* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Resonances in three-body systems S U L
Survey
Document related concepts
Canonical quantization wikipedia , lookup
Matter wave wikipedia , lookup
Scalar field theory wikipedia , lookup
Atomic orbital wikipedia , lookup
Renormalization wikipedia , lookup
Renormalization group wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Elementary particle wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Electron configuration wikipedia , lookup
Tight binding wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Transcript
S TOCKHOLM U NIVERISTY L ICENTIAT T HESIS Resonances in three-body systems Author: Muhammad Umair Supervisor: Svante Jonsell Akademisk avhandling för avläggande av licentiatexamen i fysik vid Stockholms Universitet 28 November 2014 c Muhammad Umair, Stockholm 2014 ISBN XXX-XX-XXXX-XXX-X Distributor: Department of Physics, Stockholm University Abstract Three particles interacting via Coulomb forces represents a fundamental problem in quantum mechanics whose approximate solution provides some insight into the more complex analysis associated with few-body problems. We have investigated resonance states composed of three particles interacting via Coulombic and more general potentials in non-relativistic quantum mechanics, using the complex scaling method. My calculations have been applied to two different physical systems: (i) an investigation of the possibility of resonances in the peµ system, which has been suggested as a possible reason for unexpected results from a recent measurement of the proton radius in muonic hydrogen (ii) a calculation of resonances in positron-hydrogen scattering, which shows that we can represent this system with the accuracy needed for future scattering calculations. The basis set used is built from Gaussians in Jacobi coordinates, thus automatically including mass-polarisation effects which cannot be neglected in muonic systems. My Family List of Papers The following papers, referred to in the text by their Roman numerals, are included in this thesis. PAPER I: A search for resonances in pµe system Umair M., Jonsell S., J. Phys. B. At. Mol. Opt. Phys., 47, 175003 (2014). doi:10.1088/0953-4075/47/17/175003 PAPER II: Natural and Unnatural parity resonance states in positron-hydrogen scattering Umair M., Jonsell S., J. Phys. B. At. Mol. Opt. Phys., 47, 225001 (2014). doi:10.1088/0953-4075/47/22/225001 Author’s contribution My contribution to the work reported in this thesis is substantial. In the following I will try to summarize my individual contribution to the presented work: Paper I. I have actively taken part in adapting the code, and adding the complex scaling part as well as including the correction due to the vacuum polarization. I analyzed the data and wrote the article with the close collaboration of my supervisor. Paper II. In this paper, I did all the calculations and wrote the article. Contents Abstract List of Papers i iii Author’s contribution v Abbreviations ix 1 Introduction 1 2 The Proton Radius Problem 2.1 Muonic hydrogen and the proton radius . . . . . . . . . . . . . . . . . . . . . 2.2 Why the proton radius is important . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Proton radius puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 4 3 Theory 3.1 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Linear Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Hylleraas-Undheim-MacDonald Theorem and the Generalized Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Theory of Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Stabilization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Complex Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Modelling of Quantum Three-Body Systems . . . . . . . . . . . . . . . . . . . 3.3.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Mass Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 The Couple Rearrangement Channel method . . . . . . . . . . . . . . 3.4 Vacuum Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 6 6 7 8 10 10 11 14 21 4 Negative Hydrogen Ion 4.1 The pµe Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 27 5 Positron-Hydrogen Scattering (e+ H) 29 Summary and Outlook References Abbreviations Ps QED RMS S.E µp ep CRC pee Positronium Quantum Electrodynamics Root mean square Schrödinger equation Muonic hydrogen Electron proton scattering Couple Rearrangement Channel Negative hydrogen 1. Introduction In this thesis we apply non-relativistic quantum mechanics to study three-body problems. We use interaction potentials of Coulombic or Gaussian forms. However, the formal and numerical approaches, used in this work, are in principle applicable to any three-body system. We are interested in how the energies and widths of resonances scale with the masses of the particles. In our calculations one particle is a proton, while the two other particles are negatively charged with masses between the electron and the muon mass. The origin of the distinctive energy and size of these so called exotic atoms lies in the energy difference between atomic and nuclear physics. It provides an opportunity to undertake investigations of different basic physical phenomena and quantities pertaining to atomic or nuclear physics e.g. the proton radius, the pion mass, strong interaction scattering lengths and so on. The primary motivation for our work is recent experiments on muonic hydrogen [1, 2]. Muonic hydrogen µ − p has been interest for a long time since a measurement of the Lamb shift (2S-2P energy difference) could give a precise value for the root mean squared (RMS) charge radius of the proton. An isolated (µ − p)2S is metastable with a life time mainly determined by muon decay (about 2.2 µs). The 2S-2P splitting in muonic hydrogen arises mainly through quantum electrodynamics (QED) effects such as vacuum polarization. It does however also depend on a finite-size correction of the nuclear charge distribution. Recent experiments have measured the correction and thereby determined a new value of the proton radius, 0.84184 fm [1]. The proton radius is being recognized as a basic property of the simplest nucleus, the proton, and treated in the published 2006-CODATA [4] adjustment as a fundamental physical constant, but with a 5 standard deviation away from this recent (and presumably more accurate) measurement. Various experiments have been performed to examine the formation of a longlived meta-stable state of µ p when muons are stopped in a low-pressure hydrogen target gas. In combination with the recent advances in hydrogen spectroscopy [3], such an experiment will enable assessment of bound state QED to a new level of precision. We also calculate resonances in the positron-hydrogen system (or equivalently the charge conjugate electron-antihydrogen system). These resonances could play an important role in the production of anti-hydrogen atoms through rearrangement scattering of positronium (Ps) by anti-protons [5]. Recently, attention has been drawn to the existence of chemical compounds between anti-matter and matter. The bound or quasi-bound state (resonance) of a positron or a Ps and atoms or molecules is a subject of intense studies [9–13]. Unfortunately, experiment has not provided us with data about the positron-hydrogen scattering [6–8]. The excited Ps is interesting since it probably gives a much higher formation cross-section. The process relating to cross-sections e+ + H → Ps + p (1.1) Ps + p̄ → e− + H̄ (1.2) 1 are of great importance since anti-hydrogen atom reflects an ideal system to study charge conjugation symmetries of physics. Other exotic atoms, for example, muonium, positronium and protonium have short lifetimes and are not so appropriate for high-precision spectroscopic research. Nevertheless, there is a growing interest for studying various effects of positron-hydrogen scattering, especially for comparison with electron-hydrogen scattering (though we do not really pursue this). Outline Chapter 2 gives an overview of the proton radius problem, its origin and why it is important. In chapter 3, we describe the computational method used for the calculation in Paper I-II. In chapter 4, negative hydrogen ion results are discussed and also scaling the mass of one of electrons to make this a pµe system. In chapter 5, positron-hydrogen system results are discussed. Chapter 6 gives a summary and outlook of the work. 2 2. The Proton Radius Problem Since the inception of quantum theory, the hydrogen atom has been an important system, which can be solved exactly. Due to its relatively simple atomic structure, comprising only one electron and a proton, the hydrogen atom has been important to the development of the theories of atomic structure, quantum mechanics, and quantum electrodynamics (QED). For example, the small observed deviation of the 2S-2P energy splitting from the prediction of the Dirac equation, called the Lamb shift, established the development of QED [1, 15]. The ambiguity related to the proton radius was extracted from H-spectroscopy experiments, which had been the most accurate determination of the hydrogen energy levels and accordingly were limiting the comparison between theory and measurements. So, to further examine the bound state QED representing the hydrogen energy levels it was necessary to have a more accurate verification of the proton radius. The most recent experiment to determine the proton radius study the 2S → 2P transition in muonic hydrogen [1, 15]. The very accurate results from these experiments do not agree with previous determination of the proton radius [17]. This has motivated our investigation into possible exotic states involving 2S muonic H. 2.1 Muonic hydrogen and the proton radius A muon is just like an electron in all regards with the exception of that it is heavier and has a finite life time. Since the muon is heavier than the electron (about 200 times) and muonic hydrogen is about 200 times smaller than ordinary hydrogen. Therefore, the proton is having a diameter of about 60,000 times smaller than ordinary hydrogen, only 300 times smaller than muonic hydrogen. As a consequence, the influence of the finite nuclear size on the energy levels is significantly increased. That makes the details of muonic hydrogen more sensitive to the size of the proton, and thus allows for a more precise measurement. Hence muonic atoms represent a unique laboratory for the determination of RMS radii and other nuclear properties. 2.2 Why the proton radius is important The small uncertainty in the new proton radius value (1 × 10−18 m) has opened the way for checking bound-state QED calculations in hydrogen to an unprecedented level of accuracy (3 × 10−7 ). This is very interesting, since bound state QED is challenging and both are part of technical and fundamental aspects. In summary, the new proton radius value will lead to: • An order-of-magnitude more accurate analysis of the theory explaining hydrogen energy levels. 3 • An order-of-magnitude correction to the Rydberg constant (to a relative level of 1 × 10−12 ), which is a vital part of the constants adjustment [4]. • A criterion for lattice QCD calculation aiming to model the proton, starting from quarks and their interactions [18]. • Confrontation with the electron-proton scattering domain, which is the historical way used to determine the proton radius [19]. 2.3 Proton radius puzzle The proton radius puzzle is the disagreement between the proton charge radius determined from muonic hydrogen and that determined from electron-proton scattering and hydrogen spectroscopy. Up until 2010, the accepted value for the proton radius was 0.8768(69) fm, determined from atomic hydrogen measurements in the 2006 CODATA analysis [4]. The prime result obtained from ep scattering was 0.879(9) fm, from the analysis of Sick [20]. The first experiment using muonic hydrogen by Pohl et al. [1], obtained a value of the proton radius of 0.84184(67) fm. Recently, updated results [2] give a value of 0.84087(39) fm. This new result is consistent with the first measurement [1] which virtually eliminates the possibility of experimental error. But both measurements strongly disagree with the hydrogen spectroscopy and ep scattering results. The 5σ discrepancy compared to the earlier, less precise measurement has attracted much attention. Presently, physicists all over the world are searching for the solution to this problem, generally referred as a "Proton Radius Puzzle". It is obvious from these numbers that the discrepancy is severe, and it is difficult to imagine an effect that could shift the resonance position by 5 standard deviations. Jentschura [15] suggested that the presence of an electron could result in a shift of the resonance position if the distance between the electron and the µ p(2S) atom was about one Bohr radius. He suggested that the spectroscopy might have been carried out not on a µ p(2S) atom, but on the molecular ion (pµe)− . It is therefore important to search for (pµe)− resonances using theoretical calculations. One such study was performed by Karr and Hilico [21], where they found a resonance in (pµ µ)− , but this resonance does not survive as the muon mass is scaled mµ → me . In our work, we have used another numerical technique and discovered additional resonances in the (pex)− and (pµx)− systems, where x is a muon with scaled mass. 4 3. Theory 3.1 Variational Principle The variational principle [22, 23] is a numerical minimization technique used to find the approximate solutions of many-body problems by finding the best possible approximation to the true ground state using trial wave functions of a certain form. 3.1.1 Linear Variational Principle We deal with the stationary many-body Schrödinger equation HΨ = EΨ and represent the eigenfunctions and eigenvalues by Ψn and En , respectively. The well-established Rayleigh-Ritz variational principle states that the variational energy ε evaluated with an arbitrary trial-function Ψt provides an upper bound to the exact ground state energy of the Hamiltonian H, i.e. ε= hΨt |H|Ψt i hΨ|H|Ψi ≥ E0 = , t t hΨ |Ψ i hΨ|Ψi (3.1) where E0 is the exact ground state energy. In order to find approximate variational solutions to the full Schrödinger equation, we restrict the problem to a smaller space spanned by a set of K basis functions ψk . Firstly, we expand the wave-function in this space, i.e. K Ψt = ∑ ck ψk (3.2) k=1 where ck are the expansion coefficients. These coefficients determine Ψt completely within the space {ψk }. How do we use the variational principle in practice to find the best Ψt , i.e. the values of ck which gives the lowest ε? To minimize ε we must have ∂ε =0 ∂ ck f or all k = 1, 2.....K (3.3) From eq.(3.1) and eq.(3.2), we have ε= ∑Ki=1 ∑Kj=1 ci c j Hi j ∑Ki=1 ∑Kj=1 ci c j Si j (3.4) 5 where, Hi j = hψi |H|ψ j i, (3.5) Si j = hψi |S|ψ j i Hence, K K K K ε ∑ ∑ ci c j Si j = ∑ ∑ ci c j Hi j i=1 j=1 (3.6) i=1 j=1 Taking the derivative w.r.t, cn K K ∂ε K K c c S + ε (c S + S c ) = ∑ ∑ i j i j ∑ i in ni i ∑ (ci Hin + Hni ci ) ∂ cn i=1 j=1 i=1 i=1 Since ∂ε ∂ cn (3.7) = 0 and Sin = Sni , Hin = Hni , one has K K ε ∑ ci Sin = ∑ ci Hin (3.8) H̃ c̄ = ε S̃c̄ (3.9) i=1 i=1 or where, H̃ and S̃ are the matrix representations of the Hamiltonian and overlap operators and their elements are defined in eq.(3.5). This is a generalized eigenvalue equation of size K, with K real eigenvalues ε1 ≤ .... ≤ εK and corresponding eigenvectors c1 , ..., cK . 3.1.2 Hylleraas-Undheim-MacDonald Theorem and the Generalized Eigenvalue Problem The Hylleraas-Undheim-MacDonald theorem states that for a trial wave function with linear variational parameters such as (3.2), the higher eigenvalues εi > ε0 of eqn.(3.9) are upper bounds to the excited states of the Hamiltonian. This result is based on a theorem showing that as the number of basis functions is extended from N to N +1, the new eigenvalues interleave the former, i.e. ε0N+1 ≤ ε0N ≤ ε1N+1 ≤ ε1N ≤ ...... Thus all eigenvalues, ordered according to increasing energy, must decrease as the number of basis functions N is increased. As N → ∞ they will approach the true eigenvalue of the Hamiltonian. It follows that for a finite N, εiN is an upper bound to Ei as shown in figure (3.1). 3.2 Theory of Resonances The identification of continuum and resonant states are two significant aspects in quantum physics. Generally, a resonant state is explained as a long-lived state of the system which has sufficient energy to divide up into two or more subsystems [35]. While conducting an experiment, a particle is scattered from the target. It can be an electron, atom or molecule and the 6 Figure 3.1: Approximate eigenvalues given by the Rayleigh-Ritz variational method with linear functions. Each root εiN of the determinal equation (3.9) is an upper bound on the corresponding exact eigenvalue Ei . target can be a nucleus or any of the above. Three different processes can occur; one is elastic scattering in which the energy of the particle is conserved. Another is inelastic scattering, in which energy is exchanged between the particle and the target. The last is reactive scattering in which the particle and the target collide with each other and form a different species. There are numerous methods used to calculate the energy and the lifetime of a resonance, here we discuss some of them. 3.2.1 Stabilization Method The stabilization method provides an efficient approach to many problems in atomic and molecular physics [24, 31, 32]. This method deals with the real matrices and real basis functions, which directly exploits the locality of the resonance in the interaction region. Applying this stabilization technique, we introduce a real scaling parameter α by the transformations (for the Coulombic potential): T V (3.10) , V→ , 2 α α Where T and V are the kinetic and potential energy matrix elements, respectively. When we change the value of α, we get a stabilization graph as shown in figure (3.2) from which the resonance energies can be analysed. Eigenvalues E j (α) corresponding to bound states or resonances are stable with respect to variation of α. The stabilization method diagonalizes the Hamiltonian r → rα, T→ 7 Figure 3.2: stabilization graph showing resonance states of pee below the n = 2 threshold with J=0. in a basis set of ever larger extension around the region where the wave function of the resonance is localized. If the energies decrease for each region, these states are known to be the continuum states, and do not possess any bound state properties. Also the wave-vector k scales as k → k/α, meaning that the energy of a continuum state scales as Ek = h̄k2 /2m → (1/α 2 )Ek . 3.2.2 Complex Scaling The Complex scaling method is a powerful tool in the numerical study of resonances in few electron systems [33–40]. The Complex L2 method mathematically transform the Schrödinger equation itself, avoiding the asymptotic boundary problems for resonances. This method enables us to calculate the resonance energy and width directly, as real and imaginary parts of the complex eigenvalues. It helps us to investigate all the resonances that lie inside the energy region of interest using the complex rotated Hamiltonian. In such a method the radial coordinates r j are transformed into r j → r j eiθ (3.11) ∇ j → ∇ j e−iθ (3.12) and accordingly where θ is the scaling angle which is restricted to 0 < θ < π/4. The wave-function φ (r) transforms under the complex dilatation operator U(θ ) by definition as U(θ )φ (r) = e3iθ /2 φ (eiθ r) (3.13) The factor e3iθ /2 gives the scaling of three dimensional volume element dV = (dxdydz). The scaled Hamiltonian is defined as H(θ ) 8 H(θ ) = U(θ )HU −1 (θ ) (3.14) H(θ ) = e−2iθ T + e−iθ V (3.15) and the transformed Hamiltonian is where e−i2θ , e−iθ are complex numbers which scale the kinetic and potential energy, H(θ ) has complex eigenvalues with L2 eigenfunctions corresponding to the resonant states of the system. When we apply this transformation to the Hamiltonian operator the S.E transforms to [35] U(θ )HU −1 (θ ) (U(θ )φres ) = (Er − iΓ/2) (U(θ )φres ) Where Er and Γ represent the position and width of the resonance state, respectively, and φres represents the diverging spherical out-going eigenfunction corresponding to the complex eigenvalues of the resonance. • The bound state eigenvalues obtained from H(θ ) are independent of θ and similar to those of H(θ = 0) for |θ | ≤ π/2. • The continuous spectrum at each scattering threshold is rotated downward making an angle of 2θ . • H(θ ) may have isolated complex eigenvalues corresponding to the resonances energies with L2 square integrable complex eigenfunctions. To understand this technique we start with an example for short-range potential. In such cases, the solutions for scattering states have asymptotic behaviour given by φ scatt (r → ∞) = A(k)e−ikr + B(k)e+ikr where E= (3.16) (h̄k)2 2m (3.17) The complex-scaled scattering states, r → reiθ are given by φ scatt (reiθ )r→∞ = A(k)e−ike iθ r + B(k)e+ike iθ r (3.18) Here we see in eq.(3.18), one of the exponentials diverges as r → ∞, violating the boundary condition for a finite wave function. To preserve this asymptotic form we take k as complex values k → ke−iθ , then E = (h̄k)2 /2m → e−2iθ (h̄k)2 /2m for the allowed scattering eigenenergies. The continuum is then rotated into the lower half of the complex energy plane by an angle of 2θ shown in fig. (3.3). The bound states are unaffected by the rotation and lie on the negative real energy axis. This shows that the continuous spectrum of H(θ ) is different from that of the unscaled Hamiltonian H which is the main purpose of r → reiθ transformation. 9 Dilatation Transformation σ(H(θ)) σ(H) { { Thresholds Bound states Bound states Resonance (Hidden) 2θ Resonance (Exposed) Figure 3.3: Effect of dilatation transformation r → reiθ on a spectrum "σ " of many-body Hamiltonian. Bound states and thresholds are invariant. However, as the continua rotate, complex eigenvalues may be exposed. Such eigenvalues correspond to poles but are "hidden" if θ = 0, and will be exposed if the cuts are appropriately moved. 3.3 3.3.1 Modelling of Quantum Three-Body Systems Coordinate System We know that a system with N particles has 3N degrees of freedom. If the particles are not aligned on a straight line, the degrees of freedom may be reduced to 3N − 6 [22] by separating out the center of mass motion and rotations about the center of mass. Thus, for a three-body system three coordinates are needed to describe the degrees of freedom. Various different coordinate systems are available when discussing three-body systems. Some of them are Jacobi coordinates [24–26], hyper-spherical coordinates [24, 27, 28] and Pekeris [29, 30] coordinates etc. All these coordinate systems are good when considering the bound state calculation of the Schrödinger equation. Hyper-spherical coordinates consist of a hyper-radius providing the size of the cluster and two hyper-angles describing the radial and angular correlation of the three-body system respectively. This coordinate system is particularly interesting since the three dimensional problem now reduces to a one dimensional hyperradius problem, with a set of effective potential obtained by solving a two dimensional equation [27, 28]. Pekeris coordinates are especially convenient when describing the wave function and the relative importance of different configurations of the system. The most suitable choice of the coordinate system also depends on which physical system is studied, and may be important for the efficiency of the numerical treatment. Jacobi coordinates are one of the suitable choices for describing scattering processes and work equally well in bound state calculations for three-body systems. To define a Jacobi coordinate, R = (x, y) has been used to describe the internal motion of the system shown in figure 10 Figure 3.4: The Jacobi coordinates for a three body system. (3.4). Here, x is the vector between particle 2 and 3, y is the vector between particle 1 and center of mass of the pair (2,3), θ is the angle between the vectors x, y. The inter-particle distances, ri j are associated to the Jacobi coordinates (in one possible configuration). r23 = x, r12 = y2 − r13 = y2 + m3 2m3 xy cos θ + ( x)2 m2 + m3 m2 + m3 1/2 2m2 m2 xy cos θ + ( x)2 m2 + m3 m2 + m3 1/2 , , Here m2 and m3 are the masses of particles 2 and 3 respectively. An advantage of using Jacobi coordinates is their orthogonality, i.e. the kinetic energy operator is diagonal, meaning that there is no mixing of the different derivatives. 3.3.2 Mass Polarization To understand the phenomena of mass polarization [47] consider an atom or ion containing a nucleus of mass M and charge Ze and N electrons of mass m and charge −e. The coordinate of the nucleus with respect to the fixed origin O is denoted by R0 , and R1 , R2 .....RN those of the electrons. In the absence of external fields, and neglecting all but the coulomb interactions, the non-relativistic Hamiltonian operator of this system is given by H = T +V (3.19) 11 Figure 3.5: Coordinate System for two-electron atoms. Where, the K.E operator, T is T =− h̄2 2 h̄2 N ∇R0 − ∇2Ri ∑ 2M 2m i=1 (3.20) and the Coulomb energy V is the sum of all the (N + 1) particles of the system. In order to separate the motion of the center of mass, we change our coordinates from (R0 , R1 , R2 ....RN ) to (R, r1 ....rN ) where R= 1 (MR0 + mR1 + ...... + mRN ) M + Nm (3.21) is the coordinate of the center of mass and ri = Ri − R0 , i = 1, 2, ...N (3.22) are the relative coordinate of the electron w.r.t, the nucleus. It can be shown from eq.(3.21) and eq.(3.22) that using the chain rule ∇R0 = 12 N M ∇R − ∑ ∇ri M + Nm i=1 (3.23) m ∇R + ∇ri M + Nm ∇Ri = (3.24) Hence, ∇2R0 = M M + Nm 2 N 2M ∇2R − ∑ ∇R · ∇ri + M + Nm i=1 !2 N ∑ ∇r i (3.25) i=1 and ∇2Ri = m M + Nm 2 ∇2R + 2m ∇R · ∇ri + ∇2ri M + Nm (3.26) Inserting the expression eq.(3.25) and (3.26) and eq.(3.20), we find the kinetic energy operator in the new coordinates becomes T =− h̄2 h̄2 ∇2R − 2(M + Nm) 2µ h̄2 N ∑ ∇2r − M ∑ ∇r · ∇r i i i=1 j (3.27) i> j where µ= mM m+M (3.28) is the reduced mass of the electron with respect to the nucleus. The Hamiltonian eq.(3.19) may therefore be written as H =− h̄2 h̄2 ∇2R − 2(M + Nm) 2µ N ∑ ∇2ri − i=1 h̄2 ∑ ∇ri · ∇r j +V (r1 , r2 ...rN )(3.29) M i> j The only term containing the coordinate R in eq.(3.29) is the first one, which shows the kinetic energy operator of the center of mass. The next represents the sum of the kinetic energy operator of the N electrons. The third term is due to nuclear motion, which is often called the "mass polarization" term. Note that this term is smaller than the electronic kinetic energy by a factor µ/M. Thus for normal atoms this is a small (though sometimes important) correction. For exotic systems involving heavier particles, such as muons, this term is larger and cannot be neglected. Now, we solve this three-particle problem using Jacobi coordinates defined by 1 (MR0 + mR1 + mR2 ), M + 2m x = R1 − R0 , 1 y = R2 − (MR0 + mR1 ), M+m R= (3.30) (3.31) (3.32) Here R is the position of the center of mass of the atom, x is the position of electron relative to the nucleus, and y is the position of the second electron relative to the center of mass of the 13 other two particles. The kinetic energy operator becomes T =− h̄2 2 h̄2 2 h̄2 2 ∇R0 − ∇R1 − ∇ 2M 2m 2m R2 (3.33) The derivatives in eq.(3.33) transform according to M M ∇y + ∇R ), M+m M + 2m m m = (∇x − ∇y + ∇R ), M+m M + 2m m = (∇y + ∇R ), M + 2m ∇R0 = (−∇x − ∇R1 ∇R2 Now, the kinetic energy operator gives 2 h̄ M + m M + 2m h̄2 h̄2 T= − ∇2x − ∇2y − ∇2R 2 Mm 2 m(M + m) 2(M + 2m) And, the total Hamiltonian becomes 2 h̄2 M + 2m h̄2 h̄ M + m 2 2 2 ∇x − ∇y − ∇ +V (x, y) H= − 2 Mm 2 m(M + m) 2(M + 2m) R (3.34) (3.35) Equation (3.35) has an important advantage that there is no mass-polarization term in the kinetic energy part, but at the expense of making the expression for the potential more complicated as we do not use inter-particle coordinates. We note though that for pairwise interactions the potential can be written as V (xi , yi ) = V (x1 ) +V (x2 ) +V (x3 ) (3.36) where i in xi denotes the three different ways (or rearrangement channels) in which the Jacobi coordinates can be defined. To evaluate the potential matrix elements, we thus need to be able to transform between different rearrangement channels. 3.3.3 The Couple Rearrangement Channel method In order to solve the many-body bound state problem accurately different methods are used. The Couple Rearrangement Channel (CRC) Method is one of them, and was first introduced by M. Kamimura [41] in 1988. This method has been applied to few-body problems of different types in atomic and nuclear physics. The main advantage of this method is that it includes all three possible sets of Jacobi coordinates and can thus describe different rearrangement channels. The detailed computational explanation of the CRC method is discussed below. The total three body wave function is expanded in terms of basis functions spanning the three rearrangement channels in the Jacobian coordinate system α = a, b, c shown in figure (3.6). ΨJM = ∑ cµ φµ µ 14 (3.37) r ,l a a R R ,L b b R ,L a a r c , l b b , L c r , l c c Figure 3.6: The three arrangement channels of three body system and their Jacobi coordinates. 2 2 φµ = Nα rαlα RLαα e(−rα /rαi ) e(−Rα /RαI ) Ylα (r̂α ) ⊗YLα (R̂α ) JM (3.38) where Ylα (r̂α ) ⊗YLα (R̂α ) JM ≡ ∑ hlα m Lα M|J MJ iYl αm (r̂α )YLα M (R̂α ) (3.39) m,M In eq.(3.38) lα (Lα ) stands for the angular momentum of the relative motion associated with the coordinate rα (Rα ) and the [ ]JMJ represent the vector coupling of the spherical harmonics. Also lα (Lα ) are limited as 0 ≤ lα ≤ lαmax , |J − lα | ≤ Lα ≤ J − lα . In the term on the right hand side of eq.(3.39) h i is the Clebsch-Gordan coefficient. The non-linear variational parameters rαi and RαI are chosen as a geometric progression rαi = rα1 rαn rα1 i−1 n−1 , i = 1, · · · , n (3.40) RαI = Rα1 Rαn Rα1 I−1 N−1 , I = 1, · · · , N This choice gives a good balance between many basis functions in the inner short-range region, and a few diffuse Gaussian which capture the long-range behaviour. Also, an advantage of Gaussian basis functions is that they allow analytical calculation of the kinetic, potential energy and overlap matrix elements in the Jacobian coordinate system. The requirement < φµ |φµ > =1 gives the normalization constant Nα " 2lα +2 Nα = √ π(2lα + 1)!! 2 rαi lα +3/2 2Lα +2 √ π(2Lα + 1)!! 2 RαI Lα +3/2 #1/2 (3.41) Note however that the basis functions are not orthogonal < φµ |φν >6= 0 , which makes it necessary to, in addition to H, also calculate the overlap matrix S, and solve a generalized eigenvalue problem. 15 The main difficulty in constructing the Hamiltonian and overlap matrix lies in the transformation between different sets of Jacobi coordinates, as both functions in a matrix element must be expressed in the same coordinate system. For the Coulomb potential (and more generally potential on the form rN ) and Gaussian potentials, all integrals can be evaluated analytically. In order to calculate matrix elements between basis functions of different channels α, one of the functions is projected onto the channel coordinates of the other [42, 43]. We use a coordinate transformation (rβ , Rβ ) →(rα , Rα ) in the form rβ = γβ α rα + δβ α Rα , 0 (3.42) 0 Rβ = γβ α rα + δβ α Rα 0 0 Here γ, δ , γ , δ are kinetic mass factors. The formula below transforms the factor rl RL [Yl (r̂) ⊗ YL (R̂)]JM appearing in the basis function (3.38) from the channel β to channel α [44]: p 4π(2L + 1)! rLYLM (θ , φ ) = rL−λ rλ L 2 1 ∑ (−1)λ p(2λ + 1)!(2(L − λ + 1)!) λ =0 × [YL−λ (Ω1 ) ⊗Yλ (Ω2 )]LM (3.43) where r = r1 − r2 · Thus using eq.(3.42), we have q 4π(2lβ + 1)! = l rββ Ylβ mβ (r̂β ) l −λ lβ l −λ γββα δβλα rαβ ∑q Rλα (2λ + 1)!(2(lβ − λ ) + 1)! h i × Ylβ −λ (r̂α ) ⊗Yλ (R̂α ) λ =0 lβ mβ q 4π(2Lβ + 1)! = L Rββ YLβ Mβ (R̂β ) 0 Lβ ∑q (3.44) L −Λ Λ Rα 0 γβ α Lβ −Λ δβ α Λ rαβ (2Λ + 1)!(2(lβ − Λ) + 1)! h i × YLβ −Λ (r̂α ) ⊗YΛ (R̂α ) Λ=0 Lβ Mβ (3.45) The product of these two expressions forms the angular part of the basis-function. We can combine them and using from section (5.16) in [44]: h i 0 0 Yl (Ω1 ) ⊗Yl (Ω2 ) 1 0 00 0 2 00 ∑ hL L M M LM 16 0 LM h i 00 00 Yl (Ω1 ) ⊗Yl (Ω2 ) 0 1 |LMi ∑ Bl10 l20 L0 l1 l2 2 00 00 00 l1 l2 L l1 l2 L 00 L M 00 = [Yl1 (Ω1 ) ⊗Yl2 (Ω2 )]LM (3.46) Bl10 l20 L0 1 4π q 0 00 0 00 0 00 (2l1 + 1)(2l1 + 1)(2l2 + 1)(2l2 + 1)(2L1 + 1)(2L1 + 1) 0 00 l1 l1 l1 0 00 0 00 0 00 ×hl1 l1 0 0|l1 0ihl2 l2 0 0|l2 0i · l l l 20 200 2 L L L = 00 00 00 l1 l2 L l1 l2 L (3.47) Thus: i h l L rββ Rββ Ylβ (r̂β ) ⊗YLβ (R̂β ) = JM = ∑ l mβ Mβ L hlβ Lβ mβ Mβ |JMirββ Ylβ (r̂β )Rββ YLβ (R̂β ) ∑ q hlβ Lβ mβ Mβ |JMi4π (2lβ + 1)!(2Lβ + 1)! lβ Lβ mβ Mβ × 1 ∑∑q λ =0 Λ=0 l −λ (2λ + 1)!(2Λ + 1)!(2(lβ − λ ) + 1)!(2(Lβ − Λ) + 1)! 0 l +Lβ −λ −Λ λ +Λ Rα 0 ×γββα γβ α Lβ −Λ δβλα δβ α Λ rαβ × L L ∑ hlβ Lβ mβ Mβ |L ML i ∑ Bll −λ λl α L ML lα Lα β α β Lβ −Λ Λ Lβ Ylα (r̂α ) ⊗YLα (R̂α ) LM (3.48) We now use the orthogonality of the Clebsch-Gordan coefficients ∑ hlβ Lβ mβ Mβ |J Mihlβ Lβ mβ Mβ |L ML i = δJL δMML mβ Mβ which gives h i l L rββ Rββ Ylβ (r̂β ) ⊗YLβ (R̂β ) = JM lb (2lβ + 1)(2Lβ + 1) Lb ∑∑ λ =0 Λ=0 lβ −λ 0 Lβ −Λ s (2lβ )! (2λ )!(2lβ − 2λ )! 0 s (2Lβ )! (2Λ)!(2Lβ − 2Λ)! l +L −λ −Λ ×(γβ α ) (γβ α ) (δβ α )λ (δβ α )Λ rαβ β Rλα+Λ lβ − λ Lβ − Λ lα λ Λ Lα ×∑ hl − λ Lβ − Λ 0 0|lα 0ihλ Λ 0 0|Lα 0i β lα Lα lβ Lβ J × Ylα (r̂α ) ⊗YLα (R̂α ) JM (3.49) where the nine − j symbol {} gives the coupling between the four angular momenta lα Lα , lβ Lβ . This expression can be summarized in the form defining T ≡ λ + Λ 17 i h ˆ Ylβ (rˆβ ) ⊗YLβ (Rβ ) l L rββ Rββ ∑ ≡ JM l +Lβ −T < lβ Lβ J |lα Lα T J >β α rαβ RTα Ylα (r̂α ) ⊗YLα (R̂α ) JM (3.50) lα ,Lα ,T and the transformation coefficients < lβ Lβ J |lα Lα T J >β α can calculated and stored prior to the computation. This equation is key to the transformation between channels. Now we can write the product of two different channel’s trial wave-functions in the coordinates of a single channel (channel α) by applying the transformation above to one of the basis function. φµ∗ (rα , Rα )φν (rβ , Rβ ) = † Nα rαlα RLαα exp[−(rα /rαi )2 − (Rα /RαI )2 ] Ylα (r̂α ) ⊗YLα (R̂α ) JM 0 1 0 1 2 2 ×Nβ exp − 2 (γβ α rα + δβ α Rα ) − 2 (γβ α rα + δβ α Rα ) rαi RαI h i 0 0 l +L +T × ∑ hlβ Lβ J |lα Lα T Jiβ α rαβ β RTα Yl 0 (r̂α ) ⊗YL0 (R̂α ) 0 α 0 α lα ,Lα ,T 0 JM (3.51) 0 where lα and Lα denote the auxiliary angular momenta in channel α that are used in the transformation (3.50). If we separate the radial and angular components of the above expression, the argument of the transformation Gaussian is rewritten as 0 R2α 1 rα2 1 0 − − (γ rα + δβ α Rα )2 − 2 (γβ α rα + δβ α Rα )2 ≡ −ηrα2 − 2ξ rα · Rα − ζ R2α rα2 i R2αI rα2 i β α RαI (3.52) and using the expansion − ∞ exp(i r1 · r2 ) = 4π ∑ iλ jλ (r1 r2 ) [Yλ (r̂1 ) ⊗Yλ (r̂2 )]0 0 λ =0 ∞ exp[−2ξ rα · Rα ] = ∑ 4πiλ p 2λ + 1 jλ (2iξ rR) Yλ (r̂α ) ⊗Yλ (R̂α ) 0 (3.53) λ =0 where jλ (z) are the spherical Bessel functions. Inserting into eq.(3.51), the angular dependence is given by ∑ 0 0 lα ,Lα 18 i † h Ylα (r̂α ) ⊗YLα (R̂α ) JM Yλ (r̂α ) ⊗Yλ (R̂α ) 0 Yl 0 (r̂α ) ⊗YL0 (R̂α ) α α JM (3.54) which can be simplified, again using from section (5.16.2) in [44] i h Yλ (r̂α ) ⊗Yλ (R̂α ) 0 Yl 0 (r̂α ) ⊗YL0 (R̂α ) α λ 0 × lα Σ JM α λ 0 Lα Λ = 2λ + 1 ∑ 4π Σ,Λ q 0 0 (2J + 1)(2lα + 1)(2Lα + 1) (3.55) 0 0 0 hλ lα 0 0|Σ 0ihλ Lα 0 0|Λ 0i YΣ (r̂α ) ⊗YΛ (R̂α ) JM J J Relation (3.51) then becomes φµ∗ (rα , Rα )φν (rβ , Rβ ) = Nα Nβ rαlα RLαα exp[−ηrα2 − ζ R2α ] 0 × ∑ 0 l +Lβ +T hlβ Lβ J |lα Lα T Jiβ α rαβ RTα 0 0 lα ,Lα ,T q 0 0 × ∑ (2J + 1)(2lα + 1)(2Lα + 1)(2λ + 1)3/2 jλ (2iξ rR) iλ Σ λ λ 0 0 0 0 0 ×∑ hλ lα 0 0|Σ 0ihλ Lα 0 0|Λ 0i lα Lα J Σ,Λ Σ Λ J † × YΣ (r̂α ) ⊗YΛ (R̂α ) JM Ylα (r̂α ) ⊗YLα (R̂α ) JM (3.56) Now, this can be directly used for analytical calculation of the matrix elements of the kinetic energy operator and the interaction potential. As an example, we derive the overlap integral: we start with the angular integration which gives, Z Z † dΩr dΩR Ylα (rˆα ) ⊗YLα (Rˆα ) JM Yλ (rˆα ) ⊗YΛ (Rˆα ) 00 q h i† 2λ + 1 0 0 ˆ × Yl 0 (rˆα ) ⊗YL0 (Rα ) (2J + 1)(2lα + 1)(2Lα + 1) = ∑ α α 4π Σ,Λ JM λ λ 0 0 0 0 0 × lα Lα J hλ lα 0 0|Σ 0ihλ Lα 0 0|Λ 0i Σ Λ J Z Z † × dΩrα dΩRα Ylα (rˆα ) ⊗YLα (Rˆα ) JM Yλ (rˆα ) ⊗YΣ (RˆΛ ) JM q 2λ + 1 0 0 = (2J + 1)(2lα + 1)(2Lα + 1) 4π λ λ 0 0 0 0 0 × lα Lα J hλ lα 0 0|Σ 0ihλ Lα 0 0|Λ 0i lα Lα J (3.57) The remaining radial integrations have the form: 19 Z I= Z dr 2 2 dR rlβ +Lβ +lα −T RLα +T +2 e−ηr e−ζ R iλ jλ (2iξ rR) (3.58) We start with the integration over R and use, r jλ (z) = π 1 √ J 1 (z) 2 z λ+2 (3.59) where Jλ (z) is the usual Bessel function. 1 Further, let x = ζ 2 R and M = (Lα + T − λ )/2, which gives λ + 12 I=i r Z Z λ 5 3 1 2 2 2 π dr rlβ +Lβ +lα −T + 3 e−ηr ζ −(M+ 2 + 4 ) dx x2M+λ + 2 e−x Jλ + 1 (2iξ ζ − 2 rx) (3.60) 2 4 The integral over x can be found in [45]. The result is λ − 21 r I=i π −ηr2 −(M+ λ + 5 ) 2 4 e ζ 4 Z dr r lβ +Lβ +lα −T + 23 M! ξ 2ζr2 e 2 iξ r p ζ !λ + 12 (λ + 1 ) LM 2 ξ 2 r2 ζ (3.61) (λ + 21 ) where LM is the Laguerre polynomial. Introducing N = mial [46] lβ +Lβ +lβ −T −λ , 2 Θ = ηζ − ξ 2 and using the expansion for the Laguerre polyno- M α LM (x) = ∑ (−1)k k=0 1 M+α k x k! M − k (3.62) gives √ Z M π M! M + λ + 12 − Θ r2 λ λ −(M+λ + 23 ) 2k −k I= (−1) ξ ζ ξ ζ dr r2(N+λ +k+1) e ζ ∑ M−k 4 k=0 k! (3.63) The remaining r-integral is standard and gives finally √ π I = (−1) 8 λ M 3 M! M + λ + 21 3 ∑ k! M − k Γ(N + k + λ + 2 )ξ λ +2k ζ N−M Θ−N−λ −k− 2 k=0 (3.64) The product of the angular integrals, the radial integral and the pre-factors in the big bracket ( ) gives the overlap matrix. The potential and kinetic energies can be derived in similar, but slightly more complicated ways. 20 e- + e+ + ... Figure 3.7: Vacuum polarization insertion in the photon propagator. 3.4 Vacuum Polarization In the mid 1930’s the quantum electro-dynamical notion of vacuum polarization emerged from the work of Dirac, Furry and Oppenheimer, Serber and Uehling [48–51]. Vacuum polarization is a distortion of the Coulombic interaction due to the production of virtual electron-positron pairs by a strong electromagnetic field. It can be expressed as a correction to the photon propagator as in figure (3.7). Muonic hydrogen differs from ordinary hydrogen atoms in that the effect of vacuum polarization is much larger and is the prime QED contribution. The QED shift for an ordinary hydrogen atom is called Lamb shift. The total Lamb shift of the 2S level of ordinary H is 1058 MHz, while the vacuum polarization is only −27 MHz. Thus for ordinary H, the vacuum polarization is a small part of the total QED effect. For muonic hydrogen the vacuum polarization is the dominating QED correction for the 2S-2P splitting of µ p by −0.2 eV. This is precisely the splitting which has been measured to obtain the proton radius. The importance of vacuum polarization for the energy spectrum of µ p atoms has increased with the reduction of atomic radius. The Bohr radius is 200 times smaller, hence, S-state muonic wave-functions overlap strongly with the charge distribution of the virtual e+ e− pairs. The first order Uehling potential for µ p gives a value of 205.001 meV [15] for the 2P-2S Lamb shift. The question arises: why are the p, d and f states less shifted? The reason is that only in S-states is the electron wave-function different from 0 at r = 0 (i.e. close to nucleus). The shift of the P-state is given in table (3.1). If we consider a charge distribution of the nucleus which is symmetric, the effective Uehling potential [51] giving the first order correction to the Coulomb potential V (r) = Z1 Z2 /r reads as [52] 2α Vpol (r) = 3 Z ∞ 1 × p 1 2 2 dt t − 1 2 + 4 t t "Z 0 r 0 sinh(2tcr ) e−2ctr dr r ρ(r ) + 0 r ctr 0 02 0 Z ∞ r sinh(2tcr) e−2ctr dr r ρ(r) 0 0 ctr r 0 02 0 # (3.65) 0 where ρ(r ) is the charge distribution of the nucleus, α the fine-structure constant and c the 0 0 speed of light in vacuum. For hydrogen nuclei we can with decent accuracy write ρ(r ) = δ 3 (r ), yielding 21 Table 3.1: Corrections due to vacuum polarization of the 2s and 2p states in muonic hydrogen, calculated in first-order perturbation theory. Energies in meV. The experimental 2p-2s splitting is dominated by vacuum polarization, but also includes various other small terms, including terms dependent on the proton radius. experimental Uehling potential Fitted Uehling potential E(2s) meV E(2p) meV −219.58 −219.57 −14.58 −14.57 Z1 Z2 α Vpol (r) = r 3π Z 1 0 E(2p) − E(2s) meV 206.295 205.01 205.00 s 1 −2cr/x 2 e (2 + x ) − 1 dx. x2 (3.66) This form of the potential is very difficult to treat numerically. Instead we use the approach from [43] and fit it to a sum of 20 Gaussians. We find that our fitted potential gives results agreeing very well with the results using eq.(3.66). The Vpol (r) causes a splitting ∆E pol ≡ E2p − E2s = 205.00 meV of n = 2 levels of µ p as shown in table (3.1). 22 4. Negative Hydrogen Ion To explore the pµe hypothesis, we start with the negative hydrogen ion pe− e− and change the mass of one e− to make an exotic particle. While gradually scaling the mass of the exotic particle towards the muon mass, we follow the binding energy and life times of resonances in the threebody system. Exotic particles are usually unstable and thus have short lifetimes. Here, though, we treat all particles as having infinite lifetimes, which is motivated since e.g. the muon lifetime is much longer than typical atomic time scales. In addition, exotic systems serve to examine the general theory of three body systems and analyze their inter-particle correlation. Because all these exotic particles (except the positron), are heavier than the electron and therefore more strongly bound to the nucleus than electrons, their transitions during the de-excitation are considerably more energetic than those of electrons which we will discuss later in the chapter. Negative hydrogen ions are essential in astrophysics especially for the description of the opacity of the sun’s atmosphere [66]. These resonances in H− studied both experimentally and theoretically from the first identification of the presence of the resonances in electron-hydrogen scattering studies by Burke and Schey [54], Smith and others [55–59]. Like most negative atomic ions, H− has only one stable bound state. In this ground state of − H the correlation between the two electrons is already strong. Negative hydrogen ions have no singly excited states, but there exist doubly excited states, which are embedded in the continuous part of hydrogen spectrum and can be observed as resonances. The solution to the non-relativistic Schrödinger equation for the three body problem was found using the Coupled Rearrangement Channel method (discussed in section (3.3.3)) [41], including the Complex Scaling method which was disscused in section (3.2.2). In the Complex Scaling method, the separation of bound state, resonant states and the continuum states can be performed without any ambiguity by the scaling angle θ as shown in fig. (4.1). The results for the ground states of H− are shown in Table (4.1), which shows the convergence and accuracy of the code. At low values of the real scaling parameter α (discussed in section 3.2.1) the deviation of our results from literature value is 10−9 a.u. [60]. For the resonances which have more extended wave functions, the best accuracy is achieved for α ∼ 1. We are interested in the resonance below the n = 2 threshold. Figure (4.2) shows the convergence using different basis functions for the n = 2 state. It shows that within the stabilized plateau the rotational paths (changes E as a function of θ ) meet each other at the position of a pole. Hence, at the position of a pole, the change in energy with respect to θ is minimized. In this case, their position and widths are obtained by the condition ∂ E/∂ θ ≈ 0. In Table (4.2), we compare our present results with the experimental results of Warner et al [63], Williams [64], Sanchez and Burrow [65] and theoretical results of Bürgers and Lindroth [60], Ho and Bhatia [61] and Chen [62]. Our results for the resonance energy (ER ) agree quite well with those of [60]. The deviation in the energies is about 10−7 a.u. and about 10−6 a.u. for 23 Figure 4.1: The spectrum of H θ for H− .The only ground state is marked with a circle and filled colour circles show a pseudo-continuum energies, rotated downwards by 2θ in the complex energy plane. The resonance state, however has a θ -independent complex energy shown by the square. 24 −3 0.5 x 10 2826 2985 3024 3798 5105 5319 0 Im(E)(a.u) −0.5 −1 −1.5 −2 −2.5 −3 −0.15 −0.1495 −0.149 −0.1485 Re(E)(a.u) −0.148 −0.1475 −0.147 Figure 4.2: Using different basis sets for the state n=2, J=0, showing the convergence. The value of complex scaling parameter θ changes with steps of 0.01 rad. The resonance is positioned E = 0.014877625373 − i0.00086617931. 25 Table 4.1: Study of the rate of convergence for the different basis functions for the system p+ e− e− . N is the number of configurations, α is the scaling length and the maximum value used for angular momenta L and l are 4. In the last row we show the best ground state eigenvalue of the system from the reference [60] and the present work. N α 15 1 0.9 0.3 No. of Basis functions Infinite Proton mass 2985 3798 5101 5700 5101 5101 Exact 15 Exact 1 Real Proton mass 5300 −ER -0.52775067690 -0.52775074550 -0.52775094624 -0.52775101621 -0.52775101472 -0.52775101560 -0.52775101654 -0.52744588108 -0.52744584392 the width. Table 4.2: Comparison of non-relativistic resonance widths of H− below n = 2 hydrogen threshold in (a.u). States 1s2 1 Se 2s2 1 Se 2s3s 1 Se 2s3s 3 Se 2s4s 1 Se 2s4s 3 Se 2s3p 1 Pe 2s4p 1 Pe Present results −ER 0.52775101401 0.14877625373 0.12602006146 0.127104276 0.125059737 0.125118188 0.1260498047 0.1250351840 Bürgers et al [60] −ER 0.52775101654 0.14877625394 0.12602006374 0.12505785 0.12604985948 0.125035052 Ho et al [61] −ER 0.148775 0.126021 0.127104 0.1250580 0.12511818 0.126049 - Chen [62] −ER 0.148782 0.126021 0.1271042 0.1250579 0.1260499 0.1250349 Warner et al [63] −ER 0.14908(47) Williams [64] −ER 0.14879(37) Sanchez et al [65] Burrow, −ER 0.14875(37) We thus conclude that our numerical method can represent the n = 2 resonances in H− with very good accuracy. Our next step is to start scaling the mass of one of the electrons to higher values. In this way we want to find out whether the pµe system may support any resonances under the µ p(2S) threshold. We replace the mass of one of the orbital electrons in negative hydrogen system and increase it steadily and call it the pex system, where x is a negatively charged particle with mass mx = xme . The atomic orbit of the x-particle is therefore closer to the proton than the electron’s orbit in an ordinary hydrogen atom. Since we are interested in the resonance which lie under the pµ(n = 2) threshold, we focus on resonances under the px(n = 2) threshold. We find that the resonance disappears at mx = 3.8me . We also see some extra resonances which appear under the H(1s) threshold around mx = 3me shown, in paper I. The energy and width of the resonances increase when we increase the mass of the x particle, when the and px(n = 2) thresholds cross at mx = 4me , the resonance follows to the H(1s) and px(n = 2) threshold instead. We detect a resonance at binding energy 0.14 meV at mass mx = 6.7me but the width of the resonances above mx = 6me is too small to be determined accurately. The figure and table are shown in Paper I. 26 Table 4.3: Comparison of non-relativistic resonance widths of H− below n = 2 hydrogen threshold in (a.u). States 1s2 1 Se 2s2 1 Se 2s3s 1 Se 2s3s 3 Se 2s4s 1 Se 2s4s 3 Se 2s3p 1 Pe 2s4p 1 Pe 4.1 Present results Γ/2 0.000866617931 0.000045254 0.000000332 0.00000260 0.00000003 0.0000005029 0.000000058 Bürgers et al[60] Γ/2 0.00086661817 0.00004526486 0.00000261 0.0000006841 0.000000039 Ho et al[61] Γ/2 0.00086 0.000044 0.000000335 0.00000254 0.00000066 - Chen [62] Γ/2 0.00086 0.0000447 0.000000342 0.00000258 0.00000002 0.00000061 0.0.000000035 Warner et al[63] Γ/2 0.00116 Williams [64] Γ/2 0.000825 Sanchez et al [65] Burrow, Γ/2 - The pµe Hypothesis We also employed another way to investigate this pµe hypothesis, starting with pµ µ system and decreasing the mass of one of the muon and observing where the resonance state at n = 2 disappears as we did in the pex system. To do this, we first calculated the ground state energy for pµ µ giving a good agreement with the literature values [66, 67] shown in Table (4.4). For the resonance energy at n = 2 threshold the binding energy Eb = 120.4716543 eV and width Γ = 7.146072583 eV which is slightly larger binding energy than [21, 68]. We compare our results to [21] in paper I, they used a different numerical technique. However, the results agree very well with and without including the correction due to vacuum polarization given in table (3.1). Table 4.4: The ground state energy of pµ µ (a.u). Present result -97.5669903006 Frolov [66] -97.5669834 Ancarani [67] -97.3747607 Because the muon’s orbit is close to the proton, the proton charge pµe is shielded from the outer electron. However, the muon and proton in the 2S-2P state form a neutral core which interacts with the electron via dipole interactions. As expected, when we decrease the mass of the third particle, its orbital around the pµ(n = 2) core becomes larger and its binding energy decreases. The resonances then follow the pµ(n = 2) threshold, until it disappears at mx ' 30me . On the other hand, we also find that some new resonances appear under the px(n = 1) threshold at mx ∼ 65me . For the pµx system the mechanism leading to an additional resonance below the px(n = 1) threshold is similar to pex. The figure and tables are discussed in Paper I. 27 28 5. Positron-Hydrogen Scattering (e+H) We also searched for quasi-bound states in (e− , e+ , p), a three body system including a positron, an electron, and a proton. This problem is of great interest [9–13]. The positron is the antiparticle of the electron with the same mass however having opposite charge. The electron and positron can form a bound state like an electron and a proton. The bound states of e− and e+ are called positronium states. The electromagnetic interaction is accountable for binding; thus, positronium shares many features with the hydrogen atom. An important difference, however, is the possibility of annihilation of the electron when it meets the positron, giving gamma rays. Positronium is a short-lived system. Resonance phenomena have been studied extensively in positron-hydrogen scattering. According to Mittlemen [69], an infinite sequence of resonances should exist below the n = 2 excitation threshold in positron-hydrogen scattering. Investigation of the phenomena of resonance in positron-hydrogen scattering remained an important concern for research over the last few decades. Prior research argued the persistence of resonances below the excitation of the n = 2 threshold and a variety of theoretical approximations have been used to predict the positions and widths of resonances [9–13], even though there are not yet any experimental observations [5–7]. In calculation of resonances, we use the same complex scaling method described in section (3.2.2). We find resonances for natural and unnatural parity states. We all know about natural parity, π = (−1)L where L is the total angular momentum, but in case of unnatural parity states we have, π = (−1)L+1 . When a positron collides with an excited hydrogen in the vicinity of the Ps (n = 2) threshold, the positron may pick up the electron from the H and form real or virtual positronium in its n = 2 states. Here we should mention that the S-state component does not contribute to the states with unnatural parities of π = (−1)L+1 . The states interacting with the scattering continua will manifest themselves as resonances in positron scattering with excited hydrogen. Resonance states for the e+ H system with natural parity have been calculated for S-, Pand D-waves both for physical and infinite proton mass. Most of the resonances are similar to previous literature values, a few of them have better values and some resonances not previously reported are presented in Paper II. Also resonances for unnatural parity states for P- and D-waves are calculated and discussed shown in Paper II. 29 30 Summary and Outlook In this work, resonance positions and widths have been studied for three-body systems. We have shown that for arbitrary mass values accurate results can be obtained using the CRC method with complex scaling. In addition we have been able to incorporate corrections to the Coulomb potential by expressing them as a sum of Gaussians, and applying complex scaling also to these corrections. For outlook, we are working on resonances in positron-alkali atom systems. Alkali atoms can be expressed as quasi-one-electron targets, the positron-alkali-atom scattering is very different from the positron-hydrogen scattering in different ways. The alkali atoms have large dipole polarizabilities that is strong enough to produce bound states [70]. In addition, the hydrogen atom has degeneracy in its spectrum while the alkali atoms do not have such characteristics. Another important feature of a positron-alkali system is rearrangement process, i.e. positronium(Ps) formation, which is energetically possible even at zero impact energy and is expected to have a significant effect on the scattering process. The positron-electron interaction is of course only the usual Coulomb interaction. For the positron-core interaction we can use the HartreeFock approximation for the core (considered inert), and calculate an effective shielding due to the core electrons. For the electron-core interaction we do not use the shielding given by the Hartree-Fock approximation because (i) it does not incorporate electron exchange (ii) it is very important to get the atomic threshold energies to very high accuracy. For this purpose there are instead model potentials which have been used in literature [71]. Another project is to develop our codes so that we can calculate scattering cross sections. We have shown that positron-hydrogen resonances can be calculated to very high accuracy. It would therefore be natural to look at positron scattering on hydrogen in the first instance. Our three-body code, including all sets of Jacobi coordinates, is especially well suited to calculate rearrangement processes such as positronium formation. Of particular interest is the reverse process, hydrogen formation in positronium scattering on a proton. This process (or rather its equivalent charge conjugate process) could be a way of making anti-hydrogen. References [1] R. Pohl et al., Nature (London), 466, 213 (2010). 1, 3, 4 [2] A. Antognini et al., Science 339, 417-420 (2013). 1, 4 [3] B. de Beauvoir et al., Eur. Phys. J. D 12, 61 (2000). 1 [4] P. J. Mohr, et al., Rev. Mod. Phys. 80, 633 (2008). 1, 4 [5] J. Mitroy and K. Ratnavelu J. Phys. B: At. Mol. Phys. 28, 287 (1995) 1, 29 [6] G. D. Doolen, J. Nuttall and C. Wherry Phys. Rev. Lett. 40, 33 (1978) 1 [7] Y. Zhou and C. D. Lin Phys. Rev. Lett. 75, 2296 (1995) 29 [8] J. Mitroy and K. Ratnavelu J. Phys. B: At. Mol. Phys 30, L371-L375 (1997) 1 [9] Y. K. Ho, and C. H. Greene Phys. Rev. A 35, 3169 (1987) 1, 29 [10] Y. K. Ho J. Phys. B: At. Mol. Phys. 23, L419 (1990) [11] Y. K. Ho Hyperfine Interact. 73, 109 (1992) [12] Y. K. Ho and Z. C. Yan Phys. Rev. A 70, 032716 (2004) [13] K. Varga, J. Mitroy, J. Zs. Mezei and A. T. Kruppa Phys. Rev. A 77, 044502 (2008) 1, 29 [14] A. Temkin and J. F. Walker Phys. Rev. 140, A1520 (1965). [15] U. D. Jentschura, Ann. Phys. (N.Y.) 326, 500 (2011). 3, 4, 21 [16] http:/ / physics.nist.gov/cuu/Constants. [17] P. J. Mohr et al., Rev. Mod. Phys. 84, 1527 (2012). 3 [18] P. Wang, et al., Phys. Rev. D. 79, 094001 (2009). 4 [19] P. G. Blunden et al., Phys. Rev. C.72, 057601 (2005). 4 [20] I. Sick, Phys. Lett. B 62, 576 (2003). 4 [21] J. P. Karr and L. Hilico, Phys. Rev. Lett. 109, 103401 (2012). 4, 27 [22] L. D. Landau and E. M. Lifshitz. Quantum Mechanics. Pergamon Press, Third Edition, 1991. 5, 10 [23] J. J. Sakurai. Modern Quantum Mechanics. Addison Wesley, Second Edition, 1994. 5 [24] J. Z. H. Zhang, Theory and application of quantum molecular dynamics, World Scientific, 1999. 7, 10 [25] R. Schinke, Photodissociation Dynamics, Cambridge Univeristy Press, (1995). [26] N. Elander and E. Yarevsky, Phys. Rev. A 56, 1855, (1997). 10 [27] B. D. Esry, C. D. Lin and C. H. Greene, Phys. Rev. A. 54, 394 (1996). 10 [28] E. Braaten and H. W. Hammer, Universality in few-body Systems with Large Scattering Length, cond-mat/0410417 (2004). 10 [29] P. N. Roy, J. Chem. Phys. 119, 5437 (2003). 10 [30] C. L. Pekeris, Phys. Rev. 112, 1649 (1958). 10 [31] V. Ryaboy, N. Moiseyevs, V. A. Mandelshtam, and H. S. Taylor, J. Chem. Phys. 101 5677 (1994). 7 [32] A. Hazi and H. Taylor, Phys. Rev. A. 1, 1109 (1970). 7 [33] E. Balslev and J. M. Combes, Math. Phys. 22 280 (1971). 8 [34] J. Agulilar and J. M. Combes, Math. Phys. 22 269 (1971). [35] N. Moiseyev, Physics Reports. 302, 211 (1998). 6, 9 [36] P. R. Certain and N. Moiseyev, J. Phys. Chem. 89, 2974 (1985). [37] N. Lipkin, S. Levin and N. Moiseyev, J. Chem. Phys. 98, 1888 (1993). [38] S. Chu, J. Chem. Phys. 72, 4772 (1980). [39] M. Monnerville and J. M. Robbe, J. Chem. Phys. 211, 249 (1996). [40] A. Scrinzi and N. Elander, J. Chem. Phys. 98 3866, (1993). 8 [41] M. Kamimura, Phys. Rev. A 38, 621 (1988). 14, 23 [42] E. Hiyama, Y. Kino and M. Kamimura Progress in Particle and Nuclear Physics 51, 223 (2003). 16 [43] J. Wellenius Meta Stable States of Monic Molecules and the Muon Catalyzed Fusion Cycle, Thesis, (1996). 16, 22 [44] D.A. Varsalovich, A.N. Mohalev and V.K. Khesonshii, "Quantum Theory of angular momentum", Third Edition, 1992. 16, 19 [45] I. S. Gradshtegn and I.M. Ryzhik„ "Table of integrals series and product" 20 [46] Milton Abramowitz and Irene A. Stegan, "Hand Book of Mathematical Functions", Dover Publications, Inc., New York (1970) 20 [47] B. H. Bransden And C. J. Joachain, Physics of Atoms and Molecules, Third Edition, 1992. 11 [48] P. A. M. Dirac Proc. Camb. phil. Soc. 30, 150 (1934) 21 [49] W. H. Furry and J. R. Oppenheimer Phys. Rev. 45, (1934). [50] R. Serber, Phys. Rev. 48, (1935). [51] E. Uehling, Phys. Rev. 48, 55, (1935). 21 [52] W. Greiner and B. Müller and J. Rafelski, Quantum Electrodynamics of strong Fields. Springer-Verlag, Berlin (1985). 21 [53] A. M. Frolov and V. H. Smith Jr, J. Quant. Chem. 111, 4255 (2011). 23, 27 [54] P. G. Burk and H. Schey, Phys. Rev. 126, 147 (1962). 23 [55] K. Smith, R. P. McEachran and P. A. Fraser, Phys. Rev. 125, 553 (1962). 23 [56] M. E. Hamm, R. W. Hamm, J. Donahue, P.A.M. Gram, J.C. Pratt, M.A. Yates, R.D Bolton, D.A Clark, H.C. Bryant, C.A. Frost, W.W. Smith, Phys. Rev. Lett. 43, 1715 (1979). [57] S. Cohen, H.C. Bryant, C.J. Harvey, J.E. Stewart, K.B. Butterfield , D.A Clark, J.B Donahue, D.W. MacArthur, G. Comete, W.W. Smith, Phys. Rev. A 36, 4728 (1987). [58] P.G. Harris, H.C. Bryant, A.H. Mohagheghi, R.A. Reeder, H. Sharifian, C.Y. Tang, H. Tootoonchi, J.B Donahue, C.B. Quick, D.C. Rislove, W.W. Smith, J.E. Stewart, Phys. Rev. Lett. A 42, 6443 (1990). [59] P. Balling, P. Kristensen, H.H. Haugen, U.V. Pedersen, V.V. Petrunin, L.Praestgaard, H.K.Haugen, T. Andersen, Phys. Rev. Lett. 77, 2905 (1996). 23 [60] A. Bürgers and E. Lindroth, Eur. Phys. J. D 10, 327-340 (2000). 23, 26, 27 [61] A. K. Bhatia and Y. K. Ho, Phys. Rev. Lett. A. 41, 504 (1990). 23, 26, 27 [62] Ming-Keh Chen, J. Phys. B: At. Mol. Phys. 30, 1669-1676 (1997). 23, 26, 27 [63] C. D Warner, G. C King, P. Hammond and J. Slevin, J. Phys. B: At. Mol. Phys. 19, 3297 (1986). 23, 26, 27 [64] Williams J F, Electron and Photon Interaction with atoms, 1976 (New York: Plenum) 23, 26, 27 [65] L. Sanchez and P. D. Burrow, Phys. Rev. Lett. 29, 1639 (1972). 23, 26, 27 [66] A. M. Frolov, V. H. Simth Jr and J. Komasa, J. Phys. A 26, 6507 (1993). 23, 27 [67] L. U. Ancarani, K. V. Rodriguez and G. Gasaneo, EPJ Web of Conferences 3, 02009 (2010). 27 [68] Y. K. Ho, Phys. Rev. A 19, 2347 (1979) 27 [69] M. H. Mittleman, Phys. Rev. 152, 76 (1966) 29 [70] J. Mitroy, M. W. J. Bromley and G. G. Ryzhikh J. Phys. B: At. Mol. Phys. 35 R81 (2002) [71] S. Kar and Y. K. Ho Eur. Phys. J. D 35 453 (2005)