ensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y

Size: px
Start display at page:

Download "ensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y"

Transcription

1 Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay we ll nee to move on to ensity operators so that we can inclue issipation an ephasing to arrive at the full Bloch Equations. We ve alreay seen that an arbitrary pure state for a two-level system can be represente by a vector on the Bloch Sphere. If we write the quantum state vector in the form è cos exp i z è sin exp i z è, then the corresponing Bloch vector connects the origin (center) of the Bloch Sphere to a surface point at polar (latitue) angle an azimuthal (longitue) angle. By convention we assign the Bloch Sphere a raius of 1, so that the Cartesian components of the Bloch vector v are simply v x sin cos, v y sin sin, v z cos. We have also note that the Bloch vector is proportional to S Ò/ è è Ò/ v, since ès x è Ò sin cos, ès y è Ò sin sin, ès z è Ò cos. When working with ensity operators, we can use this connection to efine a generalize Bloch vector: v x Tr x, v y Tr y, v z Tr z. In other wors, we set v è è Ò S for both pure states an ensity operators. It is worth noting that for a general mixe state, v â 1. For example, if 1 1 then v 0 since the Pauli matrices i 1 0 x, y, z 1 0 i are all traceless Tr 0. In cases where correspons to a known ensemble then p èè, 1

2 è è Tr Tr p èè p Tr èè èè p è è Tr èè p v, where v è è an the summation is a vector sum. In other wors, the generalize Bloch vector corresponing to an ensemble of pure states is the weighte vector sum of the Bloch vectors representing the members of the ensemble. Hence we fin that for any equally-weighte two-member ensemble with è 1 è 0, the ensemble Bloch vector will be zero since 1 è an è correspon to antipoal points. Inee, we know that 1 1 èè 1 èè 1 1 for any such ensemble. Similarly, we see that the generalize Bloch vector corresponing to 1 è x è, p 1, è x è, p, 3 è z è, p 3 1, will simply be v x,v y,v z 0,0,1. It is worth noting that any ensity operator may be expresse in the form 1 1 v, by virtue of the orthogonality of the Pauli matrices: Tr i 0, Tr i i. Hence, we fin that v is ust as general as for the purpose of specifying mixe quantum states! The generalize Bloch vector v has only three real parameters v x,v y,v z (or,, v ), but the same is true for since Hermiticity fixes Im 11 Im 0, Re 1 Re 1, Im 1 Im 1, an normalization Tr 1 gives us Re 1 Re 11. Hence Re 11, Re 1, an Im 1 are the only real egrees of freeom for a ensity operator on a two-imensional Hilbert space. Furthermore, Tr 1 4 Tr 1 v 1 4 Tr 1 v x x v y y v z z 1 1 v. Hence the length of the generalize Bloch vector is irectly relate to the purity of.

3 Although we first introuce the ensity operator as a means of representing uncertain preparations for an iniviual quantum system, it is equally well suite to escribing the average state of a collection of ientical systems. By ientical here we mean that each member of the ensemble lives in the same Hilbert space as every other, but the quantum states of the ifferent members of the ensemble will vary. So if we have a collection of twenty spin- 1 particles an prepare ten of them in the state z è an the other ten in x è, the ensemble ensity operator is 1 z èè z 1 x èè x. This type of ensemble ensity operator can be use to compute average values of observables such as S. In fact, the quantity M ân S correspons to the net magnetic moment (or magnetization) of a collection of N spin- 1 particles. When the magnetization vector has maximum length (here 0 â M â N Ò/), all the spins in the ensemble must be pointing in the same irection. When M 0, the irections of the spins in the ensemble are isotropically istribute. Hamiltonian ynamics for a ensity operator Assuming we have an ensemble representation for a given ensity operator, we can erive its equation of motion in the following way: p èè i Ò p è è è è p i Ò H è è è i Ò è H p H èè èè H i H,. Ò Since this erivation is clearly inepenent of the particular ensemble representation we choose for, it is vali in general. Similarly, we know that è T t,0 è exp ih t/ò è, è è T 0,t, so 3

4 Hence t p T t,0 èè T 0,t T t,0 p èè T 0,t T t,0 0 T 0,t. Tr t Tr T t,0 0 T 0,t T t,0 0 T 0,t Tr T t,0 0 T 0,t Tr 0, so Hamiltonian evolution preserves the length of the generalize Bloch vector. In fact, we can use the above methos to erive equations of motion for the Bloch vector itself: v i Tr i Tr i i Tr Ò i H, i Tr Ò i H, 1 v. Hence the evolutions of the three components will generally be couple. For Larmor precession in a static applie fiel, H S B an v i i Tr H, 1 v Ò 4 Tr i B 1 v i 1 v B 4 Tr ib x x ib y y ib z z 1 v x x v y y v z z 4 Tr 1 v x x v y y v z z B x x i B y y i B z z i 4 Tr x y B x v y v x B y x z B x v z v x B z y x B y v x v y B x 4 Tr y z B y v z v y B z z x B z v x v z B x, z y B z v y v z B y where the terms involving 1 have been roppe ue to cyclic property of the trace. Taking into account the orthogonality of the Pauli matrices, we are simply left with 4

5 v x 4 Tr x y z B y v z v y B z x z y B z v y v z B y 4 B yv z v y B z Tr x y z x z y B y v z v y B z, v y 4 B xv z v x B z Tr y x z y z x B z v x v z B x, v z 4 B xv y v x B y Tr z x y z y x B x v y v x B y. In vector notation, we thus recover v v B, S S B. We can also work with ensity operators in the rotating frame. Just as for state vectors we applie the transformation Ë è O z è, O z exp z t/, where is the angular frequency (about z) of the rotating frame, for ensity operators we can apply Ë O z O z an use with H Ë Ë i Ò H Ë, Ë O z HO z 1 iòo z O z 1 O z HO z 1 Ò z. Let s return to our holing fiel B 0 z plus rotating perturbation b 1 cos t x sin t y, H L S z B 0 b 1 cos t S x sin t S y, H 1 Ë Ò ' z b 1 x, where ' L. From this point on let s assume that we are always working in the rotating frame, so we can rop the primes (Ë). Then 5

6 v i i 4 Tr i ' z b 1 x, 1 v x x v y y v z z i 4 Tr i ' z b 1 x 1 v x x v y y v z z i 4 Tr i 1 v x x v y y v z z ' z b 1 x i 4 Tr i ' z v x x v y y b 1 x v y y v z z i 4 Tr i ' v x x v y y z b 1 v y y v z z x, so v x v y i 4 Tr x' z v y y x'v y y z ' v y, i 4 Tr y' z v x x y'v x x z i 4 Tr y b 1 x v z z y b 1 v z z x 'v x b 1v z 4 Tr y x z y z x 'v x b 1 v z, v z 4 i Tr z b 1 x v y y z b 1 v y y x Copying out the results, we have v x ' v y, v y 'v x b 1 v z, v z b 1 v y. b 1 v y 4 Tr z x y z y x b 1 v y. Given our previous results, we shoul not be surprise to iscover that these may also be written as the vector equation v v B eff, where B eff B 0 z b 1x. Don t forget that we have efine L B 0. Finally, we are reay to a relaxation (that is, amping an ephasing) terms! Phenomenologically, one can generally ientify two istinct relaxation timescales for an ensemble of spins in contact with a heat bath or reservoir. These are known as the longituinal relaxation time T 1 an the transverse relaxation time T. Roughly speaking, T 1 measures the time require for spins in the higher-energy eigenstate ( z è if 0 as we have been assuming in these notes) to ecay back own to the groun state (lower-energy eigenstate). Hence T 1 is associate with energy issipation, with longer T 1 implying a lower rate of energy loss into the environment. The transverse relaxation time T accounts for ephasing, by which we mean the tenency of environmental interactions to ranomly perturb the phase of Larmor precession. In the thir term of this course we will use concrete moels of the system-reservoir interaction to erive values for T 1 an T, but for now we will ust put them in by han. In terms of these two relaxation times, we can finally write own the full-blown Bloch Equations, 6

7 v x ' v y v x T 1, v y 'v x b 1 v z v y T 1, 0 v z b 1 v y v z v z T 1 1, where v 0 z is the value of v z at thermal equilibrium. We alreay unerstan that the Hamiltonian component of these equations (terms involving ' an b 1 ) simply escribe Larmor precession of the Bloch vector aroun the effective magnetic fiel in the rotating frame. In orer to get some feeling for the role of the issipative terms, let s look at the solutions with both ' an b 1 set to zero. That is, we maintain a holing fiel B 0 an work in a frame rotating at the Larmor frequency, but set the riving fiel to zero. The Bloch Equations become v x v x T 1, v y v y T 1, 0 v z v z v z T 1 1, for which we can easily write own the analytic solutions v x t v x 0 exp t/t, v y t v y 0 exp t/t, 0 v z t v z 0 v z exp t/t 1 v 0 z. Hence the components of the Bloch vector in the equatorial plane ecay exponentially to zero, with time constant T, an the z component of the Bloch vector ecays exponentially to its equilibrium value v 0 z with time constant T 1. A typical mechanism that woul contribute to T 1 for a collection of spins is collisional interactions. If our spins correspon to the nuclear spins of gas-phase atoms such as 19 Xe, then the spin of any given atom in the ensemble will be ranomly perturbe through collisions with other atoms in the gas. Since each atom carries a magnetic moment, any time two atoms pass close by one another they will each experience a time-epenent magnetic fiel. The envelope an irection of this fiel pulse will essentially be ranom, as it epens on the etails of the atomic spatial traectories uring the collision. If the collisions have a uration that is comparable to the inverse Larmor frequency (for whatever holing fiel is being applie), we may expect that the x an y components of the fiel pulse will have Fourier components near L an collision-inuce spin flips may occur. Appealing to general thermoynamic consierations, we may expect that the net effect of such spin flips will be to bring the energy istribution of the spins into equilibrium with the overall temperature of the gas (hence v 0 z ). Dephasing can occur if the system-reservoir interaction leas to ranomly varying B z.if the iniviual spins in the ensemble experience small perturbing fiels B in aition to the holing fiel B 0, then they will precess aroun z at slightly ifferent frequencies. Viewe in the rotating frame at the average Larmor frequency L, this means that the Bloch vectors corresponing to the members of the ensemble will graually fan out with respect to the longituinal angle. After some time, we may expect that the istribution in shoul become uniform, meaning that the ensemble-average Bloch vector woul have v x v y 0. The timescale T for such ephasing to occur can be erive explicitly for various moels of B. 7

conventions and notation

conventions and notation Ph95a lecture notes, //0 The Bloch Equations A quick review of spin- conventions and notation The quantum state of a spin- particle is represented by a vector in a two-dimensional complex Hilbert space

More information

Separation of Variables

Separation of Variables Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical

More information

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+

More information

Implicit Differentiation

Implicit Differentiation Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,

More information

Qubit channels that achieve capacity with two states

Qubit channels that achieve capacity with two states Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March

More information

Chapter 4. Electrostatics of Macroscopic Media

Chapter 4. Electrostatics of Macroscopic Media Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1

More information

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:

More information

APPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France

APPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation

More information

Solution to the exam in TFY4230 STATISTICAL PHYSICS Wednesday december 1, 2010

Solution to the exam in TFY4230 STATISTICAL PHYSICS Wednesday december 1, 2010 NTNU Page of 6 Institutt for fysikk Fakultet for fysikk, informatikk og matematikk This solution consists of 6 pages. Solution to the exam in TFY423 STATISTICAL PHYSICS Wenesay ecember, 2 Problem. Particles

More information

G4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.

G4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const. G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether

More information

Physics 115C Homework 4

Physics 115C Homework 4 Physics 115C Homework 4 Problem 1 a In the Heisenberg picture, the ynamical equation is the Heisenberg equation of motion: for any operator Q H, we have Q H = 1 t i [Q H,H]+ Q H t where the partial erivative

More information

05 The Continuum Limit and the Wave Equation

05 The Continuum Limit and the Wave Equation Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,

More information

Entanglement is not very useful for estimating multiple phases

Entanglement is not very useful for estimating multiple phases PHYSICAL REVIEW A 70, 032310 (2004) Entanglement is not very useful for estimating multiple phases Manuel A. Ballester* Department of Mathematics, University of Utrecht, Box 80010, 3508 TA Utrecht, The

More information

The Press-Schechter mass function

The Press-Schechter mass function The Press-Schechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for

More information

Quantum mechanical approaches to the virial

Quantum mechanical approaches to the virial Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from

More information

The Exact Form and General Integrating Factors

The Exact Form and General Integrating Factors 7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily

More information

AN INTRODUCTION TO AIRCRAFT WING FLUTTER Revision A

AN INTRODUCTION TO AIRCRAFT WING FLUTTER Revision A AN INTRODUCTION TO AIRCRAFT WIN FLUTTER Revision A By Tom Irvine Email: tomirvine@aol.com January 8, 000 Introuction Certain aircraft wings have experience violent oscillations uring high spee flight.

More information

Average value of position for the anharmonic oscillator: Classical versus quantum results

Average value of position for the anharmonic oscillator: Classical versus quantum results verage value of position for the anharmonic oscillator: Classical versus quantum results R. W. Robinett Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 682 Receive

More information

Lecture 2 Lagrangian formulation of classical mechanics Mechanics

Lecture 2 Lagrangian formulation of classical mechanics Mechanics Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,

More information

N may be the number of photon interactions in a given volume or the number of radioactive disintegrations in a given time. Expectation value:

N may be the number of photon interactions in a given volume or the number of radioactive disintegrations in a given time. Expectation value: DESCRIPTION OF IONIZING RADIATION FIELDS (Chapter 1 p5-19) Stochastic Vs Non-Stochastic Description Raiation interaction is always stochastic to some egree. o entities whether photons or charge particles

More information

Lecture Notes: March C.D. Lin Attosecond X-ray pulses issues:

Lecture Notes: March C.D. Lin Attosecond X-ray pulses issues: Lecture Notes: March 2003-- C.D. Lin Attosecon X-ray pulses issues: 1. Generation: Nee short pulses (less than 7 fs) to generate HHG HHG in the frequency omain HHG in the time omain Issues of attosecon

More information

Quantum Mechanics in Three Dimensions

Quantum Mechanics in Three Dimensions Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.

More information

Survey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013

Survey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013 Survey Sampling Kosuke Imai Department of Politics, Princeton University February 19, 2013 Survey sampling is one of the most commonly use ata collection methos for social scientists. We begin by escribing

More information

Chapter 2 Lagrangian Modeling

Chapter 2 Lagrangian Modeling Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie

More information

Light scattering and dissipative dynamics of many fermionic atoms in an optical lattice

Light scattering and dissipative dynamics of many fermionic atoms in an optical lattice Light scattering an issipative ynamics of many fermionic atoms in an optical lattice S. Sarkar, S. Langer, J. Schachenmayer,, an A. J. Daley, 3 Department of Physics an Astronomy, University of Pittsburgh,

More information

Lecture Introduction. 2 Examples of Measure Concentration. 3 The Johnson-Lindenstrauss Lemma. CS-621 Theory Gems November 28, 2012

Lecture Introduction. 2 Examples of Measure Concentration. 3 The Johnson-Lindenstrauss Lemma. CS-621 Theory Gems November 28, 2012 CS-6 Theory Gems November 8, 0 Lecture Lecturer: Alesaner Mąry Scribes: Alhussein Fawzi, Dorina Thanou Introuction Toay, we will briefly iscuss an important technique in probability theory measure concentration

More information

PHYS 414 Problem Set 2: Turtles all the way down

PHYS 414 Problem Set 2: Turtles all the way down PHYS 414 Problem Set 2: Turtles all the way own This problem set explores the common structure of ynamical theories in statistical physics as you pass from one length an time scale to another. Brownian

More information

Construction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems

Construction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems Construction of the Electronic Raial Wave Functions an Probability Distributions of Hyrogen-like Systems Thomas S. Kuntzleman, Department of Chemistry Spring Arbor University, Spring Arbor MI 498 tkuntzle@arbor.eu

More information

Lecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell

Lecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell Lecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Bair, faculty.uml.eu/cbair University of Massachusetts Lowell 1. Pre-Einstein Relativity - Einstein i not invent the concept of relativity,

More information

Some Examples. Uniform motion. Poisson processes on the real line

Some Examples. Uniform motion. Poisson processes on the real line Some Examples Our immeiate goal is to see some examples of Lévy processes, an/or infinitely-ivisible laws on. Uniform motion Choose an fix a nonranom an efine X := for all (1) Then, {X } is a [nonranom]

More information

Dot trajectories in the superposition of random screens: analysis and synthesis

Dot trajectories in the superposition of random screens: analysis and synthesis 1472 J. Opt. Soc. Am. A/ Vol. 21, No. 8/ August 2004 Isaac Amiror Dot trajectories in the superposition of ranom screens: analysis an synthesis Isaac Amiror Laboratoire e Systèmes Périphériques, Ecole

More information

Assignment 1. g i (x 1,..., x n ) dx i = 0. i=1

Assignment 1. g i (x 1,..., x n ) dx i = 0. i=1 Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition

More information

fv = ikφ n (11.1) + fu n = y v n iσ iku n + gh n. (11.3) n

fv = ikφ n (11.1) + fu n = y v n iσ iku n + gh n. (11.3) n Chapter 11 Rossby waves Supplemental reaing: Pelosky 1 (1979), sections 3.1 3 11.1 Shallow water equations When consiering the general problem of linearize oscillations in a static, arbitrarily stratifie

More information

The Ehrenfest Theorems

The Ehrenfest Theorems The Ehrenfest Theorems Robert Gilmore Classical Preliminaries A classical system with n egrees of freeom is escribe by n secon orer orinary ifferential equations on the configuration space (n inepenent

More information

Generalization of the persistent random walk to dimensions greater than 1

Generalization of the persistent random walk to dimensions greater than 1 PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,

More information

Strength Analysis of CFRP Composite Material Considering Multiple Fracture Modes

Strength Analysis of CFRP Composite Material Considering Multiple Fracture Modes 5--XXXX Strength Analysis of CFRP Composite Material Consiering Multiple Fracture Moes Author, co-author (Do NOT enter this information. It will be pulle from participant tab in MyTechZone) Affiliation

More information

Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian Mechanics Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical

More information

6 Wave equation in spherical polar coordinates

6 Wave equation in spherical polar coordinates 6 Wave equation in spherical polar coorinates We now look at solving problems involving the Laplacian in spherical polar coorinates. The angular epenence of the solutions will be escribe by spherical harmonics.

More information

LeChatelier Dynamics

LeChatelier Dynamics LeChatelier Dynamics Robert Gilmore Physics Department, Drexel University, Philaelphia, Pennsylvania 1914, USA (Date: June 12, 28, Levine Birthay Party: To be submitte.) Dynamics of the relaxation of a

More information

1 Heisenberg Representation

1 Heisenberg Representation 1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.

More information

The Principle of Least Action

The Principle of Least Action Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of

More information

arxiv: v1 [cond-mat.stat-mech] 9 Jan 2012

arxiv: v1 [cond-mat.stat-mech] 9 Jan 2012 arxiv:1201.1836v1 [con-mat.stat-mech] 9 Jan 2012 Externally riven one-imensional Ising moel Amir Aghamohammai a 1, Cina Aghamohammai b 2, & Mohamma Khorrami a 3 a Department of Physics, Alzahra University,

More information

arxiv: v1 [quant-ph] 14 May 2015

arxiv: v1 [quant-ph] 14 May 2015 Spin Amplification with Inhomogeneous Broaening Suguru Eno, 1 Yuichiro Matsuzaki, William J. Munro, an Shiro Saito 1 rauate School of Funamental Science an Technology, Keio University,Yokohama 3-85, Japan

More information

Vectors in two dimensions

Vectors in two dimensions Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication

More information

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x) Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)

More information

Survival Facts from Quantum Mechanics

Survival Facts from Quantum Mechanics Survival Facts from Quantum Mechanics Operators, Eigenvalues an Eigenfunctions An operator O may be thought as something that operates on a function to prouce another function. We enote operators with

More information

1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.

1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7. Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency

More information

Ψ(x) = c 0 φ 0 (x) + c 1 φ 1 (x) (1) Ĥφ n (x) = E n φ n (x) (2) Ψ = c 0 φ 0 + c 1 φ 1 (7) Ĥ φ n = E n φ n (4)

Ψ(x) = c 0 φ 0 (x) + c 1 φ 1 (x) (1) Ĥφ n (x) = E n φ n (x) (2) Ψ = c 0 φ 0 + c 1 φ 1 (7) Ĥ φ n = E n φ n (4) 1 Problem 1 Ψx = c 0 φ 0 x + c 1 φ 1 x 1 Ĥφ n x = E n φ n x E Ψ is the expectation value of energy of the state 1 taken with respect to the hamiltonian of the system. Thinking in Dirac notation 1 an become

More information

Kramers Relation. Douglas H. Laurence. Department of Physical Sciences, Broward College, Davie, FL 33314

Kramers Relation. Douglas H. Laurence. Department of Physical Sciences, Broward College, Davie, FL 33314 Kramers Relation Douglas H. Laurence Department of Physical Sciences, Browar College, Davie, FL 333 Introuction Kramers relation, name after the Dutch physicist Hans Kramers, is a relationship between

More information

QUANTUMMECHANICAL BEHAVIOUR IN A DETERMINISTIC MODEL. G. t Hooft

QUANTUMMECHANICAL BEHAVIOUR IN A DETERMINISTIC MODEL. G. t Hooft QUANTUMMECHANICAL BEHAVIOUR IN A DETERMINISTIC MODEL G. t Hooft Institute for Theoretical Physics University of Utrecht, P.O.Box 80 006 3508 TA Utrecht, the Netherlans e-mail: g.thooft@fys.ruu.nl THU-96/39

More information

G j dq i + G j. q i. = a jt. and

G j dq i + G j. q i. = a jt. and Lagrange Multipliers Wenesay, 8 September 011 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine

More information

Conservation Laws. Chapter Conservation of Energy

Conservation Laws. Chapter Conservation of Energy 20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action

More information

Nuclear Physics and Astrophysics

Nuclear Physics and Astrophysics Nuclear Physics an Astrophysics PHY-302 Dr. E. Rizvi Lecture 2 - Introuction Notation Nuclies A Nuclie is a particular an is esignate by the following notation: A CN = Atomic Number (no. of Protons) A

More information

Homework 7 Due 18 November at 6:00 pm

Homework 7 Due 18 November at 6:00 pm Homework 7 Due 18 November at 6:00 pm 1. Maxwell s Equations Quasi-statics o a An air core, N turn, cylinrical solenoi of length an raius a, carries a current I Io cos t. a. Using Ampere s Law, etermine

More information

TMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments

TMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary

More information

Free rotation of a rigid body 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012

Free rotation of a rigid body 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012 Free rotation of a rigi boy 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012 1 Introuction In this section, we escribe the motion of a rigi boy that is free to rotate

More information

Convective heat transfer

Convective heat transfer CHAPTER VIII Convective heat transfer The previous two chapters on issipative fluis were evote to flows ominate either by viscous effects (Chap. VI) or by convective motion (Chap. VII). In either case,

More information

6. Friction and viscosity in gasses

6. Friction and viscosity in gasses IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner

More information

FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS. 1. Introduction

FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS. 1. Introduction FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN 1 Introuction In this note, we stuy the fluctuations in the number

More information

and from it produce the action integral whose variation we set to zero:

and from it produce the action integral whose variation we set to zero: Lagrange Multipliers Monay, 6 September 01 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine

More information

RFSS: Lecture 4 Alpha Decay

RFSS: Lecture 4 Alpha Decay RFSS: Lecture 4 Alpha Decay Reaings Nuclear an Raiochemistry: Chapter 3 Moern Nuclear Chemistry: Chapter 7 Energetics of Alpha Decay Geiger Nuttall base theory Theory of Alpha Decay Hinrance Factors Different

More information

1 Introuction In the past few years there has been renewe interest in the nerson impurity moel. This moel was originally propose by nerson [2], for a

1 Introuction In the past few years there has been renewe interest in the nerson impurity moel. This moel was originally propose by nerson [2], for a Theory of the nerson impurity moel: The Schrieer{Wol transformation re{examine Stefan K. Kehrein 1 an nreas Mielke 2 Institut fur Theoretische Physik, uprecht{karls{universitat, D{69120 Heielberg, F..

More information

SYNCHRONOUS SEQUENTIAL CIRCUITS

SYNCHRONOUS SEQUENTIAL CIRCUITS CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents

More information

Quantum optics of a Bose-Einstein condensate coupled to a quantized light field

Quantum optics of a Bose-Einstein condensate coupled to a quantized light field PHYSICAL REVIEW A VOLUME 60, NUMBER 2 AUGUST 1999 Quantum optics of a Bose-Einstein conensate couple to a quantize light fiel M. G. Moore, O. Zobay, an P. Meystre Optical Sciences Center an Department

More information

Lagrangian and Hamiltonian Dynamics

Lagrangian and Hamiltonian Dynamics Lagrangian an Hamiltonian Dynamics Volker Perlick (Lancaster University) Lecture 1 The Passage from Newtonian to Lagrangian Dynamics (Cockcroft Institute, 22 February 2010) Subjects covere Lecture 2: Discussion

More information

Homework 3 - Solutions

Homework 3 - Solutions Homework 3 - Solutions The Transpose an Partial Transpose. 1 Let { 1, 2,, } be an orthonormal basis for C. The transpose map efine with respect to this basis is a superoperator Γ that acts on an operator

More information

4. Important theorems in quantum mechanics

4. Important theorems in quantum mechanics TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this

More information

6 General properties of an autonomous system of two first order ODE

6 General properties of an autonomous system of two first order ODE 6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x

More information

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control 19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior

More information

Numerical Integrator. Graphics

Numerical Integrator. Graphics 1 Introuction CS229 Dynamics Hanout The question of the week is how owe write a ynamic simulator for particles, rigi boies, or an articulate character such as a human figure?" In their SIGGRPH course notes,

More information

A Model of Electron-Positron Pair Formation

A Model of Electron-Positron Pair Formation Volume PROGRESS IN PHYSICS January, 8 A Moel of Electron-Positron Pair Formation Bo Lehnert Alfvén Laboratory, Royal Institute of Technology, S-44 Stockholm, Sween E-mail: Bo.Lehnert@ee.kth.se The elementary

More information

Analytic Scaling Formulas for Crossed Laser Acceleration in Vacuum

Analytic Scaling Formulas for Crossed Laser Acceleration in Vacuum October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945

More information

Formulation of statistical mechanics for chaotic systems

Formulation of statistical mechanics for chaotic systems PRAMANA c Inian Acaemy of Sciences Vol. 72, No. 2 journal of February 29 physics pp. 315 323 Formulation of statistical mechanics for chaotic systems VISHNU M BANNUR 1, an RAMESH BABU THAYYULLATHIL 2 1

More information

Total Energy Shaping of a Class of Underactuated Port-Hamiltonian Systems using a New Set of Closed-Loop Potential Shape Variables*

Total Energy Shaping of a Class of Underactuated Port-Hamiltonian Systems using a New Set of Closed-Loop Potential Shape Variables* 51st IEEE Conference on Decision an Control December 1-13 212. Maui Hawaii USA Total Energy Shaping of a Class of Uneractuate Port-Hamiltonian Systems using a New Set of Close-Loop Potential Shape Variables*

More information

Introduction to the Vlasov-Poisson system

Introduction to the Vlasov-Poisson system Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its

More information

Topic 7: Convergence of Random Variables

Topic 7: Convergence of Random Variables Topic 7: Convergence of Ranom Variables Course 003, 2016 Page 0 The Inference Problem So far, our starting point has been a given probability space (S, F, P). We now look at how to generate information

More information

Concentration of Measure Inequalities for Compressive Toeplitz Matrices with Applications to Detection and System Identification

Concentration of Measure Inequalities for Compressive Toeplitz Matrices with Applications to Detection and System Identification Concentration of Measure Inequalities for Compressive Toeplitz Matrices with Applications to Detection an System Ientification Borhan M Sananaji, Tyrone L Vincent, an Michael B Wakin Abstract In this paper,

More information

Chapter 2: Quantum Master Equations

Chapter 2: Quantum Master Equations Chapter 2: Quantum Master Equations I. THE LINDBLAD FORM A. Superoperators an ynamical maps The Liouville von Neumann equation is given by t ρ = i [H, ρ]. (1) We can efine a superoperator L such that Lρ

More information

Convergence of Random Walks

Convergence of Random Walks Chapter 16 Convergence of Ranom Walks This lecture examines the convergence of ranom walks to the Wiener process. This is very important both physically an statistically, an illustrates the utility of

More information

NOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,

NOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy, NOTES ON EULER-BOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between Euler-MacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which

More information

Open System Dynamics with Non-Markovian Quantum Trajectories

Open System Dynamics with Non-Markovian Quantum Trajectories Open System Dynamics with Non-Markovian Quantum Trajectories W. T. Strunz 1,,L.Diósi,anN.Gisin 3 1 Fachbereich Physik, Universität GH Essen, 45117 Essen, Germany Research Institute for Particle an Nuclear

More information

Gyroscopic matrices of the right beams and the discs

Gyroscopic matrices of the right beams and the discs Titre : Matrice gyroscopique es poutres roites et es i[...] Date : 15/07/2014 Page : 1/16 Gyroscopic matrices of the right beams an the iscs Summary: This ocument presents the formulation of the matrices

More information

inflow outflow Part I. Regular tasks for MAE598/494 Task 1

inflow outflow Part I. Regular tasks for MAE598/494 Task 1 MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION Long-istance coherent coupling in a quantum ot array Floris R. Braakman 1, Pierre Barthelemy 1, Christian Reichl, Werner Wegscheier, Lieven M.K. Vanersypen 1 1 Kavli Institute of Nanoscience, TU Delft,

More information

arxiv:quant-ph/ v1 29 Jun 2001

arxiv:quant-ph/ v1 29 Jun 2001 Atomic wave packet basis for quantum information Ashok Muthukrishnan an C. R. Strou, Jr. The Institute of Optics, University of Rochester, Rochester, New York 14627 (March 15, 2001) arxiv:quant-ph/0106165

More information

Parameter estimation: A new approach to weighting a priori information

Parameter estimation: A new approach to weighting a priori information Parameter estimation: A new approach to weighting a priori information J.L. Mea Department of Mathematics, Boise State University, Boise, ID 83725-555 E-mail: jmea@boisestate.eu Abstract. We propose a

More information

On the Conservation of Information in Quantum Physics

On the Conservation of Information in Quantum Physics On the Conservation of Information in Quantum Physics Marco Roncaglia Physics Department an Research Center OPTIMAS, University of Kaiserslautern, Germany (Date: September 11, 2017 escribe the full informational

More information

Lower Bounds for the Smoothed Number of Pareto optimal Solutions

Lower Bounds for the Smoothed Number of Pareto optimal Solutions Lower Bouns for the Smoothe Number of Pareto optimal Solutions Tobias Brunsch an Heiko Röglin Department of Computer Science, University of Bonn, Germany brunsch@cs.uni-bonn.e, heiko@roeglin.org Abstract.

More information

FROM THE VON-NEUMANN EQUATION TO THE QUANTUM BOLTZMANN EQUATION IN A DETERMINISTIC FRAMEWORK F. CASTELLA 1

FROM THE VON-NEUMANN EQUATION TO THE QUANTUM BOLTZMANN EQUATION IN A DETERMINISTIC FRAMEWORK F. CASTELLA 1 FROM THE VON-NEUMANN EQUATION TO THE QUANTUM BOTZMANN EQUATION IN A DETERMINISTIC FRAMEWORK F. CASTEA 1 J. Stat. Phys., Vol. 104, N. 1/2, pp. 387-447 (2001). Abstract In this paper, we investigate the

More information

Spectral Flow, the Magnus Force, and the. Josephson-Anderson Relation

Spectral Flow, the Magnus Force, and the. Josephson-Anderson Relation Spectral Flow, the Magnus Force, an the arxiv:con-mat/9602094v1 16 Feb 1996 Josephson-Anerson Relation P. Ao Department of Theoretical Physics Umeå University, S-901 87, Umeå, SWEDEN October 18, 2018 Abstract

More information

LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION

LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische

More information

Delocalization of boundary states in disordered topological insulators

Delocalization of boundary states in disordered topological insulators Journal of Physics A: Mathematical an Theoretical J. Phys. A: Math. Theor. 48 (05) FT0 (pp) oi:0.088/75-83/48//ft0 Fast Track Communication Delocalization of bounary states in isorere topological insulators

More information

Time-of-Arrival Estimation in Non-Line-Of-Sight Environments

Time-of-Arrival Estimation in Non-Line-Of-Sight Environments 2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor

More information

ON THE MEANING OF LORENTZ COVARIANCE

ON THE MEANING OF LORENTZ COVARIANCE Founations of Physics Letters 17 (2004) pp. 479 496. ON THE MEANING OF LORENTZ COVARIANCE László E. Szabó Theoretical Physics Research Group of the Hungarian Acaemy of Sciences Department of History an

More information

Physics 2112 Unit 5: Electric Potential Energy

Physics 2112 Unit 5: Electric Potential Energy Physics 11 Unit 5: Electric Potential Energy Toay s Concept: Electric Potential Energy Unit 5, Slie 1 Stuff you aske about: I on't like this return to mechanics an the potential energy concept, but this

More information

Physics 2212 GJ Quiz #4 Solutions Fall 2015

Physics 2212 GJ Quiz #4 Solutions Fall 2015 Physics 2212 GJ Quiz #4 Solutions Fall 215 I. (17 points) The magnetic fiel at point P ue to a current through the wire is 5. µt into the page. The curve portion of the wire is a semicircle of raius 2.

More information

Chapter 6: Energy-Momentum Tensors

Chapter 6: Energy-Momentum Tensors 49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.

More information

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21 Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting

More information

Can we derive Newton s F = ma from the SE?

Can we derive Newton s F = ma from the SE? 8.04 Quantum Physics Lecture XIII p = pˆ (13-1) ( ( ) ) = xψ Ψ (13-) ( ) = xψ Ψ (13-3) [ ] = x (ΨΨ ) Ψ Ψ (13-4) ( ) = xψ Ψ (13-5) = p, (13-6) where again we have use integration by parts an the fact that

More information

Non-linear Cosmic Ray propagation close to the acceleration site

Non-linear Cosmic Ray propagation close to the acceleration site Non-linear Cosmic Ray propagation close to the acceleration site The Hebrew University of Jerusalem, Jerusalem, 91904, Israel E-mail: lara.nava@mail.huji.ac.il Stefano Gabici Astroparticule et Cosmologie

More information