DOCSLIB.ORG

O Schrödinger Equation for O Two-Electron Atoms. O Multi

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
O Schrödinger Equation for O Two-Electron Atoms. O Multi o Schrödinger equation for o Two-electron atoms. o Multi-electron atoms. o Helium-like atoms. o Singlet and triplet states. o Exchange energy. o See Chapter 9 of Eisberg & Resnick PY3P05 o See http://en.wikipedia.org/wiki/Electron_configuration PY3P05 o What is probability of simultaneously finding a particle 1 at (x1,y1,z1), particle 2 at (x2,y2,z2), etc. => need joint probability distribution. o N-particle system is therefore a function of 3N coordinates: z !(x1,y1,z1; x2,y2,z2; … xN,yN,zN) ˆ ˆ ˆ ˆ ˆ ˆ ˆ o Must solve H " (r1 ,r2 ,..., rN ) = E"(r1 ,r2 ,...,rN ) (1) e2 e1 o First consider two particles which do not interact with one r 2 r1 another, but move !in potentials V1 and V2. The Hamiltonian is Hˆ = Hˆ + Hˆ (2) y 1 2 x 2 2 $ ! 2 ' $ ! 2 ' = & " #1 + V1(rˆ1 )) + & " # 2 + V2(rˆ2 )) % 2m1 ( % 2m2 ( 2 2e i =1,2 o The electron-nucleus potential for helium is Vi = " 4#$0ri ! o The eigenfunctions of H1 and H2 can be written as the product: "(rˆ1 ,rˆ2 ) =#1(rˆ1 )#2 (rˆ2 ) ! ! PY3P05 ! o Using this and Eqns. 1 and 2, ˆ ˆ ˆ H" (rˆ1 ,rˆ2 ) = (H1 + H 2 )#1(rˆ1 )#2(rˆ2 ) = (E1 + E 2 )#1(rˆ1 )#2(rˆ2 ) ˆ o That is, H" (rˆ1 ,rˆ2 ) = E " ( rˆ 1 , r ˆ 2 ) where E = E1+E2. ! o The product wavefunction is an eigenfunction of the complete Hamiltonian H, corresponding !to an eigenvalue E which is the sum of the energy eigenvalues of the two separate particles. o For N-particles, ˆ ˆ ˆ ˆ ˆ ˆ " (r1 ,r2 ,...rN ) =#1(r1 )#2 (r2 )!#N (rN ) o Eigenvalues of each particle’s Hamiltonian determine possible energies. Total energy is thus N ! E = " E i i=1 o Can be used as a first approximation to two interacting particles. Can then use perturbation theory to include interaction. ! PY3P05 o Assuming each electron in helium is non-interacting, can assume each can be treated independently with hydrogenic energy levels: 13.6Z 2 E = " Observed n n 2 o Total energy of two-electron system in ground -50 state (n(1) = n(2) = 1) is therefore -60 ! E = E1(1) + E1(2) Energy -70 # & 2 1 1 (eV) = "13.6Z % 2 + 2 ( $ n(1) n(2) ' -80 # 1 1 & = "13.6(2)2% + ( $ 12 12 ' -90 = "109 eV -100 o For first excited state, n(1) = 1, n(2) = 2 => E =-68 eV. -110 ! Neglecting electron-electron interaction PY3P05 o For He-like atoms can extend to include electron-electron interaction: z & 2 2 ) & 2 2 ) 2 ˆ "! 2 Ze "! 2 Ze e H = ( #1 " + + ( # 2 " + + 2m 4$% r 2m 4$% r 4$% r e r e ' 0 1 * ' 0 2 * 0 12 2 12 1 o The final term represents electron-electron repulsion at a r 2 r1 distance r12. ! y o Or for N-electrons, the Hamiltonian is: x N ˆ ˆ H = "H i i=1 N 2 2 N 2 ' #! 2 Ze * e = ) $ i # , + " 2m 4%& r " 4%& r i=1 ( 0 i + i> j 0 ij and the corresponding Schrödinger equation is again of the form Hˆ " (rˆ ,rˆ ,...,rˆ ) = E"(rˆ ,rˆ ,...,rˆ ) ! 1 2 N 1 2 N where N "(rˆ ,rˆ ,...,rˆ ) =# (rˆ )# (rˆ )!# (rˆ ) and E = E 1 2 N 1 1 2 2 N N " i i=1 ! PY3P05 ! ! ' 2 Ze2 e2 * ˆ ˆ #! 2 ˆ o The solutions to the equation H i"i (ri ) = ) $ i # + ," i (ri ) ( 2m 4%&0ri 4%&0rij + ˆ = E i"i (ri ) can again be written in the form (rˆ , , ) R (rˆ ) ( , ) "i i #i $i = ni li i %li mi #i $i 2 2 ! d Rn l 2 dRn l 2µ $ e ' o The radial wave functions are solutions to i i i i E R 0 2 + + 2 & + ) ni li = dr r dri ! % 4"#0ri ( ! i and therefore have the same analytical form as for the hydrogenic one-electron atom. o Allowable solutions again only exist! for Z 2µe4 E = " eff n (4#$ )22!2n 2 0 where Zeff = Z - !nl. o Zeff is the effective nuclear charge and !nl is the shielding constant. This gives rise to the shell model for multi-electron !atoms. PY3P05 o Includes He and Group II elements (e.g., Be, Mg, Ca, etc.). Valence electrons are indistinguishable, i.e., not physically possible to assign unique positions simultaneously. o This means that multi-electron wave functions must have exchange symmetry: 2 2 | "(rˆ1 ,rˆ2 ,...,rˆK ,rˆL ,...,rˆN ) | =| "(rˆ1 ,rˆ2 ,...,rˆL ,rˆK ,...,rˆN ) | which will be satisfied if "(rˆ1 ,rˆ2 ,...,rˆK ,rˆL ,...,rˆN ) = ±"(rˆ1 ,rˆ2 ,...,rˆL ,rˆK ,...,rˆN ) ! o That is, exchanging labels of pair of electrons has no effect on wave function. ! o The “+” sign applies if the particles are bosons. These are said to be symmetric with respect to particle exchange. The “-” sign applies to fermions, which are antisymmetric with respect to particle exchange. o As electrons are fermions (spin 1/2), the wavefunction of a multi-electron atom must be anti- symmetric with respect to particle exchange. PY3P05 o He atom consists of a nucleus with Z = 2 and two electrons. z o Must now include electron spins. Two-electron wave function is therefore written as a product spatial and a spin e2 r12 e1 wave functions: r " =#spatial (rˆ1 ,rˆ2 )#spin 2 r1 o As electrons are indistinguishable => ! must be anti- y Z=2 x symmetric. See table for allowed symmetries of spatial and spin wave! functions. PY3P05 o State of atom is specified by configuration of two electrons. In ground state, both electrons are is 1s shell, so we have a 1s2 configuration. o In excited state, one or both electrons will be in higher shell (e.g., 1s12s1). Configuration must therefore be written in terms of particle #1 in a state defined by four quantum numbers (called "). State of particle #2 called #. o Total wave function for a excited atom can therefore be written: " =#$ (rˆ1 )#% (rˆ2 ) o But, this does not take into account that electrons are indistinguishable. The following is therefore equally valid: ˆ ˆ " =#$ (r1 )#% (r2 ) ! o Because both these are solutions of Schrödinger equation, linear combination also a solutions: 1 ! " = (# (rˆ )# (rˆ ) +# (rˆ )# (rˆ )) Symmetric S 2 $ 1 % 2 % 1 $ 2 1 " = (# (rˆ )# (rˆ ) &# (rˆ )# (rˆ )) Antisymmetric A 2 $ 1 % 2 % 1 $ 2 where 1 / 2 is a normalisation factor. ! PY3P05 ! o There are two electrons => S = s1+ s2 = 0 or 1. S = 0 states are called singlets because they only have one ms value. S = 1 states are called triplets as ms = +1, 0, -1. o There are four possible ways to combine the spins of the two electrons so that the total wave function has exchange symmetry. o Only one possible anitsymmetric spin eigenfunction: 1 [(+1/2,"1/2) " ("1/2,+1/2)] singlet 2 o There are three possible symmetric spin eigenfunctions: ! (+1/2,+1/2) 1 [(+1/2,"1/2) + ("1/2,+1/2)] triplet 2 ("1/2,"1/2) ! PY3P05 o Table gives spin wave functions for a two- electron system. The arrows indicate whether the spin of the individual electrons is up or down (i.e. +1/2 or -1/2). o The + sign in the symmetry column applies if the wave function is symmetric with respect to particle exchange, while the - sign indicates that the wave function is anti-symmetric. o The Sz value is indicated by the quantum number for ms, which is obtained by adding the ms values of the two electrons together. PY3P05 o Singlet and triplet states therefore have different spatial wave functions. o Surprising as spin and spatial wavefunctions are basically independent of each other. o This has a strong effect on the energies of the allowed states. S ms $spin $spatial 1 Singlet 0 0 1/ 2("# $#" ) ("# (rˆ1 )"$ (rˆ2 ) +"$ (rˆ1 )"# (rˆ2 )) 1 2 1 2 2 +1 "1"2 1 1 0 1/ 2("# +#" ) (" (rˆ )" (rˆ ) %" (rˆ )" (rˆ )) Triplet ! 1 2 1 2 # 1 $ 2 $ 1 # 2 ! 2 -1 #1#2 ! ! PY3P05 o Physical interpretation of singlet and triplet states can be obtained by evaluating the total spin angular momentum (S), where ˆ ˆ ˆ S = S1 + S 2 is the sum of the spin angular momenta of the two electrons. o The magnutude of the total spin and its z-component are quantised: S = s(s +1)! ! S = m ! z s where ms = -s, … +s and s = 0, 1. z o If s1 = +1/2 and s2 = -1/2 => s = 0. m ! s triplet o Therefore ms = 0 (singlet state) +1 state s1=1/2 o If s1 = +1/2 and s2 = +1/2 => s = 1. s2=1/2 s = 1 o Therefore ms = -1, 0, +1 (triplet states) 0 -1 singlet s1=1/2 s2=-1/2 state s = 0, ms = 0 PY3P05 o Angular momenta of electrons are described by l1, l2, s1, s2. o As Z<30 for He, use LS or Russel Saunders coupling. o Consider ground state configuration of He: 1s2 o Orbital angular momentum: l1=l2 = 0 => L = l1 + l2 = 0 o Gives rise to an S term. o Spin angular momentum: s1 = s2 = 1/2 => S = 0 or 1 o Multiplicity (2S+1) is therefore 2(0) + 1 = 1 (singlet) or 2(1) + 1 = 3 (triplet) o J = L + S, …, |L-S| => J = 1, 0. 1 3 o Therefore there are two states: 1 S0 and 1 S1 (also using n = 1) o But are they both allowed quantum mechanically? PY3P05 o Must consider Pauli Exclusion principle: “In a multi-electron atom, there can never be more that one electron in the same quantum state”; or equivalently, “No two electrons can have the same set of quantum numbers”.
Load more

Recommended publications
  • Lecture 32 Relevant Sections in Text: §3.7, 3.9 Total Angular Momentum
    Physics 6210/Spring 2008/Lecture 32 Lecture 32 Relevant sections in text: x3.7, 3.9 Total Angular Momentum Eigenvectors How are the total angular momentum eigenvectors related to the original product eigenvectors (eigenvectors of Lz and Sz)? We will sketch the construction and give the results. Keep in mind that the general theory of angular momentum tells us that, for a 1 given l, there are only two values for j, namely, j = l ± 2. Begin with the eigenvectors of total angular momentum with the maximum value of angular momentum: 1 1 1 jj = l; j = ; j = l + ; m = l + i: 1 2 2 2 j 2 Given j1 and j2, there is only one linearly independent eigenvector which has the appro- priate eigenvalue for Jz, namely 1 1 jj = l; j = ; m = l; m = i; 1 2 2 1 2 2 so we have 1 1 1 1 1 jj = l; j = ; j = l + ; m = l + i = jj = l; j = ; m = l; m = i: 1 2 2 2 2 1 2 2 1 2 2 To get the eigenvectors with lower eigenvalues of Jz we can apply the ladder operator J− = L− + S− to this ket and normalize it. In this way we get expressions for all the kets 1 1 1 1 1 jj = l; j = 1=2; j = l + ; m i; m = −l − ; −l + ; : : : ; l − ; l + : 1 2 2 j j 2 2 2 2 The details can be found in the text. 1 1 Next, we consider j = l − 2 for which the maximum mj value is mj = l − 2.
    [Show full text]
  • Configuration Interaction Study of the Ground State of the Carbon Atom
    Configuration Interaction Study of the Ground State of the Carbon Atom María Belén Ruiz* and Robert Tröger Department of Theoretical Chemistry Friedrich-Alexander-University Erlangen-Nürnberg Egerlandstraße 3, 91054 Erlangen, Germany In print in Advances Quantum Chemistry Vol. 76: Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems 30th July 2017 Abstract Configuration Interaction (CI) calculations on the ground state of the C atom are carried out using a small basis set of Slater orbitals [7s6p5d4f3g]. The configurations are selected according to their contribution to the total energy. One set of exponents is optimized for the whole expansion. Using some computational techniques to increase efficiency, our computer program is able to perform partially-parallelized runs of 1000 configuration term functions within a few minutes. With the optimized computer programme we were able to test a large number of configuration types and chose the most important ones. The energy of the 3P ground state of carbon atom with a wave function of angular momentum L=1 and ML=0 and spin eigenfunction with S=1 and MS=0 leads to -37.83526523 h, which is millihartree accurate. We discuss the state of the art in the determination of the ground state of the carbon atom and give an outlook about the complex spectra of this atom and its low-lying states. Keywords: Carbon atom; Configuration Interaction; Slater orbitals; Ground state *Corresponding author: e-mail address: [email protected] 1 1. Introduction The spectrum of the isolated carbon atom is the most complex one among the light atoms. The ground state of carbon atom is a triplet 3P state and its low-lying excited states are singlet 1D, 1S and 1P states, more stable than the corresponding triplet excited ones 3D and 3S, against the Hund’s rule of maximal multiplicity.
    [Show full text]
  • Quantum Field Theory*
    Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
    [Show full text]
  • Singlet NMR Methodology in Two-Spin-1/2 Systems
    Singlet NMR Methodology in Two-Spin-1/2 Systems Giuseppe Pileio School of Chemistry, University of Southampton, SO17 1BJ, UK The author dedicates this work to Prof. Malcolm H. Levitt on the occasion of his 60th birthaday Abstract This paper discusses methodology developed over the past 12 years in order to access and manipulate singlet order in systems comprising two coupled spin- 1/2 nuclei in liquid-state nuclear magnetic resonance. Pulse sequences that are valid for different regimes are discussed, and fully analytical proofs are given using different spin dynamics techniques that include product operator methods, the single transition operator formalism, and average Hamiltonian theory. Methods used to filter singlet order from byproducts of pulse sequences are also listed and discussed analytically. The theoretical maximum amplitudes of the transformations achieved by these techniques are reported, together with the results of numerical simulations performed using custom-built simulation code. Keywords: Singlet States, Singlet Order, Field-Cycling, Singlet-Locking, M2S, SLIC, Singlet Order filtration Contents 1 Introduction 2 2 Nuclear Singlet States and Singlet Order 4 2.1 The spin Hamiltonian and its eigenstates . 4 2.2 Population operators and spin order . 6 2.3 Maximum transformation amplitude . 7 2.4 Spin Hamiltonian in single transition operator formalism . 8 2.5 Relaxation properties of singlet order . 9 2.5.1 Coherent dynamics . 9 2.5.2 Incoherent dynamics . 9 2.5.3 Central relaxation property of singlet order . 10 2.6 Singlet order and magnetic equivalence . 11 3 Methods for manipulating singlet order in spin-1/2 pairs 13 3.1 Field Cycling .
    [Show full text]
  • Bouncing Oil Droplets, De Broglie's Quantum Thermostat And
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 August 2018 doi:10.20944/preprints201808.0475.v1 Peer-reviewed version available at Entropy 2018, 20, 780; doi:10.3390/e20100780 Article Bouncing oil droplets, de Broglie’s quantum thermostat and convergence to equilibrium Mohamed Hatifi 1, Ralph Willox 2, Samuel Colin 3 and Thomas Durt 4 1 Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel UMR 7249,13013 Marseille, France; hatifi[email protected] 2 Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914 Tokyo, Japan; [email protected] 3 Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150,22290-180, Rio de Janeiro – RJ, Brasil; [email protected] 4 Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel UMR 7249,13013 Marseille, France; [email protected] Abstract: Recently, the properties of bouncing oil droplets, also known as ‘walkers’, have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behaviour. They indeed constitute a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to try to describe walker phenomenology in terms of de Broglie-Bohm dynamics and of a stochastic version thereof. In particular, we first study how a stochastic modification of the de Broglie pilot-wave theory, à la Nelson, affects the process of relaxation to quantum equilibrium, and we prove an H-theorem for the relaxation to quantum equilibrium under Nelson-type dynamics.
    [Show full text]
  • 1 the LOCALIZED QUANTUM VACUUM FIELD D. Dragoman
    1 THE LOCALIZED QUANTUM VACUUM FIELD D. Dragoman – Univ. Bucharest, Physics Dept., P.O. Box MG-11, 077125 Bucharest, Romania, e-mail: [email protected] ABSTRACT A model for the localized quantum vacuum is proposed in which the zero-point energy of the quantum electromagnetic field originates in energy- and momentum-conserving transitions of material systems from their ground state to an unstable state with negative energy. These transitions are accompanied by emissions and re-absorptions of real photons, which generate a localized quantum vacuum in the neighborhood of material systems. The model could help resolve the cosmological paradox associated to the zero-point energy of electromagnetic fields, while reclaiming quantum effects associated with quantum vacuum such as the Casimir effect and the Lamb shift; it also offers a new insight into the Zitterbewegung of material particles. 2 INTRODUCTION The zero-point energy (ZPE) of the quantum electromagnetic field is at the same time an indispensable concept of quantum field theory and a controversial issue (see [1] for an excellent review of the subject). The need of the ZPE has been recognized from the beginning of quantum theory of radiation, since only the inclusion of this term assures no first-order temperature-independent correction to the average energy of an oscillator in thermal equilibrium with blackbody radiation in the classical limit of high temperatures. A more rigorous introduction of the ZPE stems from the treatment of the electromagnetic radiation as an ensemble of harmonic quantum oscillators. Then, the total energy of the quantum electromagnetic field is given by E = åk,s hwk (nks +1/ 2) , where nks is the number of quantum oscillators (photons) in the (k,s) mode that propagate with wavevector k and frequency wk =| k | c = kc , and are characterized by the polarization index s.
    [Show full text]
  • The Statistical Interpretation of Entangled States B
    The Statistical Interpretation of Entangled States B. C. Sanctuary Department of Chemistry, McGill University 801 Sherbrooke Street W Montreal, PQ, H3A 2K6, Canada Abstract Entangled EPR spin pairs can be treated using the statistical ensemble interpretation of quantum mechanics. As such the singlet state results from an ensemble of spin pairs each with an arbitrary axis of quantization. This axis acts as a quantum mechanical hidden variable. If the spins lose coherence they disentangle into a mixed state. Whether or not the EPR spin pairs retain entanglement or disentangle, however, the statistical ensemble interpretation resolves the EPR paradox and gives a mechanism for quantum “teleportation” without the need for instantaneous action-at-a-distance. Keywords: Statistical ensemble, entanglement, disentanglement, quantum correlations, EPR paradox, Bell’s inequalities, quantum non-locality and locality, coincidence detection 1. Introduction The fundamental questions of quantum mechanics (QM) are rooted in the philosophical interpretation of the wave function1. At the time these were first debated, covering the fifty or so years following the formulation of QM, the arguments were based primarily on gedanken experiments2. Today the situation has changed with numerous experiments now possible that can guide us in our search for the true nature of the microscopic world, and how The Infamous Boundary3 to the macroscopic world is breached. The current view is based upon pivotal experiments, performed by Aspect4 showing that quantum mechanics is correct and Bell’s inequalities5 are violated. From this the non-local nature of QM became firmly entrenched in physics leading to other experiments, notably those demonstrating that non-locally is fundamental to quantum “teleportation”.
    [Show full text]
  • The Heisenberg Uncertainty Principle*
    OpenStax-CNX module: m58578 1 The Heisenberg Uncertainty Principle* OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract By the end of this section, you will be able to: • Describe the physical meaning of the position-momentum uncertainty relation • Explain the origins of the uncertainty principle in quantum theory • Describe the physical meaning of the energy-time uncertainty relation Heisenberg's uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately. 1 Momentum and Position To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x- direction. The particle moves with a constant velocity u and momentum p = mu. According to de Broglie's relations, p = }k and E = }!. As discussed in the previous section, the wave function for this particle is given by −i(! t−k x) −i ! t i k x k (x; t) = A [cos (! t − k x) − i sin (! t − k x)] = Ae = Ae e (1) 2 2 and the probability density j k (x; t) j = A is uniform and independent of time. The particle is equally likely to be found anywhere along the x-axis but has denite values of wavelength and wave number, and therefore momentum.
    [Show full text]
  • Analysis of Nonlinear Dynamics in a Classical Transmon Circuit
    Analysis of Nonlinear Dynamics in a Classical Transmon Circuit Sasu Tuohino B. Sc. Thesis Department of Physical Sciences Theoretical Physics University of Oulu 2017 Contents 1 Introduction2 2 Classical network theory4 2.1 From electromagnetic fields to circuit elements.........4 2.2 Generalized flux and charge....................6 2.3 Node variables as degrees of freedom...............7 3 Hamiltonians for electric circuits8 3.1 LC Circuit and DC voltage source................8 3.2 Cooper-Pair Box.......................... 10 3.2.1 Josephson junction.................... 10 3.2.2 Dynamics of the Cooper-pair box............. 11 3.3 Transmon qubit.......................... 12 3.3.1 Cavity resonator...................... 12 3.3.2 Shunt capacitance CB .................. 12 3.3.3 Transmon Lagrangian................... 13 3.3.4 Matrix notation in the Legendre transformation..... 14 3.3.5 Hamiltonian of transmon................. 15 4 Classical dynamics of transmon qubit 16 4.1 Equations of motion for transmon................ 16 4.1.1 Relations with voltages.................. 17 4.1.2 Shunt resistances..................... 17 4.1.3 Linearized Josephson inductance............. 18 4.1.4 Relation with currents................... 18 4.2 Control and read-out signals................... 18 4.2.1 Transmission line model.................. 18 4.2.2 Equations of motion for coupled transmission line.... 20 4.3 Quantum notation......................... 22 5 Numerical solutions for equations of motion 23 5.1 Design parameters of the transmon................ 23 5.2 Resonance shift at nonlinear regime............... 24 6 Conclusions 27 1 Abstract The focus of this thesis is on classical dynamics of a transmon qubit. First we introduce the basic concepts of the classical circuit analysis and use this knowledge to derive the Lagrangians and Hamiltonians of an LC circuit, a Cooper-pair box, and ultimately we derive Hamiltonian for a transmon qubit.
    [Show full text]
  • 5.1 Two-Particle Systems
    5.1 Two-Particle Systems We encountered a two-particle system in dealing with the addition of angular momentum. Let's treat such systems in a more formal way. The w.f. for a two-particle system must depend on the spatial coordinates of both particles as @Ψ well as t: Ψ(r1; r2; t), satisfying i~ @t = HΨ, ~2 2 ~2 2 where H = + V (r1; r2; t), −2m1r1 − 2m2r2 and d3r d3r Ψ(r ; r ; t) 2 = 1. 1 2 j 1 2 j R Iff V is independent of time, then we can separate the time and spatial variables, obtaining Ψ(r1; r2; t) = (r1; r2) exp( iEt=~), − where E is the total energy of the system. Let us now make a very fundamental assumption: that each particle occupies a one-particle e.s. [Note that this is often a poor approximation for the true many-body w.f.] The joint e.f. can then be written as the product of two one-particle e.f.'s: (r1; r2) = a(r1) b(r2). Suppose furthermore that the two particles are indistinguishable. Then, the above w.f. is not really adequate since you can't actually tell whether it's particle 1 in state a or particle 2. This indeterminacy is correctly reflected if we replace the above w.f. by (r ; r ) = a(r ) (r ) (r ) a(r ). 1 2 1 b 2 b 1 2 The `plus-or-minus' sign reflects that there are two distinct ways to accomplish this. Thus we are naturally led to consider two kinds of identical particles, which we have come to call `bosons' (+) and `fermions' ( ).
    [Show full text]
  • 8 the Variational Principle
    8 The Variational Principle 8.1 Approximate solution of the Schroedinger equation If we can’t find an analytic solution to the Schroedinger equation, a trick known as the varia- tional principle allows us to estimate the energy of the ground state of a system. We choose an unnormalized trial function Φ(an) which depends on some variational parameters, an and minimise hΦ|Hˆ |Φi E[a ] = n hΦ|Φi with respect to those parameters. This gives an approximation to the wavefunction whose accuracy depends on the number of parameters and the clever choice of Φ(an). For more rigorous treatments, a set of basis functions with expansion coefficients an may be used. The proof is as follows, if we expand the normalised wavefunction 1/2 |φ(an)i = Φ(an)/hΦ(an)|Φ(an)i in terms of the true (unknown) eigenbasis |ii of the Hamiltonian, then its energy is X X X ˆ 2 2 E[an] = hφ|iihi|H|jihj|φi = |hφ|ii| Ei = E0 + |hφ|ii| (Ei − E0) ≥ E0 ij i i ˆ where the true (unknown) ground state of the system is defined by H|i0i = E0|i0i. The inequality 2 arises because both |hφ|ii| and (Ei − E0) must be positive. Thus the lower we can make the energy E[ai], the closer it will be to the actual ground state energy, and the closer |φi will be to |i0i. If the trial wavefunction consists of a complete basis set of orthonormal functions |χ i, each P i multiplied by ai: |φi = i ai|χii then the solution is exact and we just have the usual trick of expanding a wavefunction in a basis set.
    [Show full text]
  • Path Integrals in Quantum Mechanics
    Path Integrals in Quantum Mechanics Emma Wikberg Project work, 4p Department of Physics Stockholm University 23rd March 2006 Abstract The method of Path Integrals (PI’s) was developed by Richard Feynman in the 1940’s. It offers an alternate way to look at quantum mechanics (QM), which is equivalent to the Schrödinger formulation. As will be seen in this project work, many "elementary" problems are much more difficult to solve using path integrals than ordinary quantum mechanics. The benefits of path integrals tend to appear more clearly while using quantum field theory (QFT) and perturbation theory. However, one big advantage of Feynman’s formulation is a more intuitive way to interpret the basic equations than in ordinary quantum mechanics. Here we give a basic introduction to the path integral formulation, start- ing from the well known quantum mechanics as formulated by Schrödinger. We show that the two formulations are equivalent and discuss the quantum mechanical interpretations of the theory, as well as the classical limit. We also perform some explicit calculations by solving the free particle and the harmonic oscillator problems using path integrals. The energy eigenvalues of the harmonic oscillator is found by exploiting the connection between path integrals, statistical mechanics and imaginary time. Contents 1 Introduction and Outline 2 1.1 Introduction . 2 1.2 Outline . 2 2 Path Integrals from ordinary Quantum Mechanics 4 2.1 The Schrödinger equation and time evolution . 4 2.2 The propagator . 6 3 Equivalence to the Schrödinger Equation 8 3.1 From the Schrödinger equation to PI’s . 8 3.2 From PI’s to the Schrödinger equation .
    [Show full text]