Probability Current and Current Operators in Quantum Mechanics G
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Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 December 2019 doi:10.20944/preprints201912.0044.v1 Peer-reviewed version available at Universe 2020, 6, 24; doi:10.3390/universe6020024 1 Review 2 Rotation and Spin and Position Operators in 3 Relativistic Gravity and Quantum Electrodynamics 4 R.F. O’Connell 5 Department of Physics and Astronomy, Louisiana State University; Baton Rouge, LA 70803-4001, USA 6 Correspondence: [email protected]; 225-578-6848 7 Abstract: First, we examine how spin is treated in special relativity and the necessity of introducing 8 spin supplementary conditions (SSC) and how they are related to the choice of a center-of-mass of a 9 spinning particle. Next, we discuss quantum electrodynamics and the Foldy-Wouthuysen 10 transformation which we note is a position operator identical to the Pryce-Newton-Wigner position 11 operator. The classical version of the operators are shown to be essential for the treatment of 12 classical relativistic particles in general relativity, of special interest being the case of binary 13 systems (black holes/neutron stars) which emit gravitational radiation. 14 Keywords: rotation; spin; position operators 15 16 I.Introduction 17 Rotation effects in relativistic systems involve many new concepts not needed in 18 non-relativistic classical physics. Some of these are quantum mechanical (where the emphasis is on 19 “spin”). Thus, our emphasis will be on quantum electrodynamics (QED) and both special and 20 general relativity. Also, as in [1] , we often use “spin” in the generic sense of meaning “internal” spin 21 in the case of an elementary particle and “rotation” in the case of an elementary particle. -
A Study of Fractional Schrödinger Equation-Composed Via Jumarie Fractional Derivative
A Study of Fractional Schrödinger Equation-composed via Jumarie fractional derivative Joydip Banerjee1, Uttam Ghosh2a , Susmita Sarkar2b and Shantanu Das3 Uttar Buincha Kajal Hari Primary school, Fulia, Nadia, West Bengal, India email- [email protected] 2Department of Applied Mathematics, University of Calcutta, Kolkata, India; 2aemail : [email protected] 2b email : [email protected] 3 Reactor Control Division BARC Mumbai India email : [email protected] Abstract One of the motivations for using fractional calculus in physical systems is due to fact that many times, in the space and time variables we are dealing which exhibit coarse-grained phenomena, meaning that infinitesimal quantities cannot be placed arbitrarily to zero-rather they are non-zero with a minimum length. Especially when we are dealing in microscopic to mesoscopic level of systems. Meaning if we denote x the point in space andt as point in time; then the differentials dx (and dt ) cannot be taken to limit zero, rather it has spread. A way to take this into account is to use infinitesimal quantities as ()Δx α (and ()Δt α ) with 01<α <, which for very-very small Δx (and Δt ); that is trending towards zero, these ‘fractional’ differentials are greater that Δx (and Δt ). That is()Δx α >Δx . This way defining the differentials-or rather fractional differentials makes us to use fractional derivatives in the study of dynamic systems. In fractional calculus the fractional order trigonometric functions play important role. The Mittag-Leffler function which plays important role in the field of fractional calculus; and the fractional order trigonometric functions are defined using this Mittag-Leffler function. -
Motion of the Reduced Density Operator
Motion of the Reduced Density Operator Nicholas Wheeler, Reed College Physics Department Spring 2009 Introduction. Quantum mechanical decoherence, dissipation and measurements all involve the interaction of the system of interest with an environmental system (reservoir, measurement device) that is typically assumed to possess a great many degrees of freedom (while the system of interest is typically assumed to possess relatively few degrees of freedom). The state of the composite system is described by a density operator ρ which in the absence of system-bath interaction we would denote ρs ρe, though in the cases of primary interest that notation becomes unavailable,⊗ since in those cases the states of the system and its environment are entangled. The observable properties of the system are latent then in the reduced density operator ρs = tre ρ (1) which is produced by “tracing out” the environmental component of ρ. Concerning the specific meaning of (1). Let n) be an orthonormal basis | in the state space H of the (open) system, and N) be an orthonormal basis s !| " in the state space H of the (also open) environment. Then n) N) comprise e ! " an orthonormal basis in the state space H = H H of the |(closed)⊗| composite s e ! " system. We are in position now to write ⊗ tr ρ I (N ρ I N) e ≡ s ⊗ | s ⊗ | # ! " ! " ↓ = ρ tr ρ in separable cases s · e The dynamics of the composite system is generated by Hamiltonian of the form H = H s + H e + H i 2 Motion of the reduced density operator where H = h I s s ⊗ e = m h n m N n N $ | s| % | % ⊗ | % · $ | ⊗ $ | m,n N # # $% & % &' H = I h e s ⊗ e = n M n N M h N | % ⊗ | % · $ | ⊗ $ | $ | e| % n M,N # # $% & % &' H = m M m M H n N n N i | % ⊗ | % $ | ⊗ $ | i | % ⊗ | % $ | ⊗ $ | m,n M,N # # % &$% & % &'% & —all components of which we will assume to be time-independent. -
Motion and Time Study the Goals of Motion Study
Motion and Time Study The Goals of Motion Study • Improvement • Planning / Scheduling (Cost) •Safety Know How Long to Complete Task for • Scheduling (Sequencing) • Efficiency (Best Way) • Safety (Easiest Way) How Does a Job Incumbent Spend a Day • Value Added vs. Non-Value Added The General Strategy of IE to Reduce and Control Cost • Are people productive ALL of the time ? • Which parts of job are really necessary ? • Can the job be done EASIER, SAFER and FASTER ? • Is there a sense of employee involvement? Some Techniques of Industrial Engineering •Measure – Time and Motion Study – Work Sampling • Control – Work Standards (Best Practices) – Accounting – Labor Reporting • Improve – Small group activities Time Study • Observation –Stop Watch – Computer / Interactive • Engineering Labor Standards (Bad Idea) • Job Order / Labor reporting data History • Frederick Taylor (1900’s) Studied motions of iron workers – attempted to “mechanize” motions to maximize efficiency – including proper rest, ergonomics, etc. • Frank and Lillian Gilbreth used motion picture to study worker motions – developed 17 motions called “therbligs” that describe all possible work. •GET G •PUT P • GET WEIGHT GW • PUT WEIGHT PW •REGRASP R • APPLY PRESSURE A • EYE ACTION E • FOOT ACTION F • STEP S • BEND & ARISE B • CRANK C Time Study (Stopwatch Measurement) 1. List work elements 2. Discuss with worker 3. Measure with stopwatch (running VS reset) 4. Repeat for n Observations 5. Compute mean and std dev of work station time 6. Be aware of allowances/foreign element, etc Work Sampling • Determined what is done over typical day • Random Reporting • Periodic Reporting Learning Curve • For repetitive work, worker gains skill, knowledge of product/process, etc over time • Thus we expect output to increase over time as more units are produced over time to complete task decreases as more units are produced Traditional Learning Curve Actual Curve Change, Design, Process, etc Learning Curve • Usually define learning as a percentage reduction in the time it takes to make a unit. -
Density Matrix Description of NMR
Density Matrix Description of NMR BCMB/CHEM 8190 Operators in Matrix Notation • It will be important, and convenient, to express the commonly used operators in matrix form • Consider the operator Iz and the single spin functions α and β - recall ˆ ˆ ˆ ˆ ˆ ˆ Ix α = 1 2 β Ix β = 1 2α Iy α = 1 2 iβ Iy β = −1 2 iα Iz α = +1 2α Iz β = −1 2 β α α = β β =1 α β = β α = 0 - recall the expectation value for an observable Q = ψ Qˆ ψ = ∫ ψ∗Qˆψ dτ Qˆ - some operator ψ - some wavefunction - the matrix representation is the possible expectation values for the basis functions α β α ⎡ α Iˆ α α Iˆ β ⎤ ⎢ z z ⎥ ⎢ ˆ ˆ ⎥ β ⎣ β Iz α β Iz β ⎦ ⎡ α Iˆ α α Iˆ β ⎤ ⎡1 2 α α −1 2 α β ⎤ ⎡1 2 0 ⎤ ⎡1 0 ⎤ ˆ ⎢ z z ⎥ 1 Iz = = ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ ⎢ ˆ ˆ ⎥ 1 2 β α −1 2 β β 0 −1 2 2 0 −1 ⎣ β Iz α β Iz β ⎦ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎣ ⎦ • This is convenient, as the operator is just expressed as a matrix of numbers – no need to derive it again, just store it in computer Operators in Matrix Notation • The matrices for Ix, Iy,and Iz are called the Pauli spin matrices ˆ ⎡ 0 1 2⎤ 1 ⎡0 1 ⎤ ˆ ⎡0 −1 2 i⎤ 1 ⎡0 −i ⎤ ˆ ⎡1 2 0 ⎤ 1 ⎡1 0 ⎤ Ix = ⎢ ⎥ = ⎢ ⎥ Iy = ⎢ ⎥ = ⎢ ⎥ Iz = ⎢ ⎥ = ⎢ ⎥ ⎣1 2 0 ⎦ 2 ⎣1 0⎦ ⎣1 2 i 0 ⎦ 2 ⎣i 0 ⎦ ⎣ 0 −1 2⎦ 2 ⎣0 −1 ⎦ • Express α , β , α and β as 1×2 column and 2×1 row vectors ⎡1⎤ ⎡0⎤ α = ⎢ ⎥ β = ⎢ ⎥ α = [1 0] β = [0 1] ⎣0⎦ ⎣1⎦ • Using matrices, the operations of Ix, Iy, and Iz on α and β , and the orthonormality relationships, are shown below ˆ 1 ⎡0 1⎤⎡1⎤ 1 ⎡0⎤ 1 ˆ 1 ⎡0 1⎤⎡0⎤ 1 ⎡1⎤ 1 Ix α = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = β Ix β = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = α 2⎣1 0⎦⎣0⎦ 2⎣1⎦ 2 2⎣1 0⎦⎣1⎦ 2⎣0⎦ 2 ⎡1⎤ ⎡0⎤ α α = [1 0]⎢ ⎥ -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
Two-State Systems
1 TWO-STATE SYSTEMS Introduction. Relative to some/any discretely indexed orthonormal basis |n) | ∂ | the abstract Schr¨odinger equation H ψ)=i ∂t ψ) can be represented | | | ∂ | (m H n)(n ψ)=i ∂t(m ψ) n ∂ which can be notated Hmnψn = i ∂tψm n H | ∂ | or again ψ = i ∂t ψ We found it to be the fundamental commutation relation [x, p]=i I which forced the matrices/vectors thus encountered to be ∞-dimensional. If we are willing • to live without continuous spectra (therefore without x) • to live without analogs/implications of the fundamental commutator then it becomes possible to contemplate “toy quantum theories” in which all matrices/vectors are finite-dimensional. One loses some physics, it need hardly be said, but surprisingly much of genuine physical interest does survive. And one gains the advantage of sharpened analytical power: “finite-dimensional quantum mechanics” provides a methodological laboratory in which, not infrequently, the essentials of complicated computational procedures can be exposed with closed-form transparency. Finally, the toy theory serves to identify some unanticipated formal links—permitting ideas to flow back and forth— between quantum mechanics and other branches of physics. Here we will carry the technique to the limit: we will look to “2-dimensional quantum mechanics.” The theory preserves the linearity that dominates the full-blown theory, and is of the least-possible size in which it is possible for the effects of non-commutivity to become manifest. 2 Quantum theory of 2-state systems We have seen that quantum mechanics can be portrayed as a theory in which • states are represented by self-adjoint linear operators ρ ; • motion is generated by self-adjoint linear operators H; • measurement devices are represented by self-adjoint linear operators A. -
Representations of U(2O) and the Value of the Fine Structure Constant
Symmetry, Integrability and Geometry: Methods and Applications Vol. 1 (2005), Paper 028, 8 pages Representations of U(2∞) and the Value of the Fine Structure Constant William H. KLINK Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA E-mail: [email protected] Received September 28, 2005, in final form December 17, 2005; Published online December 25, 2005 Original article is available at http://www.emis.de/journals/SIGMA/2005/Paper028/ Abstract. A relativistic quantum mechanics is formulated in which all of the interactions are in the four-momentum operator and Lorentz transformations are kinematic. Interactions are introduced through vertices, which are bilinear in fermion and antifermion creation and annihilation operators, and linear in boson creation and annihilation operators. The fermion- antifermion operators generate a unitary Lie algebra, whose representations are fixed by a first order Casimir operator (corresponding to baryon number or charge). Eigenvectors and eigenvalues of the four-momentum operator are analyzed and exact solutions in the strong coupling limit are sketched. A simple model shows how the fine structure constant might be determined for the QED vertex. Key words: point form relativistic quantum mechanics; antisymmetric representations of infinite unitary groups; semidirect sum of unitary with Heisenberg algebra 2000 Mathematics Subject Classification: 22D10; 81R10; 81T27 1 Introduction With the exception of QCD and gravitational self couplings, all of the fundamental particle interaction Hamiltonians have the form of bilinears in fermion and antifermion creation and annihilation operators times terms linear in boson creation and annihilation operators. For example QED is a theory bilinear in electron and positron creation and annihilation operators and linear in photon creation and annihilation operators. -
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. -
Second Quantization
Chapter 1 Second Quantization 1.1 Creation and Annihilation Operators in Quan- tum Mechanics We will begin with a quick review of creation and annihilation operators in the non-relativistic linear harmonic oscillator. Let a and a† be two operators acting on an abstract Hilbert space of states, and satisfying the commutation relation a,a† = 1 (1.1) where by “1” we mean the identity operator of this Hilbert space. The operators a and a† are not self-adjoint but are the adjoint of each other. Let α be a state which we will take to be an eigenvector of the Hermitian operators| ia†a with eigenvalue α which is a real number, a†a α = α α (1.2) | i | i Hence, α = α a†a α = a α 2 0 (1.3) h | | i k | ik ≥ where we used the fundamental axiom of Quantum Mechanics that the norm of all states in the physical Hilbert space is positive. As a result, the eigenvalues α of the eigenstates of a†a must be non-negative real numbers. Furthermore, since for all operators A, B and C [AB, C]= A [B, C] + [A, C] B (1.4) we get a†a,a = a (1.5) − † † † a a,a = a (1.6) 1 2 CHAPTER 1. SECOND QUANTIZATION i.e., a and a† are “eigen-operators” of a†a. Hence, a†a a = a a†a 1 (1.7) − † † † † a a a = a a a +1 (1.8) Consequently we find a†a a α = a a†a 1 α = (α 1) a α (1.9) | i − | i − | i Hence the state aα is an eigenstate of a†a with eigenvalue α 1, provided a α = 0. -
Angular Momentum 23.1 Classical Description
Physics 342 Lecture 23 Angular Momentum Lecture 23 Physics 342 Quantum Mechanics I Monday, March 31st, 2008 We know how to obtain the energy of Hydrogen using the Hamiltonian op- erator { but given a particular En, there is degeneracy { many n`m(r; θ; φ) have the same energy. What we would like is a set of operators that allow us to determine ` and m. The angular momentum operator L^, and in partic- 2 ular the combination L and Lz provide precisely the additional Hermitian observables we need. 23.1 Classical Description Going back to our Hamiltonian for a central potential, we have p · p H = + U(r): (23.1) 2 m It is clear from the dependence of U on the radial distance only, that angular momentum should be conserved in this setting. Remember the definition: L = r × p; (23.2) and we can write this in indexed notation which is slightly easier to work with using the Levi-Civita symbol, defined as (in three dimensions) 8 < 1 for (ijk) an even permutation of (123). ijk = −1 for an odd permutation (23.3) : 0 otherwise Then we can write 3 3 X X Li = ijk rj pk = ijk rj pk (23.4) j=1 k=1 1 of 9 23.2. QUANTUM COMMUTATORS Lecture 23 where the sum over j and k is implied in the second equality (this is Einstein summation notation). There are three numbers here, Li for i = 1; 2; 3, we associate with Lx, Ly, and Lz, the individual components of angular momentum about the three spatial axes. -
(Ab Initio) Pathintegral Molecular Dynamics
(Ab initio) path-integral Molecular Dynamics The double slit experiment Sum over paths: Suppose only two paths: interference The double slit experiment ● Introduce of a large number of intermediate gratings, each containing many slits. ● Electrons may pass through any sequence of slits before reaching the detector ● Take the limit in which infinitely many gratings → empty space Do nuclei really behave classically ? Do nuclei really behave classically ? Do nuclei really behave classically ? Do nuclei really behave classically ? Example: proton transfer in malonaldehyde only one quantum proton Tuckerman and Marx, Phys. Rev. Lett. (2001) Example: proton transfer in water + + Classical H5O2 Quantum H5O2 Marx et al. Nature (1997) Proton inWater-Hydroxyl (Ice) Overlayers on Metal Surfaces T = 160 K Li et al., Phys. Rev. Lett. (2010) Proton inWater-Hydroxyl (Ice) Overlayers on Metal Surfaces Li et al., Phys. Rev. Lett. (2010) The density matrix: definition Ensemble of states: Ensemble average: Let©s define: ρ is hermitian: real eigenvalues The density matrix: time evolution and equilibrium At equilibrium: ρ can be expressed as pure function of H and diagonalized simultaneously with H The density matrix: canonical ensemble Canonical ensemble: Canonical density matrix: Path integral formulation One particle one dimension: K and Φ do not commute, thus, Trotter decomposition: Canonical density matrix: Path integral formulation Evaluation of: Canonical density matrix: Path integral formulation Matrix elements of in space coordinates: acts on the eigenstates from the left: Canonical density matrix: Path integral formulation Where it was used: It can be Monte Carlo sampled, but for MD we need momenta! Path integral isomorphism Introducing a chain frequency and an effective potential: Path integral molecular dynamics Fictitious momenta: introducing a set of P Gaussian integrals : convenient parameter Gaussian integrals known: (adjust prefactor ) Does it work? i.