Applying classical geometry intuition to quantum spin
|
|
- Jeffry McCoy
- 5 years ago
- Views:
Transcription
1 European Journal o Physics PAPER OPEN ACCESS Applying classical geometry intuition to quantum spin To cite this article: Dallin S Duree and James L Archibald 06 Eur. J. Phys View the article online or updates and enhancements. Related content - Quantum Mechanics: Angular momentum M Saleem - Corrigendum: A ull quantum analysis o the Stern Gerlach experiment using the evolution operator method: analysing current issues in teaching quantum mechanics E Benítez Rodríguez, L M Arévalo Aguilar and E Piceno Martínez - An operational approach to spacetime symmetries: Lorentz transormations rom quantum communication Philipp A Höhn and Markus P Müller This content was downloaded rom IP address on 3/08/08 at :3
2 European Journal o Physics Eur. J. Phys. 37 (06) (9pp) doi:0.088/ /37/5/ Applying classical geometry intuition to quantum spin Dallin S Duree and James L Archibald Department o Physics and Astronomy, Brigham Young University, Provo, UT 8460, USA dallin_duree@byu.edu Received April 06, revised 0 June 06 Accepted or publication 0 June 06 Published 0 July 06 Abstract Using concepts o geometric orthogonality and linear independence, we logically deduce the orm o the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets o the spin-/ system. Rather than a mathematically rigorous derivation, the relationships are ound by orcing expectation values o the dierent basis states to have the properties we expect o a classical, geometric coordinate system. The process highlights the correspondence o quantum angular momentum with classical notions o geometric orthogonality, even or the inherently non-classical spin-/ system. In the process, dierences in and connections between geometrical space and Hilbert space are illustrated. Keywords: spin, quantum mechanics, orthogonality (Some igures may appear in colour only in the online journal). Introduction It is possible to ind the Pauli spin matrices and the corresponding geometrically orthogonal basis sets or the spin-/ system using arguments rom classical geometry. Like the vector model [, ], this undertaking leverages classical understanding to build intuition or quantum angular momentum. But rather than using a semi-classical model, our approach uses completely quantum spin states. It involves the application o classical principles to expectation Currently at AMS-TAOS USA Inc. Original content rom this work may be used under the terms o the Creative Commons Attribution 3.0 licence. Any urther distribution o this work must maintain attribution to the author(s) and the title o the work, journal citation and DOI /6/ $ IOP Publishing Ltd Printed in the UK
3 Eur. J. Phys. 37 (06) D S Duree and J L Archibald values, in a manner similar to Ehrenest s theorem [3, 4]. The technique can provide valuable insight or anyone with an advanced undergraduate or introductory graduate understanding o quantum mechanics. This method illuminates the correspondence between Hilbert space, which is used to mathematically describe quantum states, and the geometrical space we use in classical descriptions o angular momentum. It makes connections between the parameters used to deine a Cartesian coordinate system and the ree parameters selected when deining geometrically orthogonal sets o basis states to describe a quantum spin-/ particle. The procedure clariies some common misconceptions about the quantum picture o angular momentum, in particular the conusion students oten have distinguishing between geometric and Hilbert space, discussed in [5]. It does not explain the why o quantum spin, but ocuses on how we work with quantum angular momentum states. There are several rigorous ways to derive the well-known relationships between the x, y, and z basis sets o the spin-/ system and the corresponding Pauli matrices. These include the use o rotation operations [6, 7], ladder operators [8], direct diagonalization [9], or symmetry arguments [0, ]. Here we deduce, rather than derive, the relationships. We limit our discussion to a spin-/ system or simplicity and clarity. This also avoids dealing with dierent types o unbiased states []. For example, as discussed later, or a spin-/ particle described using the z basis, the only states with a zero expectation value o the z component o angular momentum are ones made o an equal superposition o spin up and spin down. For a spin- particle, this is not only achieved with an equal superposition o m =-and m =, but also with the m = 0 basis state, or any combination o all states or which the and basis states have amplitudes o equal size. The canonical spin-/ system illustrates the universality o the core principle behind Ehrenest s theorem, even in systems with no classical analogue [3, 4]; even though you cannot explain an electron s intrinsic angular momentum as the result o physical rotation in the classical sense, the expectation values o a spin-/ system do obey principles o classical geometry. While the origin o spin cannot be described semi-classically, classical intuition can be applied to better understand the consequences o spin.. Quantum spin-/ ormalism It is common to describe spin-/ particles using a basis consisting o the two eigenstates o the z-component o angular momentum, z and z. In vector notation, we can write these states as () () = and = 0 z z 0. ( ) We will use the z basis as our starting point to deduce the orm o the Pauli spin matrices and ind the relationships between the x, y, and z basis states. We assume that the z basis exists and that every possible state o our particle can be expressed as a sum o these two states in the orm y = a a + b = b, ( ) z z () where a and b are scalar, potentially complex, constants. In matrix notation, the operators that yield inormation about the x, y, and z components o spin can be written as times the Pauli spin matrices. The Pauli spin matrix s z can be easily ound by noting that it is the matrix or which and z are the eigenvectors with z
4 Eur. J. Phys. 37 (06) D S Duree and J L Archibald eigenvalues o + and. This gives us the matrix s = z ( ) ( ) I we measure the z-component o angular momentum or a spin-/ particle in an arbitrary state, we will always get either plus or minus, with probabilities o a* a and b* b i the state is normalised. To predict the outcome o a measurement along a dierent axis, we can write our quantum state in terms o the eigenstates o the component o angular momentum we plan to measure. Since the z axis is chosen arbitrarily, we know that a pair o eigenstates o any component o angular momentum along any axis must exist and have similar properties to the z basis states. Two special basis sets are made up o the eigenstates o angular momentum along the x and y axes, respectively. Because the z basis orms a complete set, we should be able to write these basis states in terms o the z states. We will start by writing the x basis states as x = A z + B z and ( 4) = C + D, ( 5) x z z where A, B, C, and D are constants. We can deduce what these constants must be simply by considering what properties these states should have. In the process, we will see connections between physical (geometric) space and Hilbert space. 3. Orthogonality To ind A and B in equation (4) we note that a classical particle with its angular momentum in the x direction will have no component o angular momentum in the z direction. For a quantum spin-/ particle we will always measure the z component to be, never zero, regardless o the particle s state. So rather than mapping our intuition o geometrical orthogonality onto possible measurement outcomes, we will make our x basis states geometrically orthogonal to z by setting the expectation value o the z component o angular momentum to zero. For the expectation value to be zero, it must be exactly as likely to measure the z component o spin to be - as it is to measure +, such that the two possibilities cancel each other out. As such, we intuitively expect that x should be an equal superposition o z and z. To show this more rigorously, we use the operator S z = ( z z - z z ) = s z ( 6) to ind the expectation value o the z component o angular momentum. Using this operator, we get x Sz x = ( A* A - B* B). ( 7) We require that this expectation value be zero, such that AA * - BB * = A - B = 0. ( 8) And, likewise, since the x and y basis sets will be complete, we should be able to write the z basis states in terms o the x or y basis states. 3
5 Eur. J. Phys. 37 (06) D S Duree and J L Archibald As we expected, A and B must have the same magnitude, giving an equal superposition o z and z. But the complex phases o A and B are unrestricted. So the most general orm or x, subject only to the limitation that it be normalised and geometrically orthogonal to the z basis states, is = + x [ e i z e i z ], ( 9 ) where and are arbitrary real constants. It makes sense that we get one arbitrary phase angle, since quantum mechanics always allows us an arbitrary overall phase actor. But why two? There s a geometric explanation or this reedom. Ater deining the z axis o a Cartesian coordinate system, the x axis can point in any one o an ininite number o directions which are orthogonal to z. These choices can be parametrised by an angle relative to some reerence direction, as shown in igure. We will set and to zero both or simplicity and because it gives us the conventional orm o the basis state x = [ z + z ]. ( 0) 4. Linear independence To ind x we note that it, too, must be geometrically orthogonal to the z basis. So it must have the orm = + x [ e i 3 z e i 4 z ]. ( ) We ind an additional constraint when we consider that the x basis states could have been the z basis states had we simply chosen a dierent direction or our z axis. As such, since our z basis states are orthogonal to each other, we expect the two states in the x basis to be orthogonal to each other as well. As is oten done, we have unortunately used the word orthogonal to mean two dierent things. When we say that the x states must be orthogonal to the z states, we are reerring to geometric orthogonality. But when we say that the x states must be orthogonal to each other, we reer to orthogonality in Hilbert space or linear independence. Just as a dot product o zero assures geometric orthogonality, an inner product x x = 0 guarantees that x and x are linearly independent, such that one cannot be written in terms o the other. The inner product o the x basis states is = + j + j x x [ z z ][ ei 3 z ei 4 z ] ( ) = ( j + j e i 3 e i 4). For this to be zero, the two phases must dier by π radians. I we choose the arbitrary global quantum phase o this state such that 3 = 0, both or simplicity and by convention, we ind that ei 4 =-and x = [ z - z ]. ( 3) 4
6 Eur. J. Phys. 37 (06) D S Duree and J L Archibald Figure. Once the z axis o a Cartesian coordinate system is selected, there are still an ininite number o possible choices or the x axis which are all orthogonal to z. These choices are parametrised by the variable θ in the igure. Knowing the orm o x and x, it is simple to show that the Pauli spin matrix s x is given by s = 0 x ( 0 ). ( 4 ) 5. The y basis Just as we saw or the x basis states, or the y basis states to be normalised and geometrically orthogonal to the z axis, they must have the orm = + y [ e i 5 z e i 6 z ], ( 5 ) = + y [ e i 7 z e i 8 z ]. ( 6 ) Since we have the reedom to multiply each state by an arbitrary overall phase actor, or simplicity (and to arrive at the canonical orm o the states), we can set 5 and 7 to zero. The other phase angles are constrained by that act that, in addition to being geometrically orthogonal to z, these states must also be geometrically orthogonal to the x axis we have deined. The operator which gives us inormation about the x component o spin can be ound by noting that, when written in the x basis, this operator should look similar to S z represented in the z basis: Sx = ( x x - x x ). ( 7) 5
7 Eur. J. Phys. 37 (06) D S Duree and J L Archibald Figure. Once the x and z axes are selected, there are still two possible choices or the y direction. One choice, labelled y lh in the igure, will result in a let-handed coordinate system. The other choice results in a right-handed coordinate system. To write this in the z basis we plug in equations (0) and (3) to get S x = ( z z + z z ) = s x. ( 8) We can use S x to ind the expectation value o the x component o angular momentum or a particle in the state : y = + - y Sx y ( = 4 e i 6 e i 6) cos ( 6). ( 9) I we want y to be geometrically orthogonal to the x states then this must be zero, implying that 6 =p and giving us only two unique possibilities: y = [ z i z ]. ( 0) As we discuss in the next section, the choice o whether to use the upper or lower sign is not arbitrary, so we will not select one over the other just yet. Applying the same condition to y and orcing the inner product y y to be zero gives us y = [ z i z ]. ( ) 6. Coordinate system handedness There is a geometric explanation or the two possible choices in equations (0) and (). In a Cartesian coordinate system, once the z and x axes have been selected, there are still two 6
8 Eur. J. Phys. 37 (06) D S Duree and J L Archibald possible directions or y. As illustrated in igure, one direction results in a right-handed and the other in a let-handed coordinate system. We can ind the handedness o a geometric coordinate system with cross products. For example, i nˆ j is a unit vector along the jth axis, then or a right-handed coordinate system using a right-handed cross product, nˆx nˆy = nˆ z. For a let-handed coordinate system we get a minus sign: nˆx nˆy = -nˆ z. Similarly, we can ind the handedness o the Pauli spin matrices by noting that the Pauli matrices, multiplied by a constant and combined with the ( ) identity matrix, orm the basis o the SU() Lie group [5]. Then we can evaluate Lie brackets, which are analogous to cross products. I we evaluate the Lie bracket o sz with sx, we get s s = s + = - s -, i i z x y y, ( ) where s y + /s y- is the Pauli spin matrix we get i we choose the upper/lower sign or equations (0) and (). The change in sign in equation () is just what we would expect when going rom a right- to a let-handed coordinate system. Similarly, there is a link between the Pauli matrices and quaternions [6 8], and between cross products and commutators in quaternion algebra [9]. This suggests a connection between cross products in geometric space and commutators in Hilbert space. Because there is no spatial representation o the spin-/ particle s angular momentum operator, we will make an analogy with the orbital angular momentum operator [0]: L = r p. ( 3) Here r is the position and p the momentum operator. Assuming a right-handed coordinate system, the components o L are Lx = ypz - zp y, ( 4) Ly = zpx - xp z, ( 5) Lz = xpy - yp x. ( 6) We can use these components to calculate the commutator o L x with L y. Noting that position and linear momentum operators commute with operators or orthogonal spatial dimensions and that [ z, pz ] = i, it is easy to show that [ Lx, Ly] = ilz. I we had chosen a let-handed coordinate system (but still used a right-handed cross product), we would have gotten the same result but with a minus sign. By analogy, we may suppose that a right-handed coordinate system or a spin-/ particle is the one that results in the commutation relation [ S, S ] = i S, ( 7) x y z while a let-handed coordinate system would result in a minus sign in the commutation relation. We get the commutation relation in equation (7) i we choose the upper sign in equations (0) and (): y = [ z + i z ], ( 8) y = [ z - i z ]. ( 9) 7
9 Eur. J. Phys. 37 (06) D S Duree and J L Archibald From these, the Pauli matrix s y can be ound: s = 0 - i i y ( ) ( ) This is the last piece o the puzzle, and we have now deduced the relationships between the x, y, and z basis states as well as the Pauli matrices or the spin-/ system. 7. Conclusion We deduced the Pauli matrices and the relationships between the x, y, and z basis states or a spin-/ particle using the concepts o geometric orthogonality and linear independence. By insisting that the two states in the x basis be normalized, geometrically orthogonal to the z states, and orthogonal to each other in Hilbert space (linearly independent), we arrived at expressions which were completely speciied except or three arbitrary phase angles two due to arbitrary overall phase actors, and a third related to choosing the direction or the x axis or a given selection o z axis direction in a Cartesian coordinate system. With the y basis we had less reedom because the states had to be geometrically orthogonal to both the z and the x basis states. We again had an arbitrary overall phase actor or each basis state. But we only had two possible choices or the remaining phase actors, similar to the choice o handedness in a Cartesian coordinate system. We determined handedness by making a connection between cross products and commutators. This intuitive exercise illustrates the connection between classical geometric space and quantum Hilbert space, even or spin-/ systems, which are intrinsically not classical. Acknowledgments We thankully acknowledge Jean-Franois Van Huele or very helpul discussions, and we are grateul to Christopher J Erickson and Manuel Berrondo or eedback on this manuscript. This research was unded by NSF Grant PHY and by BYU s College o Physical and Mathematical Sciences. Reerences [] Beiser A 995 Total angular momentum Concepts o Modern Physics 5th edn (New York: McGraw-Hill) ch 7.8, pp [] Loh Y L and Kim M 05 Am. J. Phys [3] Ehrenest P 97 Z. Phys [4] Thompson W J 004 Angular momentum in quantum systems Angular Momentum (Chapel Hill, NC: Wiley-VCH) ch 5, pp 69 0 [5] Zhu G and Singh C 0 Am. J. Phys [6] Merzbacher E 998 Quantum Mechanics 3rd edn (New York: Wiley) ch 6, pp 38 6 [7] Sakurai J J 994 Modern Quantum Mechanics (Revised Edition) (Reading, MA: Addison Wesley Longman) ch 3, pp [8] Griiths D J 005 Introduction to Quantum Mechanics nd edn (Englewood Clis, NJ: Pearson Prentice Hall) ch 4, pp [9] Narducci L M and Orszag M 97 Am. J. Phys [0] Dirac P A M 958 The spin o the electron The Principles o Quantum Mechanics 4th edn (Oxord: Clarendon) ch 37, pp 49 5 [] Epstein S T 966 Am. J. Phys [] de la Torre A C and Iguain J L 998 Am. J. Phys [3] Ohanian H C 986 Am. J. Phys
10 Eur. J. Phys. 37 (06) D S Duree and J L Archibald [4] Commins E D 0 Annu. Rev. Nucl. Part. Sci [5] Adams B G 994 Algebraic Approach to Simple Quantum Systems (Berlin: Springer) ch B.5, p 5 [6] liamed Y and Salingaros N 98 J. Math. Phys [7] Gough W 986 Eur. J. Phys [8] Penrose R 997 Eur. J. Phys [9] Deavours C A 973 Am. Math Mon [0] Dirac P A M 958 Angular momentum The Principles o Quantum Mechanics 4th edn (Oxord: Clarendon) ch 35, pp
arxiv: v4 [quant-ph] 9 Jun 2016
Applying Classical Geometry Intuition to Quantum arxiv:101.030v4 [quant-ph] 9 Jun 016 Spin Dallin S. Durfee and James L. Archibald Department of Physics and Astronomy, Brigham Young University, Provo,
More informationApplying classical geometry intuition to quantum spin
European Journal of Physics PAPER OPEN ACCESS Applying classical geometry intuition to quantum spin To cite this article: Dallin S Durfee and James L Archibald 0 Eur. J. Phys. 0 Manuscript version: Accepted
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 information1 Algebra of State Vectors
J. Rothberg October 6, Introduction to Quantum Mechanics: Part Algebra of State Vectors What does the State Vector mean? A state vector is not a property of a physical system, but rather represents an
More informationOPERATORS AND MEASUREMENT
Chapter OPERATORS AND MEASUREMENT In Chapter we used the results of experiments to deduce a mathematical description of the spin-/ system. The Stern-Gerlach experiments demonstrated that spin component
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
FUNCTIONS Quadratic Functions......8 Absolute Value Functions.....48 Translations o Functions..57 Radical Functions...61 Eponential Functions...7 Logarithmic Functions......8 Cubic Functions......91 Piece-Wise
More informationMathematical Formulation of the Superposition Principle
Mathematical Formulation of the Superposition Principle Superposition add states together, get new states. Math quantity associated with states must also have this property. Vectors have this property.
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 informationVector Spaces for Quantum Mechanics J. P. Leahy January 30, 2012
PHYS 20602 Handout 1 Vector Spaces for Quantum Mechanics J. P. Leahy January 30, 2012 Handout Contents Examples Classes Examples for Lectures 1 to 4 (with hints at end) Definitions of groups and vector
More informationQuantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 6 Postulates of Quantum Mechanics II (Refer Slide Time: 00:07) In my last lecture,
More informationGeometric Algebra 2 Quantum Theory
Geometric Algebra 2 Quantum Theory Chris Doran Astrophysics Group Cavendish Laboratory Cambridge, UK Spin Stern-Gerlach tells us that electron wavefunction contains two terms Describe state in terms of
More information4. Two-level systems. 4.1 Generalities
4. Two-level systems 4.1 Generalities 4. Rotations and angular momentum 4..1 Classical rotations 4.. QM angular momentum as generator of rotations 4..3 Example of Two-Level System: Neutron Interferometry
More informationReview of the Formalism of Quantum Mechanics
Review of the Formalism of Quantum Mechanics The postulates of quantum mechanics are often stated in textbooks. There are two main properties of physics upon which these postulates are based: 1)the probability
More informationi = cos 2 0i + ei sin 2 1i
Chapter 10 Spin 10.1 Spin 1 as a Qubit In this chapter we will explore quantum spin, which exhibits behavior that is intrinsically quantum mechanical. For our purposes the most important particles are
More informationIntegrable Hamiltonian systems generated by antisymmetric matrices
Journal of Physics: Conference Series OPEN ACCESS Integrable Hamiltonian systems generated by antisymmetric matrices To cite this article: Alina Dobrogowska 013 J. Phys.: Conf. Ser. 474 01015 View the
More informationParticle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision)
Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e - e + - + e - e + - + e - e + - + e - e + - Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 Non-Relativistic QM (Revision)
More informationmsqm 2011/8/14 21:35 page 189 #197
msqm 2011/8/14 21:35 page 189 #197 Bibliography Dirac, P. A. M., The Principles of Quantum Mechanics, 4th Edition, (Oxford University Press, London, 1958). Feynman, R. P. and A. P. Hibbs, Quantum Mechanics
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 informationThe quantum state as a vector
The quantum state as a vector February 6, 27 Wave mechanics In our review of the development of wave mechanics, we have established several basic properties of the quantum description of nature:. A particle
More informationNONLINEAR CONTROL OF POWER NETWORK MODELS USING FEEDBACK LINEARIZATION
NONLINEAR CONTROL OF POWER NETWORK MODELS USING FEEDBACK LINEARIZATION Steven Ball Science Applications International Corporation Columbia, MD email: sball@nmtedu Steve Schaer Department o Mathematics
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points
Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques
More information1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationParticle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation
Dr. A. Mitov Particle Physics 45 Particle Physics Dr. Alexander Mitov µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 46 Non-Relativistic
More informationPhysics 5153 Classical Mechanics. Solution by Quadrature-1
October 14, 003 11:47:49 1 Introduction Physics 5153 Classical Mechanics Solution by Quadrature In the previous lectures, we have reduced the number o eective degrees o reedom that are needed to solve
More informationSelect/Special Topics in Atomic Physics Prof. P. C. Deshmukh Department of Physics Indian Institute of Technology, Madras
Select/Special Topics in Atomic Physics Prof. P. C. Deshmukh Department of Physics Indian Institute of Technology, Madras Lecture - 9 Angular Momentum in Quantum Mechanics Dimensionality of the Direct-Product
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 informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationColumbus State Community College Mathematics Department Public Syllabus
Columbus State Community College Mathematics Department Public Syllabus Course and Number: MATH 2568 Elementary Linear Algebra Credits: 4 Class Hours Per Week: 4 Prerequisites: MATH 2153 with a C or higher
More informationPure Quantum States Are Fundamental, Mixtures (Composite States) Are Mathematical Constructions: An Argument Using Algorithmic Information Theory
Pure Quantum States Are Fundamental, Mixtures (Composite States) Are Mathematical Constructions: An Argument Using Algorithmic Information Theory Vladik Kreinovich and Luc Longpré Department of Computer
More informationTutorial 5 Clifford Algebra and so(n)
Tutorial 5 Clifford Algebra and so(n) 1 Definition of Clifford Algebra A set of N Hermitian matrices γ 1, γ,..., γ N obeying the anti-commutator γ i, γ j } = δ ij I (1) is the basis for an algebra called
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationBasic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi
Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi Module No. # 07 Bra-Ket Algebra and Linear Harmonic Oscillator - II Lecture No. # 01 Dirac s Bra and
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 informationClifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras
Journal of Physics: Conference Series PAPER OPEN ACCESS Clifford Algebras and Their Decomposition into Conugate Fermionic Heisenberg Algebras To cite this article: Sultan Catto et al 016 J. Phys.: Conf.
More informationSome Concepts used in the Study of Harish-Chandra Algebras of Matrices
Intl J Engg Sci Adv Research 2015 Mar;1(1):134-137 Harish-Chandra Algebras Some Concepts used in the Study of Harish-Chandra Algebras of Matrices Vinod Kumar Yadav Department of Mathematics Rama University
More informationNumerical Solution of Ordinary Differential Equations in Fluctuationlessness Theorem Perspective
Numerical Solution o Ordinary Dierential Equations in Fluctuationlessness Theorem Perspective NEJLA ALTAY Bahçeşehir University Faculty o Arts and Sciences Beşiktaş, İstanbul TÜRKİYE TURKEY METİN DEMİRALP
More informationDefinition 1. A set V is a vector space over the scalar field F {R, C} iff. there are two operations defined on V, called vector addition
6 Vector Spaces with Inned Product Basis and Dimension Section Objective(s): Vector Spaces and Subspaces Linear (In)dependence Basis and Dimension Inner Product 6 Vector Spaces and Subspaces Definition
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 informationAngular momentum and spin
Luleå tekniska universitet Avdelningen för Fysik, 007 Hans Weber Angular momentum and spin Angular momentum is a measure of how much rotation there is in particle or in a rigid body. In quantum mechanics
More informationAngular Momentum in Quantum Mechanics
Angular Momentum in Quantum Mechanics In classical mechanics the angular momentum L = r p of any particle moving in a central field of force is conserved. For the reduced two-body problem this is the content
More informationDot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.
Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................
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 informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechanics for Scientists and Engineers David Miller Vector spaces, operators and matrices Vector spaces, operators and matrices Vector space Vector space We need a space in which our vectors exist
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More information3.024 Electrical, Optical, and Magnetic Properties of Materials Spring 2012 Recitation 1. Office Hours: MWF 9am-10am or by appointment
Adam Floyd Hannon Office Hours: MWF 9am-10am or by e-mail appointment Topic Outline 1. a. Fourier Transform & b. Fourier Series 2. Linear Algebra Review 3. Eigenvalue/Eigenvector Problems 1. a. Fourier
More informationThis appendix provides a very basic introduction to linear algebra concepts.
APPENDIX Basic Linear Algebra Concepts This appendix provides a very basic introduction to linear algebra concepts. Some of these concepts are intentionally presented here in a somewhat simplified (not
More informationNormal modes. where. and. On the other hand, all such systems, if started in just the right way, will move in a simple way.
Chapter 9. Dynamics in 1D 9.4. Coupled motions in 1D 491 only the forces from the outside; the interaction forces cancel because they come in equal and opposite (action and reaction) pairs. So we get:
More informationSymmetry and Simplicity
Mobolaji Williams Motifs in Physics April 11, 2017 Symmetry and Simplicity These notes 1 are part of a series concerning Motifs in Physics in which we highlight recurrent concepts, techniques, and ways
More informationChapter 2 Special Orthogonal Group SO(N)
Chapter 2 Special Orthogonal Group SO(N) 1 Introduction Since the exactly solvable higher-dimensional quantum systems with certain central potentials are usually related to the real orthogonal group O(N)
More informationThe electric multipole expansion for a magic cube
INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 26 (2005) 809 813 EUROPEAN JOURNAL OF PHYSICS doi:10.1088/0143-0807/26/5/013 The electric multipole expansion for a magic cube Adam Rogers and Peter Loly Department
More informationQuantum Physics II (8.05) Fall 2002 Assignment 3
Quantum Physics II (8.05) Fall 00 Assignment Readings The readings below will take you through the material for Problem Sets and 4. Cohen-Tannoudji Ch. II, III. Shankar Ch. 1 continues to be helpful. Sakurai
More informationSpin Dynamics Basic Theory Operators. Richard Green SBD Research Group Department of Chemistry
Spin Dynamics Basic Theory Operators Richard Green SBD Research Group Department of Chemistry Objective of this session Introduce you to operators used in quantum mechanics Achieve this by looking at:
More informationQuantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 14 Exercises on Quantum Expectation Values (Refer Slide Time: 00:07) In the last couple
More informationTwo Constants of Motion in the Generalized Damped Oscillator
Advanced Studies in Theoretical Physics Vol. 10, 2016, no. 2, 57-65 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2016.511107 Two Constants o Motion in the Generalized Damped Oscillator
More informationIncompatibility Paradoxes
Chapter 22 Incompatibility Paradoxes 22.1 Simultaneous Values There is never any difficulty in supposing that a classical mechanical system possesses, at a particular instant of time, precise values of
More informationVector Spaces in Quantum Mechanics
Chapter 8 Vector Spaces in Quantum Mechanics We have seen in the previous Chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states.
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
More informationQuantum Physics 2006/07
Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp
More information2. Lie groups as manifolds. SU(2) and the three-sphere. * version 1.4 *
. Lie groups as manifolds. SU() and the three-sphere. * version 1.4 * Matthew Foster September 1, 017 Contents.1 The Haar measure 1. The group manifold for SU(): S 3 3.3 Left- and right- group translations
More informationFluctuationlessness Theorem and its Application to Boundary Value Problems of ODEs
Fluctuationlessness Theorem and its Application to Boundary Value Problems o ODEs NEJLA ALTAY İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) nejla@be.itu.edu.tr
More informationMA201: Further Mathematical Methods (Linear Algebra) 2002
MA201: Further Mathematical Methods (Linear Algebra) 2002 General Information Teaching This course involves two types of teaching session that you should be attending: Lectures This is a half unit course
More informationLecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II
Lecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II We continue our discussion of symmetries and their role in matrix representation in this lecture. An example
More informationG : Quantum Mechanics II
G5.666: Quantum Mechanics II Notes for Lecture 5 I. REPRESENTING STATES IN THE FULL HILBERT SPACE Given a representation of the states that span the spin Hilbert space, we now need to consider the problem
More information3.3 Lagrangian and symmetries for a spin- 1 2 field
3.3 Lagrangian and symmetries for a spin- 1 2 field The Lagrangian for the free spin- 1 2 field is The corresponding Hamiltonian density is L = ψ(i/ µ m)ψ. (3.31) H = ψ( γ p + m)ψ. (3.32) The Lagrangian
More informationVectors and Matrices Statistics with Vectors and Matrices
Vectors and Matrices Statistics with Vectors and Matrices Lecture 3 September 7, 005 Analysis Lecture #3-9/7/005 Slide 1 of 55 Today s Lecture Vectors and Matrices (Supplement A - augmented with SAS proc
More informationRepresentations of Lorentz Group
Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the
More informationCh 125a Problem Set 1
Ch 5a Problem Set Due Monday, Oct 5, 05, am Problem : Bra-ket notation (Dirac notation) Bra-ket notation is a standard and convenient way to describe quantum state vectors For example, φ is an abstract
More informationTotal Angular Momentum for Hydrogen
Physics 4 Lecture 7 Total Angular Momentum for Hydrogen Lecture 7 Physics 4 Quantum Mechanics I Friday, April th, 008 We have the Hydrogen Hamiltonian for central potential φ(r), we can write: H r = p
More informationPH425 Spins Homework 5 Due 4 pm. particles is prepared in the state: + + i 3 13
PH45 Spins Homework 5 Due 10/5/18 @ 4 pm REQUIRED: 1. A beam of spin- 1 particles is prepared in the state: ψ + + i 1 1 (a) What are the possible results of a measurement of the spin component S z, and
More informationLecture 4 (Sep. 18, 2017)
Lecture 4 8.3 Quantum Theory I, Fall 07 Lecture 4 (Sep. 8, 07) 4. Measurement 4.. Spin- Systems Last time, we said that a general state in a spin- system can be written as ψ = c + + + c, (4.) where +,
More informationHYDROGEN SPECTRUM = 2
MP6 OBJECT 3 HYDROGEN SPECTRUM MP6. The object o this experiment is to observe some o the lines in the emission spectrum o hydrogen, and to compare their experimentally determined wavelengths with those
More informationSelect/Special Topics in Atomic Physics Prof. P.C. Deshmukh Department Of Physics Indian Institute of Technology, Madras
Select/Special Topics in Atomic Physics Prof. P.C. Deshmukh Department Of Physics Indian Institute of Technology, Madras Lecture - 37 Stark - Zeeman Spectroscopy Well, let us continue our discussion on
More informationWhat is spin? André Gsponer Independent Scientific Research Institute Box 30, CH-1211 Geneva-12, Switzerland
What is spin? arxiv:physics/030807v3 [physics.class-ph] 0 Sep 003 André Gsponer Independent Scientific Research Institute Box 30, CH- Geneva-, Switzerland e-mail: isri@vtx.ch ISRI-03-0.3 February, 008
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 informationDISCRETE SYMMETRIES IN NUCLEAR AND PARTICLE PHYSICS. Parity PHYS NUCLEAR AND PARTICLE PHYSICS
PHYS 30121 NUCLEAR AND PARTICLE PHYSICS DISCRETE SYMMETRIES IN NUCLEAR AND PARTICLE PHYSICS Discrete symmetries are ones that do not depend on any continuous parameter. The classic example is reflection
More informationThe experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field:
SPIN 1/2 PARTICLE Stern-Gerlach experiment The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field: A silver atom
More informationarxiv:quant-ph/ v1 3 Apr 2001
Kronecer product/direct product/tensor product in Quantum Theory arxiv:quant-ph/0104019v1 3 Apr 2001 Z. S. Sazonova Physics Department, Moscow Automobile and Road Construction Institute (Technical University),
More information(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results
(C) The rationals and the reals as linearly ordered sets We know that both Q and R are something special. When we think about about either o these we usually view it as a ield, or at least some kind o
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 information2. ETA EVALUATIONS USING WEBER FUNCTIONS. Introduction
. ETA EVALUATIONS USING WEBER FUNCTIONS Introduction So ar we have seen some o the methods or providing eta evaluations that appear in the literature and we have seen some o the interesting properties
More information9 Electron orbits in atoms
Physics 129b Lecture 15 Caltech, 02/22/18 Reference: Wu-Ki-Tung, Group Theory in physics, Chapter 7. 9 Electron orbits in atoms Now let s see how our understanding of the irreps of SO(3) (SU(2)) can help
More informationChapter 2. Matrix Arithmetic. Chapter 2
Matrix Arithmetic Matrix Addition and Subtraction Addition and subtraction act element-wise on matrices. In order for the addition/subtraction (A B) to be possible, the two matrices A and B must have the
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p
More informationLagrangian Description for Particle Interpretations of Quantum Mechanics Single-Particle Case
Lagrangian Description for Particle Interpretations of Quantum Mechanics Single-Particle Case Roderick I. Sutherland Centre for Time, University of Sydney, NSW 26 Australia rod.sutherland@sydney.edu.au
More information1 Measurement and expectation values
C/CS/Phys 191 Measurement and expectation values, Intro to Spin 2/15/05 Spring 2005 Lecture 9 1 Measurement and expectation values Last time we discussed how useful it is to work in the basis of energy
More informationQuantum mechanics in one hour
Chapter 2 Quantum mechanics in one hour 2.1 Introduction The purpose of this chapter is to refresh your knowledge of quantum mechanics and to establish notation. Depending on your background you might
More informationAlgebra I Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary
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 informationIntroduction to Group Theory
Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)
More informationEuclidean Spaces. Euclidean Spaces. Chapter 10 -S&B
Chapter 10 -S&B The Real Line: every real number is represented by exactly one point on the line. The plane (i.e., consumption bundles): Pairs of numbers have a geometric representation Cartesian plane
More informationFACULTY OF SCIENCES SYLLABUS FOR. B.Sc. (Non-Medical) PHYSICS PART-II. (Semester: III, IV) Session: , MATA GUJRI COLLEGE
FACULTY OF SCIENCES SYLLABUS FOR B.Sc. (Non-Medical) PHYSICS PART-II (Semester: III, IV) Session: 2017 2018, 2018-2019 MATA GUJRI COLLEGE FATEHGARH SAHIB-140406, PUNJAB ----------------------------------------------------------
More informationQuantum system symmetry is not the source of unitary information in wave mechanics context quantum randomness
Quantum system symmetry is not the source of unitary information in wave mechanics context quantum randomness Homogeneity of space is non-unitary Steve Faulkner 5th September 015 Abstract The homogeneity
More informationWhat is spin? Thomas Pope and Werner Hofer. School of Chemistry Newcastle University. Web: wernerhofer.eu
What is spin? Thomas Pope and Werner Hofer School of Chemistry Newcastle University Web: wernerhofer.eu Email: werner.hofer@ncl.ac.uk 1 Overview Introduction 2 Overview Introduction Standard model 3 Overview
More informationPlan for the rest of the semester. ψ a
Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and
More informationStern-Gerlach Experiment and Spin
Stern-Gerlach Experiment and Spin 1 Abstract Vedat Tanrıverdi Physics Department, METU tvedat@metu.edu.tr The historical development of spin and Stern-Gerlach experiment are summarized. Then some questions
More informationLeast-Squares Spectral Analysis Theory Summary
Least-Squares Spectral Analysis Theory Summary Reerence: Mtamakaya, J. D. (2012). Assessment o Atmospheric Pressure Loading on the International GNSS REPRO1 Solutions Periodic Signatures. Ph.D. dissertation,
More informationThe Geometry of R n. Supplemental Lecture Notes for Linear Algebra Courses at Georgia Tech
The Geometry of R n Supplemental Lecture Notes for Linear Algebra Courses at Georgia Tech Contents Vectors in R n. Vectors....................................... The Length and Direction of a Vector......................3
More informationQuantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 4 Postulates of Quantum Mechanics I In today s lecture I will essentially be talking
More informationSECOND QUANTIZATION PART I
PART I SECOND QUANTIZATION 1 Elementary quantum mechanics We assume that the reader is already acquainted with elementary quantum mechanics. An introductory course in quantum mechanics usually addresses
More information