]. Since the only difference between hermitian and self-adjoint operators is the requirement that domain of the adjoint must be equal to the domain of the operator itself, it is sometimes possible to find an operator which is both an extension of a hermitian operator and is itself fully self-adjoint. It is this idea that defines a self-adjoint extension. In most quantum systems, these extensions can arise due to the definition of the inner product

Z (ψ, φ) = dµ ψ∗φ, (2.0.3) where dµ is simply the measure associated with the space. When dealing with any

8 differential operator, showing that the operator is at least hermitian will usually result in a surface term after integration by parts. That is, for some differential operator T , the hermitian definition (2.0.2) has an extra term

(T ψ, φ) = (ψ, T φ) + surface terms.

These are normally forced to be zero through the imposition of boundary conditions on the surface. However these conditions may be over-restrictive on the domain of the operator itself, causing D(T ) ⊂ D(T ∗). The following two systems give concrete examples of the implementation and eﬀects of the self-adjoint extensions to hermitian operators.

2.1 Particle in a box

To better illustrate these concepts, consider the one dimensional free particle re- stricted to the closed interval [0, 1]. The quantization of the momentum for the system is fairly straightforward; to begin, the diﬀerential equation given by the momentum operator (P = −id/dx),

d −i ψ(x) = pψ(x), (2.1.1) dx has the solution of ψ(x) = C exp(ipx). Imposing periodic boundary conditions of ψ(0) = ψ(1) gives rise to the quantization of the momentum. These force the momentum of the system to be p = 2πn, where n ∈ N. The boundary conditions determine whether the operator is self-adjoint or merely hermitian. As the inner product for the spaces is given by

Z 1 (φ, ψ) = dxφ∗ψ, (2.1.2) 0

9 it is easy to show using integration by parts that the momentum operator P gives rise to

∗ 1 (φ, P ψ) = (P φ, ψ) − i φ ψ|0 . (2.1.3)

If the boundary condition of ψ = 0 at the boundary points x = 0 and x = 1 is adopted, the surface term in (2.1.3) clearly vanishes and the operator P satisﬁes (φ, P ψ) = (P φ, ψ). The operator P is then clearly hermitian under the standard boundary conditions for a particle in the box. However, the operator is not self-adjoint. When the condition of ψ = 0 at the boundary is imposed, the surface term vanishes regardless of the behavior of φ at the boundary. Thus, D(P ) consists of the functions that vanish at the boundary whereas D(P ∗) consists of functions that do not have to respect any boundary conditions. The domains are not equal and P is not self-adjoint under the standard particle in a box boundary conditions. Indeed one can see that P has no eigenfunctions with these boundary conditions, far from the complete set of eigenfunctions that every self-adjoint operator should possess. Now consider the boundary condition of ψ(0) = λψ(1), where λ is some arbitrary complex number. Using this new condition, the requirement for surface term of φ∗(1)ψ(1) − φ∗(0)ψ(0) to vanish is now φ∗(0) = (1/λ)φ∗(1). For the domains of P and P ∗ to coincide, it must be that λ = (1/λ∗), which can be expressed in the simpler form of λ = eiα, where α is some real number. Thus the momentum operator P is self-adjoint on the unit interval if and only if the twisted boundary condition, ψ(0) = eiαψ(1), is imposed.

2.2 Delta Function

As a second example, consider a two dimensional system with a contact interaction at the origin. Here the Hamiltonian would be of the form

H = ∇2 + vδ(x).

10 Due to the rotation invariance the eigenfunctions may expressed as ψ = R(r) exp(i`θ) where R is the radial wavefunction and ` is an integer corresponding to the angular momentum of the system. Away from the origin of the system, the contact interaction can be eﬀectively ignored. Expressing the Laplacian in polar coordinates

∂2 1 ∂ 1 ∂ ∇2 = + + ∂r2 r ∂r r2 ∂θ2

it is apparent that the radial wave function obeys Bessel’s equation and has solutions

which are linear combinations of the Bessel function, J|`|(kr), and Neumann function,

|`| Y|`|(kr). The small r behavior of the Neumann function is Y|`|(kr) ≈ 1/r for ` 6= 0 and

Y|`|(kr) ≈ ln r for ` = 0, while the Bessel function remains regular at the origin. The inner product for two wave functions in the same angular momentum channel is given by

Z ∞ (φ, ψ) = dr rS∗(r)R(r), (2.2.1) 0 where S and R are the radial wave functions corresponding to φ and ψ respectively. It can be quickly noted that that outside the ` = 0 channel the Neumann functions are not square integrable and thus unacceptable as quantum mechanical wavefunctions for the system. The Y0(kr), on the other hand, is allowable; despite diverging at the origin, the divergence is weak enough so that the function remains square integrable. Thus, in the ` = 0 channel, the boundary condition are not determined by square integrability alone, suggesting the existence of self-adjoint extensions. Continuing to employ the polar form of the Laplacian and integrating by parts, it is evident that

11 dS∗ dR∞ (φ, Hψ) = (Hφ, ψ) + r R − rS∗ . (2.2.2) dr dr 0

The Hamiltonian is formally self-adjoint, in the sense that it obeys (φ, Hψ) = (Hφ, ψ) up to surface terms. By locality, it is assumed that separate boundary conditions are applied at r = 0 and r = ∞. As these boundary conditions at inﬁnity are generally determined by square integrability, the focus should remain at the origin. Motivated by the small r behavior of the Bessel and Neumann functions in the ` = 0 channel, the radial components can be expressed as

R = a0 + a1r + ... + b0 ln r + b1r ln r + ...,

S = A0 + A1r + ... + B0 ln r + B1r ln r + ..., (2.2.3)

near the origin. Substitution of these forms into the surface term seen in (2.2.2) yields

∗ dS ∗ dR ∗ ∗ lim r R − rS = a0B0 − b0A0, (2.2.4) r→0 dr dr

where the omitted terms vanish in the r → 0 limit. Imposing the boundary condition of

∗ a0 = λb0 on ψ, then the similar condition of A0 = λ B0 must be imposed on φ for the surface term of (2.2.4) to vanish. By restricting λ to the real numbers, the two boundary conditions will coincide and the domain of H will be the same as for its adjoint. To summarize, in the ` = 0 channel, the radial function R for a non-relativistic particle in two dimensions experiencing a contact or Dirac potential at the origin will have the leading asymptotic behavior of R = a0 + b0 ln r at small r and must obey the boundary condition of a0 = λb0, with λ ∈ R. This condition may be expressed more elegantly as

12 R(r0) R(r) lim R(r) − lim ln r = λ lim . (2.2.5) r→0 r0→0 ln r0 r→0 ln r

In the case of λ = 0, only the non-singular Bessel function will be used for the wavefunction near the origin. This corresponds to the case of a free particle. The case of non-zero λ corresponds to the contact potential. Physically the interaction with the contact potential will result in scattering processes and the possibility of a bound state near the origin. These matters are more fully discussed in Jackiw [8].

2.3 Scale Invariance

Aside from rotational invariance the delta function potential also exhibits scale invariance. Scale invariance powerfully constrains the scattering of the particle. Due to the contact nature of the potential the particle should only exhibit s-wave scattering and scale invariance would imply that the scattering phase shift should be independent of energy. However, the self-adjoint extension analysis shows that the phase shift exhibits a complicated energy dependence. Following the methods of Jackiw, analysis of the contact interaction problem using a renormalization group method shows that the scale invariance is anomalously broken and that a scale enters the problem via “dimensional transmutation” [8]. As will be discussed in Chapter 5, a similar analysis of the physics of a Dirac particle bound to a charged impurity, relevant to graphene, also exhibits an anomalous breaking of scale invariance. The scattering solution for the delta potential is given by

−∇2 + V¯ − k2 ψ = 0, (2.3.1)

13 which has the solutions of

Z ψ = exp(ik · r) − dr0G(0)(r, r0; k2)V¯ ψ, (2.3.2)

where G(0) is the Green’s function, obeying the relation that

−∇2 − k2 G(0)(r, r0; k2) = δ (r − r0) , (2.3.3)

which has the real space solution of

i G(0)(r, r0; k2) = [J (k|r − r0|) + iY (k|r − r0|)] . (2.3.4) 4 0 0

The Fourier transform of the Green’s function is then

1 G˜(0)(q, k2) = . (2.3.5) q2 − k2

This can be employed after rewriting the scattering solution of (2.3.2) in momentum space. The Fourier transform of the scattering solutions is then

Z dp ψ˜(q) = (2π)d δ(q − k) − G˜(0)(q, k2) V˜ (q − p)ψ˜(p) (2.3.6) (2π)d

Taking the potential as V¯ = λδ, where λ is the strength of the potential, the Fourier transform is V˜ = λ is independent of the momenta. To avoid any divergences in the momenta integration, the momentum space cutoﬀ Λ is introduced so that V˜ (p) = λ for

14 0 ≤ p < Λ and V˜ = 0 otherwise. As in quantum ﬁeld theory, the limit of Λ → ∞ will eventually be employed. Employing the regulated potential in (2.3.6), and assuming that q Λ,

ψ˜(q) = (2π)d δ(q − k) − G˜(0)(q, k2)λΨ(0), (2.3.7) with

Z dp ˜ Ψ(0) = d ψ(p). (2.3.8) Λ (2π)

Solving for the quantity λΨ(0) results in the consistency equation

Z −1 1 dp ˜(0) 2 λΨ(0) = + d G (p, k ) . (2.3.9) λ Λ (2π)

Using the form of G˜(0) in (2.3.5), integration yields

Z 2 p ˜(0) 2 1 Λ i d G (p, k ) = ln 2 + . (2.3.10) Λ (2π) 4π k 4

Introducing a new ﬁxed momentum µ, where µ < Λ, and using the result of the integration, a quick rewrite of (2.3.9) gives

1 1 Λ 1 k i −1 λΨ(0) = + ln − ln + . (2.3.11) λ 2π µ 2π µ 4

Naively, applying the limit of Λ → ∞ yields λΨ(0) = 0, which, from equation (2.3.7), has that the eigenstates of the systems are just plane waves. This implies that the eﬀect of delta function is trivial in two dimensions, a result called the Beg-Furlong-Huang theorem. However, it is possible to get a non-trivial contact potential in two dimensions

15 by altering the limit. By introducing a renormalized coupling constant g, deﬁned in equation (2.3.12), and the limit of Λ → ∞, and assuming λ varies in such a way as to keep the renormalization coupling g ﬁxed,

1 1 1 Λ = + ln , (2.3.12) g λ 2π µ which allows for (2.3.11) to be expressed as

1 1 k i −1 λΨ(0) = − ln + . (2.3.13) g 2π µ 4

This development proves extremely useful in the calculation of the scattering states of the system. Using the results of regulated Green’s function, as well as the asymptotic behavior of the Bessel functions, the scattering states are

1 π 2 1 k i −1 ψ(r) = exp(ikr) + √ exp ikr + i − ln + . (2.3.14) 2πkr 4 g π µ 2

The scattering is clearly isotropic, and the only non-vanishing scattering amplitude is in the ` = 0 channel. The associated scattering amplitude is

π 2 1 k i −1 f = exp i − ln + . (2.3.15) 0 4 g π µ 2

This implies that the phase shift for the s-wave is

2 k 2 cot δ = ln − . (2.3.16) 0 π µ g

Despite the fact that the λ coupling is dimensionless the phase shift for the contact

16 potential is clearly dependent on k, which even exists in the Λ → ∞ limit. Similarly, the phase shift of the Dirac Coulomb problem, explored in chapter 5, is also scale dependent when self-adjoint extensions are employed.

17 Chapter 3

Experimental systems

This chapter focuses on the review of three separate experimental systems: electrons on liquid helium, Rydberg atoms and charge impurities in graphene. In each of these systems, the interest remains on how accurately the Coulomb model, with appropriate boundary conditions, can describe the system. An electron placed on a pool of liquid helium becomes bound by its image charge, essentially the one dimensional non relativistic Coulomb problem. As Rydberg atoms have one electron in a highly excited state which experiences a Coulomb potential for a small, positively charged atomic core, they are modeled very similar to hydrogen. The motion of the electron is also non relativistic. The motion of electrons in graphene is eﬀectively governed by the massless Dirac equation [18, 19, 20]. Impurities with controllable amounts of charge can be assembled using scanning force microscopy near graphene sheets thereby providing a realization of the two dimensional Dirac Coulomb problem.

3.1 Liquid Helium

Electrons placed in the vapor outside of liquid helium surface have been observed to be attracted to the surface due to the existence of an image charge. The liquid helium can be seen as essentially impenetrable to the electrons, so it follows that electrons are conﬁned

18 to the half space above the surface, x > 0, if the x axis is taken to be perpendicular to the surface. The motion of the electron perpendicular to the surface may then be described to a ﬁrst approximation by the Schr¨odinger equation [14, 15]

2 d2ψ Ze2 − ~ − = Eψ, (3.1.1) 2m dx2 x

Here, the electron charge is −e and the image charge is −Ze, where // Z = (0 −

g)/4(0 + g), a result which should be familiar from any graduate electromagnetism course. 0 and g being the dielectric constants for liquid and gaseous helium and x the distance from the surface. For temperatures below 1.4 K, Grimes and Brown used the

−4 −3 values of 0 = 1.0572, g − 1 < 10 , and Z = 6.95 × 10 [14, 15]. Note that the equation (3.1.1) is identical to the radial equation for three dimensional hydrogen in the s-wave channel. According to the textbook solution, the energy levels corresponding to eq (3.1.1) are given by the Balmer formula but with a Rydberg constant reduced by the factor of Z2 to approximately a half milli-electron volt. Transitions between these energy levels can therefore be driven by electromagnetic radiation in the microwave band. Although the simplified model described above provides a good qualitative picture, the measure transitions do not agree exactly with the Balmer formula. Cole [1] proposed a model that took into account several important physical effects that are glossed over in the simplified model. Notably, helium is not completely impenetrable and is better modeled as a finite barrier of height ≈ 1 eV. In addition, once the electron is within a distance b comparable to the interatomic separation of the helium atoms, it no longer makes sense to model the liquid as a continuous dielectric. Cole took this into account by assuming that the 1/x behavior would be cut off at a distance b from the surface of helium and near to that should be treated as a constant. Grimes and Brown attempted to model the same physics by suggesting that the surface should be assumed shifted by

19 Figure 3.1: Sample spectra of the transition in the electron bound states made by Grimes and Brown in 1976 [15]. an amount β, so that the potential in eq (3.1.1) was of the form 1/|x − β| [15]. Yet another model was proposed by Huang [21], but was less successful in ﬁtting the data.

3.2 Rydberg Atoms

Rydberg atoms. or atoms that exhibit Rydberg states, are those with one valence electron in a high n state. In this situation, the system can then be seen as modiﬁed hydrogen; the core atom has a charge of +e, with only slight deviations arising from the interactions between the electron in the high n state and the core electrons. This lead to the postulation by Rydberg that the energy spectrum for such atoms could be given by

R E = − , (3.2.1) (n + δ)2

20 with R being the Rydberg constant and δ the quantum defect [10, 11]. For single electron species such as H, He+, Li2+, the defect remains essentially at zero. Systems with more electrons will lead to non-zero values for the defect. A full examination shows that the defect has an n dependence of

δ δ δ = δ + 2 + 4 + ... (3.2.2) 0 n2 n4

This formula may also be written in the implicit form suggested by Ritz [10],

R E = − 2 . (3.2.3) (n + δ1 + δ2E)

Not only atoms but also sufficiently excited molecules exhibit Rydberg behavior and have levels described by the above formula. Rydberg states are useful in astronomical observations. The transition between n = 108 and n = 109 has a frequency of 2.4 GHz, favorable for radio astronomy. Rydberg states have also proven useful for atomic clocks [22]. These authors find that transitions between states with 20 ≤ n ≤ 25 can yield clock accuracies of 1 part in 1018. Chapter 4 will describe an attempt to describe Rydberg atoms using self-adjoint extensions. Such states were first analyzed in the framework of old quantum theory, notably by Bohr and Born, and this analysis is outlined in the next subsection. Later studies developed the quantum defect model based on perturbation theory and WKB analysis [10].

3.2.1 First Order Quantum Defect

The derivation of the ﬁrst order quantum defect can be easily accomplished following the method of Born [23]. Here, the normal Coulomb potential is perturbed in such a way that the asymptotics of the system will be the same as the non-perturbed potential, while having the perturbations play a role at small r. To satisfy these requirements, the

21 potential can be written

e2Z a a2 V (r) = − 1 + c + c + ... (3.2.4) r 1 r 2 r

where a is some length scale, which can be taken as the Bohr radius, aH . As the perturbations preserve the spherical symmetry of the original potential, the system can treated in the same fashion as the Kepler problem. To simplify matters from

the onset, the reduced mass can eﬀectively be ignored; as µ = mpme/(mp +me) ≈ mp(1+

me/mp), where me and mp are the mass of the electron and proton, respectively, the correction to the mass of the proton is well under 0.1%. This allows for the Hamiltonian to be written as

1 p2 p2 H = p2 + θ + φ + V (r) (3.2.5) r 2 2 2 2me r r sin θ

Following the Hamilton-Jacobi procedure, Hamilton’s characteristic function of

W = Wr(r) + Wθ(θ) + Wφ(φ) (3.2.6)

is introduced. As the Hamiltonian is independent of φ, φ is clearly cyclic, and

Wφ = αφφ (3.2.7)

where αφ is a constant of integration. Using the characteristic function, the Hamilton- Jacobi equation

" # ∂W 2 1 ∂W 2 α2 r + θ + θ + 2mV (r) = 2mE (3.2.8) ∂r r2 ∂θ sin2 θ with E as the total energy of the system. As the θ dependence of (3.2.8) is relegated to the terms inside the square brackets, the rest of the equation is independent of θ, implying that the term is constant:

22 ∂W 2 α2 θ + θ = α (3.2.9) ∂θ sin2 θ θ

This relation can then be substituted into (3.2.8), separating all variables and given the radial equation of

∂W 2 α r + θ = 2m (E − V (r)) (3.2.10) ∂r r2

As the perturbations only eﬀect the radial equation, the angular integrals, Jθ and Jφ retain the expected forms of

s I I α2 J = dφ α and J = dθ α2 − φ φ φ θ θ sin2 θ with the solutions of Jφ = 2φαφ and Jθ = 2π(αθ − αφ). Using these results, the radial integral is of the form

s I (J + J )2 J = dr 2m (E − V (r)) − θ φ (3.2.11) r 4π2r2

Following the quantization procedure, each of the action variables Ji must be a multiple of the Planck constant h. Replacing the sum of Jθ + Jφ with the kh, where k ∈ Z, allows for the radial action to be written as

r I C C C J = dr −C + 2 1 − 2 + 3 + ... (3.2.12) r 0 r r2 r3 with the coeﬃcients from (3.2.12) related to the potential (3.2.4) through

23 C0 = −2mE,

2 C1 = me Z, k2h2 C = − 2me2Za c , 2 4π2 H 1 e 2 C3 = 2me ZaH c2. (3.2.13)

As the effects from the perturbation are already apparent in the coefficient C2, it is expected that first order effects should be apparent when keeping to quadratic order. Using both complex integration, and using n as the total quantum number yields

p C1 Jr = (n − k)h = 2π − C2 + √ C0

The energy of the system can then be determined as

4π2C C = −2mE = 1 0 √ 2 (n − k) h + 2π C2

Substitution of (3.2.13) gives the desired form of the energy

RhZ2 E = − (3.2.14) (n + δ)2 with

r 8π2me2Z p δ = −k + k2 − a c = −k + k2 − 2Zc (3.2.15) h2 H 1 1

Assuming that the perturbations are small, the defect (3.2.15) can be written in the simpliﬁed form Zc δ = − 2 k

24 3.2.2 Higher Order Corrections

Higher order corrections can be derived from basic perturbation theory [23]. The Hamiltonian for the Rydberg state is written as

H = H0 + H1 (3.2.16)

where H0 is the usual Coulomb Hamiltonian, with the energy levels of

RhZ2 E = − (3.2.17) n n2

The second part of the hamiltonian (3.2.16), H1, is the perturbation, which is of the same form of the expanded potential in (3.2.4), written as

e2Z a a 2 H = − c H + c H + ... (3.2.18) 1 r 1 r 2 r

where aH is the Bohr radius. The ﬁrst order corrections to the energy are then given by

1 1 1 E = −e2Z c a + c a2 + c a3 + ··· (3.2.19) 1 1 H r2 2 H r3 3 H r4

Determination of the average values for the 1/rn

1 Z2 2 = 2 3 , r aH n k 1 Z2 3 = 3 3 , r aH n k 1 (3n2 − k2) Z4 4 = 4 5 5 , r 2aH n k 1 (5n2 − 3k2) Z5 5 = 5 5 7 . (3.2.20) r 2aH n k

25 2 −1 Using the Rydberg constant, R = e (2aH h) , as well as substituting the average values back into (3.2.19), the energy of the system is then

RhZ2 2Zc 2Z2c Z3 (3n2 − k2) c Z4 (5n2 − 3k2) c E = − 1 + 1 + 2 + 3 + 4 + ··· n2 nk nk3 n3k5 n3k7 (3.2.21) This can be written to ﬁt the normal form of the Rydberg equation, namely

RhZ2 E = − (n + δ)2

Comparison of the Taylor series obtained by expanding around δ = 0 and comparison to the energy in (3.2.21) gives that the quantum defect is

Zc Z2c Z3 (3n2 − k2) c Z4 (5n2 − 3k2) c δ = − 1 − 2 − 3 − 4 − ... k k3 2n2k5 2n2k7 collecting like terms in n allows for δ to be written as an expansion in n, taking the desired form of

δ δ(n) = δ + 2 + ..., (3.2.22) 1 n2

where the δi are given as

Zc Z2c 3Z3c 5Z4c δ = − 1 − 2 − 3 − 4 − ..., 1 k k3 2k5 2k7 Z3c 3Z4 δ = 3 + + .... 2 2k3 2k5

The dependency of the corrections δ on n can also be expressed in the recursive relation originally used by Ritz RhZ2 E = 2 . (n + δ1 + δ2E)

26 3.3 Graphene

Current studies of graphene systems have shown that one of the more peculiar properties of the substance is the existence of quasiparticle charge carriers. These arise from the interactions of the electrons with the periodic nature of the graphene itself. The result of the electron interactions are these quasiparticles acting as charge carriers. Un- like most condensed matter systems though, these excitations are best described by the (2 + 1) Dirac equation instead of the Schr¨odinger equation. These so-called massless

6 −1 Dirac fermions have the effective speed of light vf = 10 ms . Thus, these fermions are either viewed as electrons with zero rest mass, or alternatively as neutrinos with an effective charge of e. Initially, arguments against the existence of 2-D crystalline put forth by Landau and were based on the thermal stability of the system, but were ultimately shown to be incorrect. In 2006, Geim and Novoselov were able to isolate monolayers using the (also refered to as the ’Scotch Tape’ method). This work was the basis for their 2010 Nobel prize. Measurements of the conductivity against various gate voltages of graphene have agreed with the theoretical behavior of the conductivity based on scattering off potentials. Conductivity measurements by Novoselov, Geim, et. al. [20] have shown a linear relation between the conductivity of the sample and the applied gate voltage. Using samples with thicknesses of less than 10 nm, and applying gate voltages across the sample, the group found that the resistivity ρ of the samples peaked at several kilo-ohms, and decayed the to ≈ 100 ohms at high gate voltages. The conductivity, σ = 1/ρ exhibited a linear increase on both sides of the resistivity peak, which coincided with the reversal of sign for the Hall coefficient RH . Studies of the scattering states off charged impurities in graphene have been able to replicate the linear behavior of the conductivity. Analysis preformed by Hwang, Adam and Das Sarma [19], determined the conductivity of the system from the Boltzmann

27 Figure 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].

2 relation of σ = (e /h)(2EF hτi/~), where EF is the fermi energy associated with the R R system and hτi/ is the averaged scattering time, hτi/ = dkτ(k)∂f/ dkk∂f, where f(k) is the fermi distribution function. The energy dependent scattering time τ(k)

(a) 2 1 π X (a) ν (q) 2 = ni (1 − cos θ) δ(k − k0 ), (3.3.1) τ(k) ε(q) ~ a,k0

0 where q = |k − k |, θ is the angle between the wave vectors, ni is the charge impurity

(a) concentration, k = ~γ|k| and ν (q) is the matrix element of the scattering potential. Using the assumption of Random Phase Approximation(RPA) screening, the conductivity curve exhibited the same linear behavior of the graphene, and can be seen in the inset of ﬁgure 3.3. Using slightly diﬀerent methodology, Shytov, Katsnelson, and Levitov [18] were able to determine the behavior of the Local Density of States (LDOS) which again had the same linear behavior as observed in the experiments. This behavior was found using the solutions of the scattering equation (which are discussed in Chapter 5), which are related

28 Figure 3.3: Calculated conductivity in graphene based on Random Phase Approximation screening [19]. The conducivity curve exhibits the same linear pattern observed in the experiments of Geim et. al. in ﬁgure 3.2

to the Local Density of States ν by

N X ν(, ρ) = |ψ(k , ρ)|2, k = − , (3.3.2) π ν ν ~ F m ~ f where νf is the Fermi velocity, and ρ is the radial distance. They found that large deviations from the expected relation of ν ∝ || only occurred in the supercritical case;

2 where coupling β = Ze /κ~νF > 1/2. These results can be seen in ﬁgure 3.4. However, these results were based on discarding the irregular solutions to the Coulomb scattering in the subcritical case (β < 1/2); this allows for opportunity to apply the method of self-adjoint extensions to the subcritical case.

29 Figure 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 Chapter 4

Non-Relativistic Coulomb Problem

Our original goal of investigating self-adjoint extensions was to apply them to muonic hydrogen in hope that the Lamb shift would be affected. While the analysis of the shift is still being explored, exploration of the effects on a Coulombic potential yielded several interesting results. To begin, the time-independent Schrödinger equation for this system is

2 Ze2 − ~ ∇2 + ψ = Eψ. (4.0.1) 2m 4π0r

The system described above exhibits two major symmetries: the obvious rotational symmetry and the symmetry due to the conservation of the Runge-Lenz vector associated with 1/r potentials [4]. The rotational invariance is responsible for the degeneracy of energy levels that diﬀer only in the quantum azimuthal angle m. The conservation of the Runge-Lenz vector implies an SO(4) symmetry of the system, which in turn is associated with the degeneracy of states with diﬀerent values of `. However, this SO(4) symmetry present at the classical level is anomalously broken in the quantum system due to the introduction of boundary conditions, similar to the breaking of scale invariance in the delta potential seen in (Section 2.3). In particular, the ` = 0 energy levels are no longer

31 degenerate with states of higher ` even if they share the same principle quantum number n. Exploiting the rotational symmetry of the Coulomb potential, (4.0.1) can be broken

down into its radial and angular parts. Using the usual ψ = R(r)Ylm(θ, φ), where R(r)

is the radial function and Ylm are the spherical harmonics, then using the substitution of u = rR, the Schr¨odingerequation (4.0.1) can be rewritten as

1 d2 1 `(` + 1) 1 − u − u + u = − κu. (4.0.2) 2 dr2 r r2 2

Between the equations (4.0.1) and (4.0.2), Hartee units have been introduced, which

2 2 2 have ~ = 1, m = 1, and e /4π0 = 1, and the energy is E = −~ κ /2m is written for convenience while examining the bound states. By considering only the s wave systems, so ` = 0 the equation (4.0.2) behaves as the Coulomb problem in one dimension, greatly simplifying the process. For the one-dimensional case, the self-adjoint extensions are determined much like in Chapter 2. To begin, the inner product is deﬁned by

Z ∞ (φ, ψ) = dx φ(x)∗ψ(x). (4.0.3) 0

To show that Hamiltonian is fully self-adjoint, H is required to be symmetric with respect to the inner product, (φ, Hψ) = (Hφ, ψ). Substitution of the Hamiltonian, followed by integration by parts gives

∗ ∞ ∗ dψ dφ (φ, Hψ) = (Hφ, ψ) − φ − ψ . (4.0.4) dx dx 0

32 Caution must be employed during the determination of the boundary conditions, due to the singular behavior of the Coulomb potential at the origin. Solving equation (??), yields two independent solutions, one of which is regular at the origin and the other singular. In every ` channel other than ` = 0, the singular solution contains divergences that are not square integrable. In these channels the boundary condition at the origin is that only regular solutions are accepted. This is the classic boundary condition that is presented in most quantum texts and hence inclusion of self-adjoint extensions will not change the behavior in the ` 6= 0 channels. However, in the ` = 0 channel, the divergences associated with the singular solution are logarithmic, and thus square integrable. The implication is that consideration of the self-adjoint extensions are important for the s-wave channel. Employing the Frobenius method to equation (??), suggests that the wavefunctions will have the forms

φ(x) = c x − x2 + ... + d (1 − 2x ln x + ...) ,

ψ(x) = C x − x2 + ... + D (1 − 2x ln x + ...) (4.0.5) near the origin, where c, C, d, D ∈ C and the ellipses indicate higher orders of x or higher orders of x with logarithmic terms, such as x ln x or x2 ln x. Using these forms in equation (4.0.4), reduces the surface term to

∗ ∗ dψ dφ ∗ ∗ φ − ψ = (cD − dC ), (4.0.6) dx dx 0 where the omitted terms are those that vanish in the limit of x → 0. Consider the boundary condition on φ that c = λd; for the surface term to vanish, the similar condition of C = λ∗D must be imposed on ψ. To ensure that the domains of H and its adjoint coincide, the boundary conditions imposed on φ and ψ need to be the same. This is ensured by restricting λ to the real numbers. Thus, for one dimensional hydrogen or three dimensional hydrogen in the s-wave

33 channel, the boundary wave function ψ must have an asymptotic form of equation (4.0.5) near the origin as well as satisfy the condition that c = λd, with λ ∈ R. This condition can be likewise be expressed as

ψ(0) = λ lim [ψ0(x) + 2ψ(x) + 2ψ(x) ln x] . (4.0.7) x→0

As stated before, the one dimensional analysis may be applied to the full three dimensional formulation for states in the l = 0 channel. Aside from the Hamiltonian being the same, the inner product for the u(r) will be of the exact same form as in equation (4.0.3). This is due to the deﬁnition of U = rR, so the inner product of two wavefunctions in the l = 0 channel φ(r) = Rφ(r) = ru(r) and ψ(r) = Rψ(r) = rv(r) is

Z ∞ Z ∞ Z ∞ 2 ∗ 2 ∗ ∗ (φ, ψ) = dr r φ (r)ψ(r) = dr r Rφ(r)Rψ(r) = dr u (r)v(r) 0 0 0

Thus, the conditions needed to have (u, Hv) = (Hu, v) will be exactly those discussed in the one dimensional case. Having established the equivalency between one dimensional hydrogen and the ` = 0 channel for three dimensional hydrogen, as well as determining the most general boundary conditions which may be applied at the origin in either case, it is now possible to solve the Schr¨odingerequation (4.0.1), and the associated energy levels. Knowing the solutions to the radial equation (4.0.8) in the ` = 0 channel will have the small r behavior

u(r) = c r − r2 + ... + d (1 − 2r ln r + ...) where the coeﬃcients are related through c = λd and λ ∈ R.

34 With the substitutions in place, the Hamiltonian for the system is

1 d2 1 1 − u − u = − κu, (4.0.8) 2 dr2 r 2

−1 where u is related to the wave function by ψ(r, θ, φ) = r u(r)Ylm(θ, φ). Using the ansatz

l+1 − 1 s u = s ξ(s)e 2 s = 2κr (4.0.9) in equation (4.0.8), it is clear that the function ξ(s) satisﬁes Kummer’s equation

d2 d s ξ + (b − s) ξ + aξ = 0. (4.0.10) ds2 ds

Here the parameters a and b are given by

1 a = 1 − , b = 2. (4.0.11) κ

Solutions to Kummer’s equation (4.0.10) are the conﬂuent hypergeometric functions, described in more detail in appendix A. Imposing the boundary condition that the solution vanishes as r → ∞, has u being described by

−iπa u(r) = N e κrU2(a, b, 2κr) exp(−κr), (4.0.12)

where N is a normalization factor, a and b are the same as described in (4.0.11) and U2 is a conﬂuent hypergeometric equation of the fourth kind. The factor of e−iπa is included to keep the solution real.

35 The imposition of the boundary conditions at r = 0 yields the quantization condition for the energy. To determine this, the behavior of (4.0.12) is examined near the origin. Expansion has that

u ∝ 1 − 2r ln r, 1 + r κ + 2 − 2 ln(2κ) − 4γ − 2Ψ − , (4.0.13) κ

where a common factor of N /2Γ(1 − 1/κ) on the right side has been neglected, and Ψ is the digamma function. Comparison to the expected results from the self-adjoint analysis agrees as u remains of O(r ln r), and that the coeﬀecient d = 1 and c is equal to the term in the square bracket on r. Then, using the condition that c = λd, the quantization condition is

1 −1 λ = κ + 2 − 2 ln(2κ) − 4γ − 2Ψ − . (4.0.14) κ

Solving this transcendental equation allows for the determination of κ, which can then be

1 2 used to determine the quantized energy E = − 2 κ . Figure (4) shows a plot of the λ(κ), with the intersections with the horizontal line of constant λ signifying the quantization at that value of λ A quick check of the condition is the case of λ = 0. This implies that digamma function on right side of (4.0.14) must diverge. This divergence occurs for any negative integer valued argument; this leads to the quantization

1 1 = n ⇒ E = − , n ∈ , (4.0.15) κ n 2n2 N

which is well known Balmer formula.

36 0.2

0.1

0.036 0.038 0.040 0.042

-0.1

-0.2

-0.3

Figure 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 For small values of λ, it is possible to solve equation (4.0.14) using perturbative methods. Setting

1 = n − δ (4.0.16) κn and using the approximation of

1 Ψ(−n + δ) = − . (4.0.17) δ

Inserting this into the quantization condition of (4.0.14), the self-adjoint parameter is approximated as

1−1 δ λ ≈ κ + 2 − 2 ln(2κ) − 4γ − 2 ≈ . (4.0.18) δ 2

Using the result of δ = 2λ, the Balmer formula is then modiﬁed for small λ in the ` = 0 channel to

1 1 En = . (4.0.19) 2 (n − 2λ)2

4.1 Helium

As mentioned in section 3.1, Grimes and Brown [14, 15] were able to deposit electrons onto the surface of liquid helium as well as drive transitions between the sub bands using microwaves. These electrons were bound not only by attraction to the image charge but also due to the presence of a uniform holding electric ﬁeld. Measuring the transitions as a function of the strength of the holding ﬁeld, Grimes and Brown [14, 15] were able

38 to extrapolate to the zero ﬁeld transition frequencies. In natural units, these transitions were found to be

ω1→2(exp) = 0.3941 and ω1→3(exp) = 0.4652. (4.1.1)

The Balmer approximation to these transitions

ω1→2(theory) = 0.375 and ω1→3(theory) = 0.4444. (4.1.2) is clearly inadequate, but the modified Balmer equation (4.0.19) with λ = 0.0105 is able to reproduce the experimental results to the given number of significant figures. The relative strengths of the 1 → 2 and 1 → 3 transitions were also observed by Grimes and Brown in the experiment. Similar to transitions themselves, this ratio is significantly different from the one calculated using only the regular wave functions at the origin. It would be of great interest to determine if the modified wave functions with λ = 0.0105 are able to more accurately reproduce this ratio. This problem currently remains open for future work. Originally the data collected by Grimes and Brown were interpreted by a model of the liquid helium surface put forth by Cole [1]. It is instructive to demonstrate the equivalence between the Cole model and the self-adjoint analysis for an appropriate range of parameters. In the Cole model, the electrons were assumed to be subjected to a potential described through the piecewise relation

39   V0 for x < 0   V (x) = − Ze2 1 for 0 < x < b. (4.1.3)  8π0 b   2 − Ze 1 for x > b  8π0 x

The 1/x potential is cut oﬀ near the surface due to the fact that the continuum dielectric model breaks down once the electron is close enough to the surface of the liquid. Using Cole’s values, b = 3.6 A,˚ or in Hartree units b = 0.047. The potential barrier for the

liquid helium is given at V0 = 1.3 eV, which corresponds to V0 ≈ 982 in natural units. To

2 2 exploit the form of the bound states in Coulomb potential, deﬁne V0 = ~ Q /2m, so that in natural units Q ≈ 44. Q−1 has the physical interpretation of the penetration depth of the electron. If, instead of the Cole model, the helium is assumed to be impenetrable and the continuum model is allowed to extend to the surface, the self-adjoint method can be applied. Here, the physics associated with the interaction between the electron and the surface of the helium are encoded in the extension parameter at the origin. Despite now having a singularity at x = 0, the self-adjoint method is able to accurately reproduce the results of the Cole model, provided that the parameters b and Q−1 are small enough. To demonstrate this, the solutions of the Cole model must be investigated. Solving the model is straight forward. The solutions are then in the form of

 A exp xpQ2 + κ2 for x < 0    q q ψ = B sin x 2 − κ2 + C cos x 2 − κ2 for 0 < x < b. (4.1.4)  b b   2 c(1 − 2x ln x + ...) + d(x − x + ...) for x > b

Simpliﬁcation can be done by applying the provisions outlined above, namely that both

40 Q κ and 1/b κ. This allows for the κ dependence of (4.1.4) to be neglected. The wave function near the surface of the helium is energy independent for a wide range of energies. Furthermore, requiring that both the wave function and its derivative to be continuous at x = 0 and x = b gives the relation between the c and d coeﬃcients as c = λd, with

1 λ = . (4.1.5) Q + 2 ln b

Thus the results of the Cole model can be reproduced exactly using the self-adjoint method, with the extension parameter set to (4.1.5).

4.2 Alkali atoms

As previously described, alkali atoms oﬀer the ideal experimental systems to compare the quantum defect method for Rydberg states to the self-adjoint extension. Energy transitions for the Rydberg states are easily calculated using the Rydberg equation, while the energy predicted by the self-adjoint extensions must be determined using numerical methods. The exact details of these methods are outlined in Appendix B. Recent measurements of s state transitions in cesium in the range of n = 27 to 45, done by Goy, Raimond, Vitrant and Haroche [24], give a wide range of values to compare the quantum defect method to the self-adjoint extension. The National Bureau of Standards also provides a large source of data for the s orbital transitions in Alkali metals [25]. Comparison to the quantum defects reported in Gallagher [10]. This also allows for direct comparison to the extension parameter using the ﬁtting methods described above. The energy levels that are of interest, along the with the corresponding states, are reproduced in tables 4.1 to 4.4. To keep the units consistent with the cesium data listed in table 4.2, the conversion from eV to Hz was

41 State Energy State Energy State Energy 3 4.881 × 108 6 1.04919 × 108 9 4.44792 × 107 4 2.5409 × 108 7 7.5526 × 107 10 3.56949 × 107 5 1.5553 × 108 8 5.6956 × 107 11 2.92775 × 107

Table 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. Measurements that included orbitals with ` 6= 0 have been excluded. All units are MHz, converted from the original cm−1. State Energy State Energy State Energy 4 4.74424 × 108 8 7.46009 × 107 12 2.90511 × 107 5 2.48641 × 108 9 5.63465 × 107 13 2.42742 × 107 6 1.52854 × 108 10 4.40572 × 107 14 2.05855 × 107 7 1.03416 × 108 11 3.53907 × 107

Table 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. Measurements that included orbitals with ` 6= 0 have been excluded. Units are in MHz, converted from the original cm−1

done on the source data. Following a similar procedure to the cesium data, ﬁtting the extension parameter λ to the data allows for a direct comparison to the quantum defect model.

Table (4.2) has both the ﬁtted λ values as well as the defect values δn, where the

subscripts correspond to the order in the expanded Rydberg equation: δ(n) = δ0 +

2 4 2 δ2/n +δ4/n +.... The residuals refer to the average square residual, expressed in MHz . To simplify the comparison, the quantum defect model was taken only to second order. Moreover, it can be seen that both models have better ﬁts for heavier alkali atoms. This is most likely due to the fact that the spectral data used for the potassium and cesium ﬁtting began at a much higher excited state than either the lithium or sodium. This would then allow for a better approximation of the core as a point charge.

Comparison of the values of the ﬁrst order defects (δ0) to the values of the extension parameter shows the expected relation that δ = 2λ, at least in the Cesium data set. Some care needs to be exercised during this comparison, as λ cannot distinguish the

42 State Energy State Energy State Energy 9 7.07949 × 107 19 1.16294 × 107 29 4.57384 × 106 10 5.38303 × 107 20 1.03608 × 107 30 4.25093 × 106 11 4.23083 × 107 21 9.28891 × 106 32 3.69986 × 106 12 3.41264 × 107 22 8.37519 × 106 34 3.24937 × 106 13 2.81078 × 107 23 7.58994 × 106 36 2.87645 × 106 14 2.35518 × 107 24 6.91019 × 106 38 2.56421 × 106 15 2.00201 × 107 25 6.31782 × 106 40 2.3002 × 106 16 1.72272 × 107 26 5.79849 × 106 42 2.07495 × 106 17 1.49805 × 107 27 5.34065 × 106 44 1.88125 × 106 18 1.31463 × 107 28 4.93499 × 106 46 1.71346 × 106

Table 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. Measure- ments 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.

Orbital Initial State Final State Transition Frequency (MHz) n n0 nS1/2 → (n + 1)S1/2 27 28 510 708.98 ± 0.2 32 33 285 901.94 ± 0.2 33 34 257 744.27 ± 0.2 34 35 233 166.19 ± 0.2 35 36 211 616.43 ± 0.2 36 37 192 643.05 ± 0.2 37 38 175 872.09 ± 0.2 38 39 160 992.71 ± 0.2 39 40 147 745.64 ± 0.2 40 41 135 912.97 ± 0.2 42 43 115 783.74 ± 0.2 44 45 99 441.22 ± 0.2

Table 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 Atom δ0 δ2 λ Residual (δ) Residual (λ) Li7 0.3994 0.0302 3.2738(3) 3.14 × 108 1.23 × 1010 Na23 1.3479 0.0613 0.72057(2) 1.313 × 1012 5.00 × 1010 K39 2.1802 0.136 0.213163(1) 15103.1 2.75 × 107 Cs133 4.0493 0.246 0.0245725(1) 3.502 61.16

Table 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 integer part of the defect. This behavior due to how the models are set up - namely, the quantum defect has the ground state at something other than n = 1. This is also reason for the larger δ0 values in the higher alkali atoms is from the ground state of the valence electron being at a higher principle quantum number; for lithium n = 2, sodium n = 3, and so forth. It is because of this the integer part of the first order quantum defect is n−2; this forces the energy spectrum to have the effective ground state lower than n = 2, which allows the Rydberg equation to mimic the Balmer formula despite ground states at higher n values. As evident in table (4.2), the fitting to the extension parameter λ produces residuals much higher than those done with the quantum defect. To quantitatively compare the models, the Akaike Information Criterion (AIC) is applied. A larger discussion of AIC is contained in Appendix C. The AIC allows for direct comparison of models with different sets of parameters, making it ideal for a more detailed comparison of the extension parameter results to those from the defect. By using the number of parameters into the AIC, the method is able to avoid the inherent bias towards models with a higher number of parameters. Table (4.2) compares the calculated AIC of the two models for each of the alkali atoms studied. With the exception of Na23, the quantum defect model is clearly favorable to the self-adjoint in predicting the energy levels in the Alkali atoms. Even if the second order criterion is used to correct for any small sized samples, the correction is only of 2.8 for Li7, 1.93 for Na23, 0.5 for K39, and 1.6 for Cs133 (here the correction was simply the difference

44 Atom AICδ AICλ ∆ Favored Model Li7 182.09 213.15 31.06 Quantum Defect Na23 312.94 274.99 -37.95 Self-Adjoint K38 294.68 517.93 223.25 Quantum Defect Cs133 29.55 61.16 31.61 Quantum Defect

Table 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. between the correction factor for the quantum defect model and that of the self-adjoint model). None of these come close to having the self-adjoint model being better than the quantum defect for predicting the energy levels in Alkali Atoms.

45 Chapter 5

Graphene

The Hamiltonian for the two dimensional Dirac Coulomb problem is

2 2 Ze 1 HD = −i~cα · ∇ + βmc − , (5.0.1) 4πr0 r where c is the speed of light when describing a fundamental Dirac electron. However, in application to actual graphene systems c = 1.0 × 106 m/s and m = 0 [18, 19]. These parameter values are determined by the double conical intersection in the π band of graphene. The final term in (5.0.1) represents the Coulomb interaction of the Dirac electron (charge −e with a fixed positive charge of Ze). In fundamental QED r = 1. In the application to graphene this corresponds to either the relative permittivity of graphene or the effective relative permeability if the graphene layer is sufficiently close to a substrate. The precise numerical value of the permittivity of graphene is not know, values range from 3.5 to 30 have been proposed in literature. Working in natural units wherein ~ = 1 and c = 1, the Hamiltonian (5.0.1) becomes

κ H = −iα · ∇ + βm − , (5.0.2) D r

46 2 with κ = Ze /4πr0~c being a dimensionless coupling, the ﬁne-structure constant. The parameter m has mass dimension unity, but in the massless limit m = 0 relevant to graphene, the Hamiltonian (5.0.1) has no dimensionful parameters at all. Apart from the obvious symmetry of two dimensional rotational invariance, the Hamiltonian (5.0.1) is also scale invariant in the massless limit m = 0 which is relevant to graphene. Scale invariance would be a powerful constraint on the solutions to eq (5.0.1), but this invariance can be anomalously broken in a way that can be parameterized by the method of self-adjoint extensions. Numerically the ﬁne structure constant is famously κ = Z/137 in fundamental QED, but in graphene κ ≈ 2.2Z/r. Due to this, atomic physics has rel- atively little access to the strong coupling regime (κ ∼ 1 through heavy ion colliders, but in constrast, graphene allows for access to strong coupling regime for easily attainable values of Z.

5.1 Separation of Variables

To completely formulate the model it is necessary to specify the boundary condition.

Introducing the representation where αx = σx, αy = σy and β = σz, where σi are the Pauli matrices. In cylindrical co-ordinates (r, θ), the Hamiltonian (in the limit of m = 0), has the form

  −iθ −κ/r e [−i∂r − (1/r)∂θ] HD =   . (5.1.1)  iθ  e [−i∂r + (1/r)∂θ] −κ/r

Rotational invariance suggests that the solutions should be of the form

47   f   i`θ ψ =   e , (5.1.2) igeiθ with ` ∈ Z and the factor of i in the lower component has been introduced in order to ensure that the functions f and g are both real. The time-independent Dirac equation,

HDψ = Eψ now has the form of two coupled ordinary diﬀerential equations for f and g

∂ ` + 1 κ g + g + − − E f = 0, ∂r r r ∂ ` κ f − f + + E g = 0. (5.1.3) ∂r r r

Using the Frobenius method to solve, the series solutions are given by the forms

−1/2+s −1/2+s+1 −1/2−s −1/2−s+1 g = a0r + a1r + ... + c0r + c1r ,

−1/2+s −1/2+s+1 −1/2−s −1/2−s+1 f = b0r + b1r + ... + d0r + d1r , (5.1.4)

where the coeﬃcients a0 and c0 may be regarded as independent. The other coeﬃcients

are determined in terms of a0 and c0 by means of recurrence relations, which are omitted for the sake of brevity, but it is important to note that the two solutions are related by

` + 1/2 + s ` + 1/2 + s b = a and d = c , (5.1.5) 0 κ 0 0 κ 0

with s being the Frobenius index given by

48 s 12 s = ` + − κ2. (5.1.6) 2

It is best to remain focused on the weak coupling regime, 0 ≤ κ < 1/2, so that s is real valued in all ` channels. There are then evidently two solutions: one corresponding to c0 = 0, and the other a0 = 0. From examination of the leading term in (5.1.4), it is clear that the first solution is less singular than the second at r = 0. Closer inspections reveals that for all values of `, with the expection of ` = 0 and ` = −1, the second solution diverges more rapidly than 1/r at the origin and is unacceptable as a quantum mechanical wave function, as it is not square integrable. This implies that in all channels other than ` = 0 and ` = −1, the boundary conditions at the origin is unambiguous, namely that only the less singular solution is acceptable. In the case of ` = 0 and ` = −1, the square integrability requirement is not sufficient to determine the boundary conditions. The method of self-adjoint extensions is able to parameterize the family of allowed boundary conditions. Which boundary condition is obtained in practice is then determined by experiment. Alternatively, the development of a complete theory to which the Dirac Kepler model is a low energy effective approximation, it would be possible to derive the boundary conditions from this more complete theory. As for the time being, it is best to simply parametrize the possible boundary conditions using self-adjoint extensions.

5.2 Self-Adjoint Extensions

Concentrating on the ` = 0 or ` = −1 case, the Frobenius index simpliﬁes to

r1 s = − κ2. (5.2.1) 4

49 Right away it is evident that the case of κ = 0 corresponds to the free particle, as only the less singular solution remains square integrable: the more singular solution diverges exactly as 1/r. For 0 < κ < 1/2, both solutions are square integrable, which will be analyzed shortly. In the case of κ > 1/2, both solutions are also square integrable. To analyze the self-adjointness of the Hamiltonian (5.0.1), consider two states φ and ψ both in the same ` channel. Take ψ to have the form in eq (5.1.2) and φ to have a similar form

  F   i`θ φ =   e . (5.2.2) iGeiθ

The inner product is

Z 2π Z ∞ Z Z ∞ (φ, ψ) = dθ dr rφ†ψ = 2π dr r (F ∗f + G∗g) . (5.2.3) 0 0 0

It is because of the factor of r that is part of the measure that the wave functions that have a divergence weaker than 1/r are square integrable. Using integration by parts

∗ ∗ ∞ (φ, HDψ) = (HDφ, ψ) + r (F g − fG ) |0 = 0. (5.2.4)

HD is formally self-adjoint: it obeys the self-adjointness condition (φ, HDψ) = (HDφ, ψ) up to surface terms. By locality, separate boundary conditions must be imposed at the origin and at infinity. Square integrability is usually sufficient to determine the boundary conditions at infinity, so the focus remains at the origin. Again, from the Frobenius method, the form of ψ should be (5.1.4), and likewise φ will have a behavior of

50 −1/2+s −1/2+s+1 −1/2−s −1/2−s+1 G = A0r + A1r + ... + C0r + C1r + ...,

−1/2+s −1/2+s+1 −1/2−s −1/2−s+1 F = B0r + B1r + ... + D0r + D1r + .... (5.2.5)

Using these small r expansion of the functions and only keeping the non-zero terms in the limit of r → 0, the surface term has that

∗ ∗ ∗ ∗ −s−s∗ ∗ ∗ s∗−s r (F g − fG ) = (D0c0 − C0d0) r + (B0 c0 − C0 b0) r

∗ ∗ s+s∗ ∗ ∗ s−s∗ + (B0 a0 − A0 b0) r + (D0 a0 − A0 d0) r = 0

The coeﬃcient relations from (5.1.5) can be substituted and simpliﬁed to show that subcritical regime, κ < 1/2,

2s r (F ∗g − fG∗) = (A∗ c − C∗ a ) = 0. (5.2.6) κ 0 0 0 0

As only the coupling range 0 < κ < 1/2 is being considered, s is completely real, and the term proportional to r−2s vanishes. In order to make the second term vanish, the appropriate boundary conditions must be determined. As the boundary condition on ψ

must take the form of a linear relation between a0 and c0, the condition can be written

∗ as a0 = λc0. Imposing this condition forces the condition of A0 = λ C0 if the surface term is to vanish. For the Hamiltonian to be fully self-adjoint, the boundary conditions must be the same for both φ and ψ, which as that λ must be real. Returning to the supercritical regime, κ > 1/2, s is imaginary, the surface term is

51 2s r (F ∗g − fG∗) = (A∗ a − C∗ c ) = 0. (5.2.7) κ 0 0 0 0

Again, the linear boundary condition a0 = λc0 is imposed. This leads to the boundary

∗ condition on φ of A0 = 1/λ C0. Thus for the Hamiltonian to be self-adjoint, the self- adjoint parameter must be a pure phase, a0 = exp(iα)c0.

5.3 Solutions and Phase Shift

To determine the behavior of the full solutions, it is useful to introduce the new functions ω(r) and ν(r) related to ψ through

  ω(r) + ν(r)   s−1/2 i`θ ikr ψ =   r e e , (5.3.1) (ω(r) − ν(r)) eiθ

where k = E/~c, and ω(r) and ν(r) can be taken as the in going and out going waves respectively [18]. Inserting this new ansatz into the Hamiltonian (5.0.1) yields the new coupled diﬀerential equations of

1 r∂ ω − (s + iκ + 2ikr) ω − ` + ν = 0, r 2 1 r∂ ν − (s − iκ) ν − m + ω = 0, (5.3.2) r 2 which have the solutions of

52 −2s ν(r) = A 1F1 (s − iκ, 2s + 1, r) + Br 1F1 (−s + iκ, 1 − 2s, r) , (5.3.3) s − iκ ω(r) = A F (s + 1 − iκ, 2s + 1, r) ` + 1/2 1 1 −s − iκ + B r−2s F (−s − iκ + 1, 1 − 2s, r) , (5.3.4) ` + 1/2 1 1

where A and B are normalization factors and 1F1(a, b, r) is the solution to Kummer’s equation, explained in greater detail in Appendix A. By using the expansion of 1F1 for small r arguments (the expansion is covered in Appendix A), and matching powers, the

relations between the coeﬃcients A and B for ω and ν, and the a0 and c0 for f and g are

s − iκ a s + iκ c k−2s A = 1 + 0 and B = 1 − 0 ` + 1/2 κ(m + 1/2) ` + 1/2 κ(` + 1/2)

Division has that again A/B = λk−2s, with λ the extension parameter for system. Determination of the scattering phase diﬀerence is done through analysis of the

r a−b asymptotic forms equations (5.3.3) and (5.3.4). At large r, 1F1(a, b, r) ≈ Γ(b)[e r /Γ(a)+ (−r)−a/Γ(b − a)], and applying this to the ω and ν, the dominant powers of the radial functions are

Γ(2s + 1) Γ(1 − 2s) eiκ ln 2kr ν = A + λk−2s i−s+iκ Γ(1 + s − iκ) Γ(1 − s − iκ) (2kr)s s − iβ Γ(2s + 1) −s − iκ Γ(2s + 1) ω = A + λk−2s m + 1/2 Γ(s − iκ + 1) ` + 1/2 Γ(−s − iβ + 1) e−2ikre−iκ ln 2kr × (−i)−s−iκ (2kr)s

Again it is evident that rs−1/2ωeikr and rs−1/2νeikr describe incoming and outgoing waves respectively. The phase diﬀerence, given by the ratio of the two, is then

53 ω = e2iδλ(k)−2ikr, δ (k) = β ln 2kr + Arg(λ), (5.3.5) ν λ where

Γ(2s+1) + λk−2s Γ(1−2s) λ = Γ(s+iκ+1) Γ(−s+iκ+1) . (5.3.6) s−iκ Γ(2s+1) −2s −s−iκ Γ(1−2s) `+1/2 Γ(s−iκ+1) + λk m+1/2 Γ(−s−iκ+1)

Here λ is the self-adjoint parameter. By including the irregular solution, the phase diﬀerence now has a much more complex dependence on s,κ, and `, as compared to the phase diﬀerence obtained by taking only the regular solution,

ν = e2iδm(k)+2ikr, (5.3.7) ω

where

1 s − iκ Γ(s − iκ + 1) δ (k) = β ln(2kr) + Arg . (5.3.8) m 2 ` + 1/2 Γ(s + iκ + 1)

This original phase shift is recovered by setting λ = 0, while the largest deviation from the original phase shift can be seen in the limit of λ → ∞. It is interesting to note that in both limits (only regular and only irregular), the phase shift becomes k independent.

54 Chapter 6

Conclusions

The Coulomb problem presented an ideal opportunity to apply and test the method of the self-adjoint extensions on known systems. In the one dimensional and three dimensional non-relativistic Coulomb systems, results of the models were compared to experimental systems of liquid helium and Rydberg states respectively. Spectra collected by Grimes and Brown on electron transitions bound to liquid helium were used to test the self-adjoint method, whereas spectra from alkali metals used for the Rydberg case. Moreover, in the three dimension non-relativistic Coulomb problem, an anomalous breaking of the SO(4) symmetry group, associated with the Runge-Lenz vector conservation, was observed when the self-adjoint method was applied. For the relativistic limit, recent experiments with graphene provided a realization of the two dimensional Dirac Coulomb problem. The self-adjoint method was able to replicate results already known, as well as introduce the possibility for the anomalous breaking of scale invariance. The self-adjoint analysis of the liquid helium was able to reproduce the results of Grimes and Brown [14, 15] as well as the Cole model [1]. Moreover, the self-adjoint analysis is able to do so with one parameter, with the relation between the parameters seen in (6.0.1), where λ is the self-adjoint parameter and Q and b relate the penetration depth and distance from the surface, respectively.

55 1 λ = . (6.0.1) Q + 2 ln b

This is not entirely surprising as the two parameters employed in the Cole model are not completely independent. While the method was successful in the replication of the first order Rydberg-Ritz equation, fitting to known spectra suggest that the Rydberg equation is heavily favored. Although the self-adjoint method did provide a better fit to the Na23 spectrum, the Rydberg formula produced smaller residuals on all other alkali atoms tested. While it is possible that numerical errors exist in the fitting algorithm, it is unlikely that such errors alone could explain the large differences in the residual spectrum. Even without the fitting results, it seems likely that the physical basis for the models would favor the Rydberg equation. The derivation of the Rydberg equation was based off perturbations to the Coulomb potential, whereas the self-adjoint method assumed that the system could be modeled as an electron in a Coulomb potential along with a contact potential at the center. Due to the large electron cloud associate with core of the alkali metals, it seems unlikely that just a contact potential would be able to encapsulate all of the associate interactions between the valence electron and the core electrons. The application of the extensions to graphene shows that the method of self-adjoint extensions does not need to be limited to the Schrödingercase. Analysis of the scattering states off an impurity in graphene including the self-adjoint extension was able to succe- fully reconstructed the results of Shytov, Katsnelson, and Levitov [18]. In addition, the inclusion of the irregular solution in the subcritical case (κ < 1/2) yielded a scattering phase that was energy dependent; implying that the scale invariance was broken, which is not seen the traditional handling of the Coulomb scattering problem. The focus on a Coulomb potential was mainly motivated by the ability to quickly

56 compare results to known systems combined with the broad use of a 1/r potential in various physical models. One of the more promising systems to apply this method to is strong interactions in meson systems. Various models use a linear potential v ∝ r for small distances, as well as a contact potential. As the solutions to such systems are the Airy functions, with only one being accepted as the solution, these systems with linear potentials do hold some promise for the application of the self-adjoint extensions. There is also no reason that this analysis needs to be constrained to only spherically symmetric wave functions. As the main concern for the self-adjoint criterion is that the surface term vanishes, there is no apriori reason to limit this method to spherically symmetric wave functions. In the Schr¨odinger case, the surface term due to the kinetic energy operator can be expressed in the general form of

Z Z dV ∇ · (ψ∗∇φ − (∇ψ∗)φ) = dΩ · (ψ∗∇φ − (∇ψ∗)φ)

due to Stoke’s Theorem, where Ω is the boundary of V . Diﬃculties do arise in how the surface is determined; the most likely solution is to still employ a spherical boundary, and evaluate the various terms in the limit of r → 0. The other possibility, using a surface which exploits the symmetry of the system, would introduce the need to carefully examine the limit of vanishing volume.

57 Appendix A

Special Functions

The discussion of the Hypergeometric equations is based of the work of Morse and Feshbach [27]. Kummer’s equation is of the form

d2f df x + (b − x) − af = 0. (A.0.1) dx2 dx

The solutions are conﬂuent hypergeometric equations. Solutions of the ﬁrst type, which

are regular at the origin have the form 1F1(a, b, x) and can be expressed as the power series of

a 1 a(a + 1) F (a, b, x) = 1 + + x2 + .... (A.0.2) 1 1 b 2! b(b + 1)

Solutions of the second type are singular at the origin may be expanded in a Frobenius series

1−b G(a, b, x) = x 1F1(a − b + 1, 2 − b, x). (A.0.3)

58 Problems arise for the second type when b is an integer value; for the case of b = 1 the two solutions coincide, while for b = 2 the power series in (A.0.2) diverges for each term. These difficulties may be rectified by using a limiting procedure to define the second solution. Defining

1 − a G(a, b, x) = G(a, b, x) − F (a, b, x) (A.0.4) b − 2

as the second solution to Kummers equation, it is well behaved in the limit of b → 2.

Using the series expansions of 1F1 this allows for (A.0.4) to be written as

a − b + 1 1 (a − b + 1)(a − b + 2) G(a, b, x) = x1−b 1 + x + + ... 2 − b 2! (2 − b)(2 − b + 1) 1 − a a 1 a(a + 1) − 1 + x + x2 + ... . (A.0.5) b − 2 b 2! b(b + 1)

Taking the limit of b → 2, the small x behavior of the second solution is

1 G(a, 2, x) = + [1 − (1 − a) ln x] x x 7 3 + a(a − 1) ln x − 1 + a − a2 .... (A.0.6) 2 2 2

The solutions to Kummer’s equation (A.0.1) may also be expressed as the conﬂuent

hypergeometric equations of the third and fourth kind, written as U1 and U2 respectively. Of these, only the solution of the fourth kind decays exponentially as x → ∞, and thus is the only solution of interest. Furthermore it is related to the ﬁrst and second solutions through

59 Γ(1 − a) F (a, b, x) Γ(b − 1)G(a, b, x) U (a, b, x) = eiπx 1 1 + . (A.0.7) 2 Γ(a + b − 1) Γ(a)

In the case of b = 2, G is substituted with G from (A.0.4), and taking the limit of b → 2 has the behavior of U2 is

Ψ(a − 1) + 2γ − 1 U (a, 2, x) = eiπx F (a, 2, x) 2 Γ(a − 1) 1 1 1 + eiπx G(a, 2, x), (A.0.8) Γ(a)

where Ψ(x) = d/dx ln Γ(x) is the digamma function and γ = −Ψ(1) = 0.57721 ... is the Euler gamma, or the Euler-Mascheroni constant. Substitution of (A.0.2) and (A.0.6) into

(A.0.8) give the small x approximation for U2(a, 2, x) In the case of b = 1, it is possible to deﬁne a solution which is well-deﬁned in the limit of b → 1. The solution is given as

1 H(a, b, x) = [ F (a, b, x) − G(a, b, x)] . (A.0.9) b − 1 1 1

The small x behavior is again determined by using the power series of F and G in (A.0.9), then taking the limit as b → 1). This result has small x behavior of the new solution is given as

H(a, 1, x) = ln x + ax ln x + (1 − 2a) ln x 1 3 1 1 + a(a + 1)x2 ln x + − a2 − a + x2. (A.0.10) 4 4 4 4

60 Solutions of the fourth kind in the limit of b = 1 are given by the joining formula (A.0.7),

with the replacement of G by H. U2 can now be expressed as

2γ − Ψ(a) U (a, 1, x) = eiπx F (a, 1, x) 2 Γ(a) 1 1 1 − eiπx H(a, 1, x). (A.0.11) Γ(a)

Finally, the asymptotic behavior of 1F1(a, b, x) is given by

exxa−b (−x)−a F (a, b, x) ∼ Γ(b) + . (A.0.12) 1 1 Γ(a) Γ(b − a)

Large x behavior of the other solutions can be determined by substituting (A.0.12) into the relations given in (A.0.3), (A.0.4), (A.0.7) and (A.0.9).

61 Appendix B

Numerical Fitting

Since the quantization condition derived in (Hydrogen section) was very nonlinear, a numerical fitting algorithm was employed to find the value for extension λ. A sample code is provided, with the data for Cesium being fitted. The header mostly defines constants needed for either unit conversion or counters and limits in the algorithm itself, as well as defining the function relating λ and the energy. The fitting algorithm used minimizes the residual errors under the assumption that there is only one minima possible for the system. The sum of the square residuals between the energies predicted by a test value of λ and the measured values are calculated. The next value of λ is then generated by a step dλ and the sum of residuals is again calculated. The two sums are compared; if the sum at λ + dλ is less than the sum at λ the λ + dλ is taken as the new value. If not, the the step size dλ is redefined as dλ = −dλ/10. The process is repeated until the step size is on the order of 10−15. Initialization of λ was based on the minimized value from the residuals

N d X 2 (λ(E ) − λ ) = 0, dλ n 0 0 n=0 N 1 X λ = λ(E ) = hλi, (B.0.1) 0 N n n=0

62 which has that minimized value is simply the averaged value. However, this minimization process is interested in the prediction of λ from the energies, whereas the predicted energies from a given λ are desired. The average λ is then simply used as the initial value for the minimaztion process. Energies are calculated from the test value using the Newtonian method. Since the derivative of the function is much larger than the function itself, usage of the Newtonian method should be stable for any test value of lambda. The ﬁve iterations used have the precision on the order of O(10−10) or lower, which was observed in all test runs of the program, independent of the data used. The error in λ is calculated by generating random sets of data based on the errors in the measured energy values. The random set is generated from a Gaussian distribution with the mean set to the measured value and the error used as the standard deviation. The ﬁtting process is then used to calculate the best λ for the generated set. This is repeated multiple times, at which point statistics are preformed on the generated λ values; the mean is taken to be the measured value and the standard deviation as the error. Sample code are provided in the next three pages. It uses the spectral data from Potassium33; The energy levels were measured with respect to a ground state of zero. To account for this, the ionization energy is subtracted from these measured energies before comparison to either model.

63 Setting Constants h 4.135667662 10^ 9; plank's constant in Ev MHz c(* 29979245800 ; speed*) of light in cm s Ry= 3.2898284196* 10^9;- (* Rydberg Constant, MHz *) delta= 2.18020 ; (*quantum defect for ns/ in*) K39 d2 = 0.136; second* order(* quantum defect *) hart =27.22138505(* Ev to Hartee ; *) ion= 35 009.8138;(* *) counter= 0; (* *) sum = 0; sumtest = 0; dir = 1; index 1;= = self= adjoint relation lambda x : x 2 2 PolyGamma 1 x 2 Log 2 x 4 EulerGamma ^ 1; lam(* 0; *) e 0;[ _] =  + - * -   - * [ * ] - *  - Raw= data measured= 32 648.3511, 33214.2267, 33598.5597, 33871.4788, 34072.2393, 34 224.2113, 34 342.0150, (*34 435.1762,*) 34510.1190, 34571.3017, 34621.8976, 34664.2161, 34699.9692, 34730.4476, 34756.6407, 34779.3147,= { 34799.0740, 34816.3971, 34831.6690, 34845.2004, 34857.2470, 34868.0180, 34886.3999, 34901.4264, 34913.8657, 34924.2808, 34933.0874, 34940.6009, 34947.0620, 34952.6589 ; deviation 0.0007 c 10^6; } Conversion= for fitting*  For i 1, i Length measured , i , (*measured i ion measured*) i c 10^6 ;  = ≤ [ ] ++ averages[[ extension,]] = ( - intializes[[ energy]]) *  and determines appropriate energy shifts n 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, (* 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 42, 44, 46 ; *) Print= { Length measured , " ", Length n Array k, Length measured ; } [ [ ] [ ]] Initialization[ [ of lambda]] For j 1, j Length n , j , (*e Ry h n j delta*)d2 n j delta ^2 ^2 hart; k j Sqrt 2 e ; Print = k j ≤^2 hart[ ] 2 ++h , " ", measured j ; sum= - k* j^2[[hart]]- 2 h- (measured[[ ]]- j ^2;)   [ ] = [- * ] lam  lambda[ ] * k j ; ;*  [[ ]] Print sum+= ;[ ] *  *  - [[ ]] lam lam+= Length[ [n ]];  Print[lam]; = / [ ] [ ] Generates Random error data num 1000; Array(* meas, Length measured ;*) Array= aiff, Length measured ; Array[lams, num ; [ new lambda]] values Array[Resid, num ;[ ]] [ ] (* *) [ ]

64 2 For i 1, i num, i ,

For =j 1,≤ j Length++ measured , j , meas j RandomVariate NormalDistribution measured j , deviation ; ; [ = ≤ [ ] ++ counter[ ] =0; [ [ [[ ]] ]] ] sum 0; While counter= 15, If =counter 0, dl 0, dl dir 10^ counter ;  < [Finds apporpriate = energy= shifts* based-( on the)] extension index 1; For(* j 1, j Length n , j , *) = For l= 0, l<= 5, l ,[ ] ++ k j lambda k j lam dl lambda' k j ; [ = < ++ aiff[ ]j -= ( k j ^2[ [ hart]]-( 2 + h ;))/; [ [ ]]]

Determination[ ] =  [ ] of* the Residuals *   For m 1, m Length n , m , (*sumtest aiff m meas m ^2; *); [ = ≤ [ ] ++ Ensures+= step ( for[ ]- first run[ ]) ] If counter 0, sum sumtest 1; counter ; (* *) [Comparison of old= and new+ residuals ++] If sumtest sum, lam dl; sum sumtest , counter ; dir 1 ; sumtest(* 0; *) ; [ < += = ++ *= - ] Resid i = sum Length n ; lams i lam; ; [ ] = / [ ] Calculation[ ] = of AIC and mean square Residual for lambda Print "Lambda: ", Mean Table lams i , i, num , (*" ", StandardDeviation Table lams i , i, num *) Print["Avg Residual: ",[ Mean [Table[ Resid] { i , }]]i, num , "AIC: ", Length+/- measured Log Mean [Table [Resid[ i] ,{ i, num}]]] 4 [ [ [ [ ] { }]] Calculation[ of]* AIC[ and mean[ square[ Residual[ ] { for}]]] defect + ] sum 0; For(* j 1, j Length n , j , *) e = Ry h n j delta d2 n j delta ^2 ^2 hart; kj = Sqrt≤ 2 e[ ;] ++ sum= - k* j^2[[hart]]- 2 h- (measured[[ ]]- j ^2;) ;  Print[ ]"Avg = Residual[- * ] defect : ", sum Length n , " AIC: ", Length n Log sum Length n 6 +=  [ ] *  *  - [[ ]]  [ ( ) / [ ] [ ] * [ / [ ]] + ]

65 3

; Resid i sum Length n ; lams i lam; ; [ ] = / [ ] Calculation[ ] = of AIC and mean square Residual for lambda Print "Lambda: ", Mean Table lams i , i, num , (*" ", StandardDeviation Table lams i , i, num *) Print["Avg Residual: ",[ Mean [Table[ Resid] { i , }]]i, num , "AIC: ", Length+/- measured Log Mean [Table [Resid[ i] ,{ i, num}]]] 4 [ [ [ [ ] { }]] Calculation[ of]* AIC[ and mean[ square[ Residual[ ] { for}]]] defect + ] sum 0; For(* j 1, j Length n , j , *) e = Ry h n j delta d2 n j delta ^2 ^2 hart; kj = Sqrt≤ 2 e[ ;] ++ sum= - k* j^2[[hart]]- 2 h- (measured[[ ]]- j ^2;) ;  Print[ ]"Avg = Residual[- * ] defect : ", sum Length n , " AIC:+= ",  Length[ ] * n Log sum*  -Length n [[6]]  [ ( ) / [ ] [ ] * [ / [ ]] + ]

66 Appendix C

Akaike’s Information Criterion

The Akaike Information Criterion (AIC) is a method for comparing models with differing numbers of parameters. AIC was introduced in a series of papers published between 1973 and 1977 [?, 28, 29]. This appendix focuses on how AIC can be used to discriminate between least squares fits to data based on two models potentially with different numbers of parameters.

Suppose that the data consist of a sequence of numbers xi with i = 1, 2, . . . , n. These numbers might for example be energy levels or transition frequencies. Now suppose that there is a model that predicts the values of these numbers in terms of model parameters

denoted θ. The predicted values are denoted as gi(θ). Assuming there are K parameters and θ is just a composite symbol for all of them. We now consider the squared deviation

n 1 X σ2 = [x − g (θ)]2. (C.0.1) n i i i=1

The least squares estimate of the parameters is to minimize σ2 with respect to θ. The best ﬁt value of the parameters is denoted θˆ and the corresponding squared deviationσ ˆ2 is the residual. The AIC for this this model is then given by

67 AIC = n lnσ ˆ2 + 2K. (C.0.2)

Note that the AIC is a score that describes the quality of the fit [28, 30, 31, 32]. It takes into account the number of data points, the residual and the number of parameters. Ob- viously a smaller value of the AIC implies a better fit. Different models can be compared by calculating the difference in their AIC. These differences are generally interpreted according to the following rules of thumb. A difference in the range 0 − 2 means that the empirical support for the two models is essentially comparable. A difference in the range 4 − 7 is considered to favor one model and a difference greater than 10 is generally taken to mean that there is essential no empirical support for the less favored model [32]. For a derivation of AIC the reader is referred to the literature but it is worth noting here that the AIC assumes that deviations of the data from the model have a normal distribution [28, 30, 31, 32]. There is a variant on the AIC called the second order information criterion that is useful when models with large numbers of parameters are being used to fit a small number

of data points [32]. In this variant one computes a quantity AICc given by

2K(K + 1) AIC = AIC + . (C.0.3) c n − K − 1

The second order was not used in this work.

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