Notes on Spin Operators and the Heisenberg Model. Physics : Winter, David G. Stroud
|
|
- Karin Hancock
- 5 years ago
- Views:
Transcription
1 Notes on Spin Operators and the Heisenberg Model Physics : Winter, David G. Stroud In these notes I give a brief discussion of spin-1/ operators and their use in the Heisenberg model. 1. Spin Operators for Spin-1/ Particles. We denote the spin angular momentum operator for a single spin by S = (S x, S y, S z ). These operators satisfy the commutation relations [S x, S y ] = i hs z (1) and cyclic permutations of these relations. For spin-1/ particles, these can be written S i = h σ i, () where i = x, y, z, and the σ is are the Pauli spin matrices. These spin matrices have the convenient representations, σ x = , 1
2 0 i σ y =, i 0 σ z = We can also introduce the raising and lowering operators S ± = S x ± is y. (3) In terms of these, S x = 1 (S + + S ) (4) S y = 1 i (S + S ) (5) S = 1 (S +S + S S + ) + S z. (6) S z has two eigenstates, which may be denoted and, with eigenvalues ± h/. Thus and S z = h (7) S z = h. (8) If the spin operators are represented by Pauli matrices, then and are column vectors: = 1 0
3 and = 0 1. It is also easily shown, using the Pauli spin matrices, that S + and S satisfy S + = (9) S = (10) S + = 0 (11) S = 0. (1) Using all these results, one can show that the states and are also eigenstates of the operator S :. Spin Operators for Two Spin-1/ Particles. S = 3 h (13) 4 S = 3 h. (14) 4 If one has two identical spin-1/ particles, one may introduce the spin operators S 1i and S i (i = x, y, z) for spins 1 and. The i th component of total spin is The spin operators satisfy the commutation relations S i = S 1i + S i. (15) [S αi, S βj ] = i hs αk δ αβ, (16) 3
4 where i, j, k are cyclic permutations of x, y, z and α and β label the spins. One can also introduce the square of the total spin, S = S x + S y + S z. (17) It can easily be shown that S = 1 (S 1+S + S 1 S + ) + S 1z S z. (18) If there are two spins, then there are four possible spin states. A convenient basis, in obvious notation, is the four states,,, and. These can be arranged into a set of three triplet states, which we denote for short 10 and 1 ± 1, and a singlet state, which we denote 00. The triplet states are given by 11 = (19) 10 = 1 ( + ) (0) 1 1 =. (1) The singlet state is 00 = 1 ( + ). () The first number is the total spin quantum number and the second is the S z quantum number. A state with total spin quantum number S has a square of total spin angular momentum h S(S + 1), and one with quantum number 4
5 S z has z component of total spin angular momentum hs z. Thus, the state 10 satisfies S 10 = h 10 (3) S z 10 = 0. (4) Note that the S = 1 states are symmetric under spin exchange, while the singlet S = 0 state is antisymmetric. But the total wave function (space + spin) for fermions must be antisymmetric, and all spin-1/ particles are fermions. Therefore, if we write the total wave function as a product of a spin wave function and a space wave function, then a spin-1 state must go with an antisymmetric space wave function, and a spin-0 with a symmetric wave function. To deal with the Heisenberg model (discussed below) it is useful to calculate expectation values of the operator S 1 S. The operator S 1 S can be written as S 1 S = 1 ( ) S S1 S. (5) Using this representation, it is easily shown that 1i S 1 S 1i = (1/) h ( 3/4 3/4) = (1/4) h (6) where i is the S z quantum number, and 00 S 1 S 00 = (3/4) h. (7) 5
6 3. Heisenberg Model. The Heisenberg model is defined by the Hamiltonian H = ij J ij S i S j i S i B. (8) Here J ij is the exchange interaction, and the first sum runs over distinct pairs of spins. The external field is denoted B. In what follows, we consider the nearest-neighbor Heisenberg model, where J ij = J if i and j are nearest neighbor sites, and J ij = 0 otherwise. We shall also consider the spin-1/ Heisenberg model, so that the quantities S i are operators for a spin-1/ particle. The model is called ferromagnetic if J > 0, because parallel spins are favored energetically in this case; it is called antiferromagnetic if J < 0. We generally consider the spin sites i to form a Bravais lattice, though of course one could consider a Heisenberg model on a lattice with a basis, or even on a non-periodic collection of sites. Finally, one need not consider only a spin-1/ Heisenberg model; one could also consider higher spins, or even a classical Heisenberg model, in which the spins are classical vectors. What is the origin of the interaction energy J? As implied in class, it is a purely quantum-mechanical interaction arising from the fact that electrons are fermions, and is actually a certain integral involving single-particle wave functions of two electrons, known as the exchange integral. The true magnetic dipole-dipole interaction is much smaller than the usual size of J in ferromagnetic materials, and can usually be neglected in comparison to the 6
7 exchange energy. Let us consider the spin-1/ nearest-neighbor ferromagnetic Heisenberg model with zero external field. The Hamiltonian can be expressed in terms of raising and lowering operators for spin, as H = J ij [ 1 (S i+s j + S i S j+ ) + S iz S jz ]. (9) The ground state of H is the state with all spins pointed down, and is denoted.... It is readily verified that this state is an eigenstate of H with eigenvalue E g = (Nz/)( h /4). (Note that we are retaining the factor of h in the definition of the spin operator here.) If we denote this state by 0 for short, then we have H 0 = E g 0. (30) 4. Excited states of the Heisenberg model; spin waves. One might think that the state with all spins but one pointed down, and one spin (say the l th ) pointed up, would be an excited eigenstate of H, but one would be wrong, because this state is not an eigenstate of H at all. To see this, first introduce a notation for this state: call it l. Then, operating on l with H and using the properties of the raising and lowering operators, one finds that H l = (E g + J h J h ) l l + δ, (31) δ 7
8 where δ denotes a nearest neighbor lattice vector, and the sum runs over all nearest neighbors. For example, in a simple cubic lattice with lattice constant a, the six possible values of δ are δ = ±aˆx, ±aŷ, ±aẑ. (3) The form of eq. (31) suggests that perhaps a suitably chosen linear combination of the states l might be an eigenstate of H. Indeed, this is the case. In fact, the suitable linear combination is just k = 1 exp(ik R l ) l. (33) N To prove this, we just operate on k with H, to get l H k = (E 0 + ɛ k ) k, (34) where ɛ(k) = J h [z δ cos(k δ)]. (35) This equation is derived by using the properties of the raising and lowering operators together with the definition of the state k. We also make use of the fact that for every Bravais lattice vector δ there is another Bravais lattice vector δ. Note that the above spin waves have a wave-vector/frequency relationship ω k at small values of k. 5. Physical Interpretation of Spin Waves. 8
9 The above definition of a spin wave may seem a bit strange: a linear combination of states, each with one spin up and all the rest down. You can think of this as a result of quantum mechanics. If you have a spin-1/ particle, and you measure its spin in the z direction, you must measure it as being either up or down. In a spin wave state, any given spin has only a 1/N probability of pointing down. Thus, we have to represent this state using a linear combination as mentioned above. The idea of spin wave also exists classically, and this may be easier to understand conceptually. Think of the ground state of the Heisenberg model as a collection of pendula, all at rest in their down position. In a state with a spin wave of wave vector k, any given spin is precessing with small amplitude around its equilbrium down position, and each pendulum has a definite phase with respect to the others. Finally, the wave is traveling with a frequency ω. 9
Introduction to Heisenberg model. Javier Junquera
Introduction to Heisenberg model Javier Junquera Most important reference followed in this lecture Magnetism in Condensed Matter Physics Stephen Blundell Oxford Master Series in Condensed Matter Physics
More informationMagnetism. Andreas Wacker Mathematical Physics Lund University
Magnetism Andreas Wacker Mathematical Physics Lund University Overview B=μ0(H+M) B: Magnetic field (T), satisfies div B=0 M: Magnetization (density of magnetic moments) H: H-field (A/m), satisfies curl
More informationS j H o = gµ o H o. j=1
LECTURE 17 Ferromagnetism (Refs.: Sections 10.6-10.7 of Reif; Book by J. S. Smart, Effective Field Theories of Magnetism) Consider a solid consisting of N identical atoms arranged in a regular lattice.
More informationd 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5)
LECTURE 12 Maxwell Velocity Distribution Suppose we have a dilute gas of molecules, each with mass m. If the gas is dilute enough, we can ignore the interactions between the molecules and the energy will
More informationIntermission: Let s review the essentials of the Helium Atom
PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the
More informationChemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron):
April 6th, 24 Chemistry 2A 2nd Midterm. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (-electron): E n = m e Z 2 e 4 /2 2 n 2 = E Z 2 /n 2, n =, 2, 3,... where Ze is
More informationAngular Momentum Algebra
Angular Momentum Algebra Chris Clark August 1, 2006 1 Input We will be going through the derivation of the angular momentum operator algebra. The only inputs to this mathematical formalism are the basic
More informationPhysics 505 Homework No. 8 Solutions S Spinor rotations. Somewhat based on a problem in Schwabl.
Physics 505 Homework No 8 s S8- Spinor rotations Somewhat based on a problem in Schwabl a) Suppose n is a unit vector We are interested in n σ Show that n σ) = I where I is the identity matrix We will
More informationSolutions Final exam 633
Solutions Final exam 633 S.J. van Enk (Dated: June 9, 2008) (1) [25 points] You have a source that produces pairs of spin-1/2 particles. With probability p they are in the singlet state, ( )/ 2, and with
More informationFor example, in one dimension if we had two particles in a one-dimensional infinite potential well described by the following two wave functions.
Identical particles In classical physics one can label particles in such a way as to leave the dynamics unaltered or follow the trajectory of the particles say by making a movie with a fast camera. Thus
More informationPhysics 125 Course Notes Identical Particles Solutions to Problems F. Porter
Physics 5 Course Notes Identical Particles Solutions to Problems 00 F. Porter Exercises. Let us use the Pauli exclusion principle, and the combination of angular momenta, to find the possible states which
More informationLuigi Paolasini
Luigi Paolasini paolasini@esrf.fr LECTURE 4: MAGNETIC INTERACTIONS - Dipole vs exchange magnetic interactions. - Direct and indirect exchange interactions. - Anisotropic exchange interactions. - Interplay
More informationψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions.
1. Quantum Mechanics (Fall 2004) Two spin-half particles are in a state with total spin zero. Let ˆn a and ˆn b be unit vectors in two arbitrary directions. Calculate the expectation value of the product
More informationMAT265 Mathematical Quantum Mechanics Brief Review of the Representations of SU(2)
MAT65 Mathematical Quantum Mechanics Brief Review of the Representations of SU() (Notes for MAT80 taken by Shannon Starr, October 000) There are many references for representation theory in general, and
More informationConsequently, the exact eigenfunctions of the Hamiltonian are also eigenfunctions of the two spin operators
VI. SPIN-ADAPTED CONFIGURATIONS A. Preliminary Considerations We have described the spin of a single electron by the two spin functions α(ω) α and β(ω) β. In this Sect. we will discuss spin in more detail
More informationPotential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form
Lecture 6 Page 1 Atoms L6.P1 Review of hydrogen atom Heavy proton (put at the origin), charge e and much lighter electron, charge -e. Potential energy, from Coulomb's law Potential is spherically symmetric.
More informationBasic Physical Chemistry Lecture 2. Keisuke Goda Summer Semester 2015
Basic Physical Chemistry Lecture 2 Keisuke Goda Summer Semester 2015 Lecture schedule Since we only have three lectures, let s focus on a few important topics of quantum chemistry and structural chemistry
More informationSpins and spin-orbit coupling in semiconductors, metals, and nanostructures
B. Halperin Spin lecture 1 Spins and spin-orbit coupling in semiconductors, metals, and nanostructures Behavior of non-equilibrium spin populations. Spin relaxation and spin transport. How does one produce
More information(1.1) In particular, ψ( q 1, m 1 ; ; q N, m N ) 2 is the probability to find the first particle
Chapter 1 Identical particles 1.1 Distinguishable particles The Hilbert space of N has to be a subspace H = N n=1h n. Observables Ân of the n-th particle are self-adjoint operators of the form 1 1 1 1
More information26 Group Theory Basics
26 Group Theory Basics 1. Reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian.
More informationSolution Set of Homework # 6 Monday, December 12, Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second Volume
Department of Physics Quantum II, 570 Temple University Instructor: Z.-E. Meziani Solution Set of Homework # 6 Monday, December, 06 Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second
More informationLecture 19 (Nov. 15, 2017)
Lecture 19 8.31 Quantum Theory I, Fall 017 8 Lecture 19 Nov. 15, 017) 19.1 Rotations Recall that rotations are transformations of the form x i R ij x j using Einstein summation notation), where R is an
More informationThe Quantum Heisenberg Ferromagnet
The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,
More informationQuantum Physics II (8.05) Fall 2002 Outline
Quantum Physics II (8.05) Fall 2002 Outline 1. General structure of quantum mechanics. 8.04 was based primarily on wave mechanics. We review that foundation with the intent to build a more formal basis
More informationPreliminary Quantum Questions
Preliminary Quantum Questions Thomas Ouldridge October 01 1. Certain quantities that appear in the theory of hydrogen have wider application in atomic physics: the Bohr radius a 0, the Rydberg constant
More informationLecture #13 1. Incorporating a vector potential into the Hamiltonian 2. Spin postulates 3. Description of spin states 4. Identical particles in
Lecture #3. Incorporating a vector potential into the Hamiltonian. Spin postulates 3. Description of spin states 4. Identical particles in classical and QM 5. Exchange degeneracy - the fundamental problem
More informationDepartment of Physics, Princeton University. Graduate Preliminary Examination Part II. Friday, May 10, :00 am - 12:00 noon
Department of Physics, Princeton University Graduate Preliminary Examination Part II Friday, May 10, 2013 9:00 am - 12:00 noon Answer TWO out of the THREE questions in Section A (Quantum Mechanics) and
More informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation
Lecture 17 Page 1 Lecture 17 L17.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent
More informationwhich can be distinguished from a normal paramagnet oberying Curie law [5] :
Jiayi Hu PHYS 211A Final Paper Nov. 15 Famous for the Heisenberg uncertainty principle, Werner Heisenberg contributed hugely to the development of quantum mechanics, especially with his work on the matrix
More informationAngular Momentum set II
Angular Momentum set II PH - QM II Sem, 7-8 Problem : Using the commutation relations for the angular momentum operators, prove the Jacobi identity Problem : [ˆL x, [ˆL y, ˆL z ]] + [ˆL y, [ˆL z, ˆL x
More informationClassical behavior of magnetic dipole vector. P. J. Grandinetti
Classical behavior of magnetic dipole vector Z μ Y X Z μ Y X Quantum behavior of magnetic dipole vector Random sample of spin 1/2 nuclei measure μ z μ z = + γ h/2 group μ z = γ h/2 group Quantum behavior
More informationG : Quantum Mechanics II
G5.666: Quantum Mechanics II Notes for Lecture 7 I. A SIMPLE EXAMPLE OF ANGULAR MOMENTUM ADDITION Given two spin-/ angular momenta, S and S, we define S S S The problem is to find the eigenstates of the
More information221B Lecture Notes Spontaneous Symmetry Breaking
B Lecture Notes Spontaneous Symmetry Breaking Spontaneous Symmetry Breaking Spontaneous Symmetry Breaking is an ubiquitous concept in modern physics, especially in condensed matter and particle physics.
More informationNumerical diagonalization studies of quantum spin chains
PY 502, Computational Physics, Fall 2016 Anders W. Sandvik, Boston University Numerical diagonalization studies of quantum spin chains Introduction to computational studies of spin chains Using basis states
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 22, March 20, 2006
Chem 350/450 Physical Chemistry II Quantum Mechanics 3 Credits Spring Semester 006 Christopher J. Cramer Lecture, March 0, 006 Some material in this lecture has been adapted from Cramer, C. J. Essentials
More informationQuantization of the Spins
Chapter 5 Quantization of the Spins As pointed out already in chapter 3, the external degrees of freedom, position and momentum, of an ensemble of identical atoms is described by the Scödinger field operator.
More informationv(r i r j ) = h(r i )+ 1 N
Chapter 1 Hartree-Fock Theory 1.1 Formalism For N electrons in an external potential V ext (r), the many-electron Hamiltonian can be written as follows: N H = [ p i i=1 m +V ext(r i )]+ 1 N N v(r i r j
More informationProblem Set No. 3: Canonical Quantization Due Date: Wednesday October 19, 2018, 5:00 pm. 1 Spin waves in a quantum Heisenberg antiferromagnet
Physics 58, Fall Semester 018 Professor Eduardo Fradkin Problem Set No. 3: Canonical Quantization Due Date: Wednesday October 19, 018, 5:00 pm 1 Spin waves in a quantum Heisenberg antiferromagnet In this
More informationLecture 11: Long-wavelength expansion in the Neel state Energetic terms
Lecture 11: Long-wavelength expansion in the Neel state Energetic terms In the last class we derived the low energy effective Hamiltonian for a Mott insulator. This derivation is an example of the kind
More informationWORLD SCIENTIFIC (2014)
WORLD SCIENTIFIC (2014) LIST OF PROBLEMS Chapter 1: Magnetism of Free Electrons and Atoms 1. Orbital and spin moments of an electron: Using the theory of angular momentum, calculate the orbital
More informationHomework assignment 3: due Thursday, 10/26/2017
Homework assignment 3: due Thursday, 10/6/017 Physics 6315: Quantum Mechanics 1, Fall 017 Problem 1 (0 points The spin Hilbert space is defined by three non-commuting observables, S x, S y, S z. These
More informationAngular momentum. Quantum mechanics. Orbital angular momentum
Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular
More informationAn introduction to magnetism in three parts
An introduction to magnetism in three parts Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D-76131 Karlsruhe 0. Overview Chapters of the three lectures
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationThe Postulates of Quantum Mechanics Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case: 3-D case
The Postulates of Quantum Mechanics Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about the
More information1. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants.
Sample final questions.. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants. 2. A one-dimensional harmonic oscillator, originally in the ground state,
More informationwave functions PhD seminar- FZ Juelich, Feb 2013
SU(3) symmetry and Baryon wave functions Sedigheh Jowzaee PhD seminar- FZ Juelich, Feb 2013 Introduction Fundamental symmetries of our universe Symmetry to the quark model: Hadron wave functions q q Existence
More informationEE 223 Applied Quantum Mechanics 2 Winter 2016
EE 223 Applied Quantum Mechanics 2 Winter 2016 Syllabus and Textbook references Version as of 12/29/15 subject to revisions and changes All the in-class sessions, paper problem sets and assignments, and
More informationM04M.1 Particles on a Line
Part I Mechanics M04M.1 Particles on a Line M04M.1 Particles on a Line Two elastic spherical particles with masses m and M (m M) are constrained to move along a straight line with an elastically reflecting
More informationCoupling of Angular Momenta Isospin Nucleon-Nucleon Interaction
Lecture 5 Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction WS0/3: Introduction to Nuclear and Particle Physics,, Part I I. Angular Momentum Operator Rotation R(θ): in polar coordinates the
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS
A11046W1 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS TRINITY TERM 2015 Wednesday, 17 June, 2.30
More information3 Symmetry Protected Topological Phase
Physics 3b Lecture 16 Caltech, 05/30/18 3 Symmetry Protected Topological Phase 3.1 Breakdown of noninteracting SPT phases with interaction Building on our previous discussion of the Majorana chain and
More informationLuigi Paolasini
Luigi Paolasini paolasini@esrf.fr LECTURE 7: Magnetic excitations - Phase transitions and the Landau mean-field theory. - Heisenberg and Ising models. - Magnetic excitations. External parameter, as for
More informationAre these states normalized? A) Yes
QMII-. Consider two kets and their corresponding column vectors: Ψ = φ = Are these two state orthogonal? Is ψ φ = 0? A) Yes ) No Answer: A Are these states normalized? A) Yes ) No Answer: (each state has
More informationLecture 4, January 12, 2015 Bonding in H2
Lecture 4, January 12, 2015 Bonding in H2 Elements of Quantum Chemistry with Applications to Chemical Bonding and Properties of Molecules and Solids Course number: Ch125a; Room 147 Noyes Hours: 11-11:50am
More informationMolecular Term Symbols
Molecular Term Symbols A molecular configuration is a specification of the occupied molecular orbitals in a molecule. For example, N : σ gσ uπ 4 uσ g A given configuration may have several different states
More informationarxiv:quant-ph/ v2 24 Dec 2003
Quantum Entanglement in Heisenberg Antiferromagnets V. Subrahmanyam Department of Physics, Indian Institute of Technology, Kanpur, India. arxiv:quant-ph/0309004 v2 24 Dec 2003 Entanglement sharing among
More information3.024 Electrical, Optical, and Magnetic Properties of Materials Spring 2012 Recitation 3 Notes
3.024 Electrical, Optical, and Magnetic Properties of Materials Spring 2012 Outline 1. Schr dinger: Eigenfunction Problems & Operator Properties 2. Piecewise Function/Continuity Review -Scattering from
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 20, March 8, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer Lecture 20, March 8, 2006 Solved Homework We determined that the two coefficients in our two-gaussian
More informationSommerfeld (1920) noted energy levels of Li deduced from spectroscopy looked like H, with slight adjustment of principal quantum number:
Spin. Historical Spectroscopy of Alkali atoms First expt. to suggest need for electron spin: observation of splitting of expected spectral lines for alkali atoms: i.e. expect one line based on analogy
More informationSt Hugh s 2 nd Year: Quantum Mechanics II. Reading. Topics. The following sources are recommended for this tutorial:
St Hugh s 2 nd Year: Quantum Mechanics II Reading The following sources are recommended for this tutorial: The key text (especially here in Oxford) is Molecular Quantum Mechanics, P. W. Atkins and R. S.
More informationMagnetism in ultracold gases
Magnetism in ultracold gases Austen Lamacraft Theoretical condensed matter and atomic physics April 10th, 2009 faculty.virginia.edu/austen/ Outline Magnetism in condensed matter Ultracold atomic physics
More information4 Matrix product states
Physics 3b Lecture 5 Caltech, 05//7 4 Matrix product states Matrix product state (MPS) is a highly useful tool in the study of interacting quantum systems in one dimension, both analytically and numerically.
More informationBerry s phase in Hall Effects and Topological Insulators
Lecture 6 Berry s phase in Hall Effects and Topological Insulators Given the analogs between Berry s phase and vector potentials, it is not surprising that Berry s phase can be important in the Hall effect.
More informationAddition of Angular Momenta
Addition of Angular Momenta What we have so far considered to be an exact solution for the many electron problem, should really be called exact non-relativistic solution. A relativistic treatment is needed
More informationQuantum Phase Transitions
Quantum Phase Transitions Subir Sachdev Department of Physics Yale University P.O. Box 208120, New Haven, CT 06520-8120 USA E-mail: subir.sachdev@yale.edu May 19, 2004 To appear in Encyclopedia of Mathematical
More informationPhysics 129B, Winter 2010 Problem Set 4 Solution
Physics 9B, Winter Problem Set 4 Solution H-J Chung March 8, Problem a Show that the SUN Lie algebra has an SUN subalgebra b The SUN Lie group consists of N N unitary matrices with unit determinant Thus,
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationLS coupling. 2 2 n + H s o + H h f + H B. (1) 2m
LS coupling 1 The big picture We start from the Hamiltonian of an atomic system: H = [ ] 2 2 n Ze2 1 + 1 e 2 1 + H s o + H h f + H B. (1) 2m n e 4πɛ 0 r n 2 4πɛ 0 r nm n,m Here n runs pver the electrons,
More informationBilliard ball model for structure factor in 1-D Heisenberg anti-ferromagnets
Billiard ball model for structure factor in 1-D Heisenberg anti-ferromagnets Shreyas Patankar 1 Chennai Mathematical Institute August 5, 2010 1 Project with Prof. Kedar Damle, TIFR and Girish Sharma, Satyasai
More informationMagnets, 1D quantum system, and quantum Phase transitions
134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider
More informationParticle Physics. Michaelmas Term 2009 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims
Particle Physics Michaelmas Term 2009 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2009 205 Introduction/Aims Symmetries play a central role in particle physics;
More informationAlkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split
Electron Spin Electron spin hypothesis Solution to H atom problem gave three quantum numbers, n,, m. These apply to all atoms. Experiments show not complete description. Something missing. Alkali metals
More informationLecture 24 Origins of Magnetization (A number of illustrations in this lecture were generously provided by Prof. Geoffrey Beach)
Lecture 4 Origins of Magnetization (A number of illustrations in this lecture were generously provided by Prof. Geoffrey Beach) Today 1. Magnetic dipoles.. Orbital and spin angular momenta. 3. Non-interacting
More information1 The Heisenberg model
1 The Heisenberg model 1.1 Definition of the model The model we will focus on is called the Heisenberg model. It has the following Hamiltonian: H = 1 J ij S i S j. (1) 2 i,j i j Here i and j refer to sites
More informationThe Λ(1405) is an anti-kaon nucleon molecule. Jonathan Hall, Waseem Kamleh, Derek Leinweber, Ben Menadue, Ben Owen, Tony Thomas, Ross Young
The Λ(1405) is an anti-kaon nucleon molecule Jonathan Hall, Waseem Kamleh, Derek Leinweber, Ben Menadue, Ben Owen, Tony Thomas, Ross Young The Λ(1405) The Λ(1405) is the lowest-lying odd-parity state of
More informationThe Λ(1405) is an anti-kaon nucleon molecule. Jonathan Hall, Waseem Kamleh, Derek Leinweber, Ben Menadue, Ben Owen, Tony Thomas, Ross Young
The Λ(1405) is an anti-kaon nucleon molecule Jonathan Hall, Waseem Kamleh, Derek Leinweber, Ben Menadue, Ben Owen, Tony Thomas, Ross Young The Λ(1405) The Λ(1405) is the lowest-lying odd-parity state of
More informationLecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators
Lecture Spin, orbital, and total angular momentum 70.00 Mechanics Very brief background MATH-GA In 9, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform
More informationPath Integral for Spin
Path Integral for Spin Altland-Simons have a good discussion in 3.3 Applications of the Feynman Path Integral to the quantization of spin, which is defined by the commutation relations [Ŝj, Ŝk = iɛ jk
More informationMagnetic ordering of local moments
Magnetic ordering Types of magnetic structure Ground state of the Heisenberg ferromagnet and antiferromagnet Spin wave High temperature susceptibility Mean field theory Magnetic ordering of local moments
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More informationQM and Angular Momentum
Chapter 5 QM and Angular Momentum 5. Angular Momentum Operators In your Introductory Quantum Mechanics (QM) course you learned about the basic properties of low spin systems. Here we want to review that
More informationLinear spin wave theory
Linear spin wave theory Sándor Tóth Paul Scherrer Institut August 17, 2015 Sándor Tóth (Paul Scherrer Institut) Linear spin wave theory August 17, 2015 1 / 48 Motivation Presentation Outline 1 Motivation
More informationRotations in Quantum Mechanics
Rotations in Quantum Mechanics We have seen that physical transformations are represented in quantum mechanics by unitary operators acting on the Hilbert space. In this section, we ll think about the specific
More informationSolutions to chapter 4 problems
Chapter 9 Solutions to chapter 4 problems Solution to Exercise 47 For example, the x component of the angular momentum is defined as ˆL x ŷˆp z ẑ ˆp y The position and momentum observables are Hermitian;
More informationQuantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid
Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Announcement This handout includes 9 problems. The first 5 are the problem set due. The last 4 cover material from the final few lectures
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Advocacy chit Chemistry 350/450 Final Exam Part I May 4, 005. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle
More informationPhysics 4022 Notes on Density Matrices
Physics 40 Notes on Density Matrices Definition: For a system in a definite normalized state ψ > the density matrix ρ is ρ = ψ >< ψ 1) From Eq 1 it is obvious that in the basis defined by ψ > and other
More informationPhysics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I
Physics 342 Lecture 26 Angular Momentum Lecture 26 Physics 342 Quantum Mechanics I Friday, April 2nd, 2010 We know how to obtain the energy of Hydrogen using the Hamiltonian operator but given a particular
More informationPHY331 Magnetism. Lecture 8
PHY331 Magnetism Lecture 8 Last week. We discussed domain theory of Ferromagnetism. We saw there is a motion of domain walls with applied magnetic field. Stabilization of domain walls due to competition
More informationExact diagonalization methods
Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Exact diagonalization methods Anders W. Sandvik, Boston University Representation of states in the computer bit
More informationPHY305: Notes on Entanglement and the Density Matrix
PHY305: Notes on Entanglement and the Density Matrix Here follows a short summary of the definitions of qubits, EPR states, entanglement, the density matrix, pure states, mixed states, measurement, and
More informationCHM Physical Chemistry II Chapter 9 - Supplementary Material. 1. Constuction of orbitals from the spherical harmonics
CHM 3411 - Physical Chemistry II Chapter 9 - Supplementary Material 1. Constuction of orbitals from the spherical harmonics The wavefunctions that are solutions to the time independent Schrodinger equation
More informationPHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10: Solutions
PHYS851 Quantum Mechanics I, Fall 009 HOMEWORK ASSIGNMENT 10: Solutions Topics Covered: Tensor product spaces, change of coordinate system, general theory of angular momentum Some Key Concepts: Angular
More informationThe 1+1-dimensional Ising model
Chapter 4 The 1+1-dimensional Ising model The 1+1-dimensional Ising model is one of the most important models in statistical mechanics. It is an interacting system, and behaves accordingly. Yet for a variety
More informationLecture III: Higgs Mechanism
ecture III: Higgs Mechanism Spontaneous Symmetry Breaking The Higgs Mechanism Mass Generation for eptons Quark Masses & Mixing III.1 Symmetry Breaking One example is the infinite ferromagnet the nearest
More informationFor the case of S = 1/2 the eigenfunctions of the z component of S are φ 1/2 and φ 1/2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 03 Excited State Helium, He An Example of Quantum Statistics in a Two Particle System By definition He has
More informationLecture 10. The Dirac equation. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 10 The Dirac equation WS2010/11: Introduction to Nuclear and Particle Physics The Dirac equation The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist
More informationNANOSCALE SCIENCE & TECHNOLOGY
. NANOSCALE SCIENCE & TECHNOLOGY V Two-Level Quantum Systems (Qubits) Lecture notes 5 5. Qubit description Quantum bit (qubit) is an elementary unit of a quantum computer. Similar to classical computers,
More information(relativistic effects kinetic energy & spin-orbit coupling) 3. Hyperfine structure: ) (spin-spin coupling of e & p + magnetic moments) 4.
4 Time-ind. Perturbation Theory II We said we solved the Hydrogen atom exactly, but we lied. There are a number of physical effects our solution of the Hamiltonian H = p /m e /r left out. We already said
More information