DOCSLIB.ORG

Einstein Notation and Commutators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Einstein Notation and Commutators Spring 2017 Physics 115C Group 1 115C: Quantum Group 1 Commutators, Vector Operators, and Measurement Jared Pagett April 26, 2017 1 Index Notation and Einstein Notation As you discovered on the last homework, writing out huge lists of components can make commutators tedious { especially when dealing with cross products. You can use Einstein notation to make life easier on yourself. For an introduction to index notation and Einstein notation, see the following: This may be obvious to some of you, but in case you haven't seen it, a vector v represented in the basis fe^ig (which, for convenience, we will take as orthonormal) with components fvig can be written X v = vie^i i Note that for two vectors A and B, we can write X X X X A · B = (Aie^i) · (Bje^j) = Ai(e^i · e^j)Bj = AiδijBj = AiBi ij ij ij i where we sum over i = x; y; z. In Einstein notation we suppress the summation sign, and take on the convention that if an index is repeated there is an implied sum over it. For example, we would write for the dot product above A · B ≡ AiBi In case you aren't familiar with cross products in summation notation, note that the cross product of vectors A and B takes the form A × B = ijkAjBke^i where ijk is the Levi-Civita symbol. This symbol is antisymmetric under the exchange of any two of its indices, meaning that ijk = −ikj = kij, and so on. th th Finally, some side notes about matrices. We write the entry of a matrix A in the i row, j column as Aij (or aij). The trace (Tr) of a matrix is the sum of its diagonal components, and so can be written Tr[A] = Aijδij = Aii The product of two matrices A and B is generally written AB = (AB)ik = AijBjk 2 Index Notation Practice These identities might prove useful: ijkimn = δjmδkn − δjnδkm & jmnimn = 2δij [AB; C] = A [B; C] + [A; C; B] [A; BC] = B [A; C] + [A; B] C For each of the following, try not to write out individual components { the idea here is to practice using index notation. 1 Spring 2017 Physics 115C Group 1 2.1 Proving the Bread & Butter Given that L = r × p, and knowing that [ri; rj] = 0, [pi; pj] = 0, and [ri; pj] = i~δij, show that [Li;Lj] = i~ijkLk This may not be the best problem to illustrate that index notation makes life easy { still, practice is good. Next, verify that [Li; xj] = i~ijkxk and [Li; pj] = i~ijkpk 2.2 Again! Remember that lovely problem from your homework where you computed [L · S; L]? If you found yourself writing out components, try it again with Einstein notation. Hopefully you'll find it much easier this time. You may use the fact that [Li;Sj] = 0 and that [Li;Lj] = i~ijkLk. 2.3 Quirks of Vector Operators On the last homework set, you dealt a lot with L and S { vectors which have operators as their components. Sometimes this can introduce quirks that you're not used to dealing with { if you're not careful, the simplest vector identities can take on a far more sinister nature. To do the following, make use of the fact that you can switch the order of any two operators at the cost of those terms' commutator: aibj = bjai + [ai; bj] 1. For general vector operators a and b, (a) How is a · b related to b · a? (b) How is a × b related to b × a? Be careful not to assume anything about the way components of a and b behave! 2. Compute the following: (a) L × L. It's not what you think! (b) L · (L × S). Again { it's not what you think! You may use the result of the previous part. 3. Convince yourself that [A · B; C] 6= A · [B; C] + [A; C] · B. If you don't see it, write out one of the commutators on the right side and ask yourself if what you've written makes sense. 2.4 More Practice Show the following using Einstein notation. 1. Ah, your good friends the Pauli matrices. Given that [σi; σj] = 2iijkσk and that fσi; σjg = σiσj + σjσi = 2δijI, (a) Show that 1 σ σ = (fσ ; σ g + [σ ; σ ]) i j 2 i j i j (b) With the given information and the result of the previous part, show that (σ · a) · (σ · b) = (a · b)I + iσ · (a × b) for σ = (σ1; σ2; σ3) and general 3-vectors a and b. (c) Use the results of the previous part to determine the result of (σ · n^)2 for general unit vector n^. 2 Spring 2017 Physics 115C Group 1 2. Show that the scalar triple product is invariant under cyclic permutation { that is, that for vectors a, b, and c with real coefficients, a · (b × c) = c · (a × b) 3. Show that for n × n matrices Ω, Γ, and Λ, (a) Tr[ΩΓ] = Tr[ΓΩ] (b) Tr[ΩΓΛ] = Tr[ΓΛΩ] = Tr[ΛΩΓ] { i.e., that the trace is invariant under cyclic permutation. 4. Without using the result of the last part, show that the trace is invariant under unitary transformations { that for general matrix Ω and unitary U, Tr[U yΩU] = Tr[Ω]. Recall that a matrix U is unitary if and only if U yU = I. 3 Measurement Something really important to quantum mechanics is the concept of measurement. This section is intended to be review, but hopefully it proves helpful. First, some clarification. The result of a measurement is an eigenvalue. Yes, the measurement causes your wavefunction to collapse to an eigenstate of the measured observable, but the thing you get back from your measurement is the eigenvalue corresponding to the eigenstate it collapsed to. Let's say we start with a state jΨi and make a measurement of an observable A^. Suppose that our state collapses to eigenstate jai of A^ such that A^jai = ajai. • The result of our measurement will be a. • The probability of having obtained this result from a measurement of A^ on jΨi is given by P (a) = jhajΨij2 Unless it's incredibly convenient (say, if you can use a Clebsch-Gordon table or something), there's no need to go rewrite jΨi in the basis of eigenstates of A^ and pick out the coefficient of the jai component to square and obtain the probability. Don't do it! You can simply compute the inner product right off the bat. Some of you have a fixation on the coefficients of the expansion and do unimaginable amounts of algebra to obtain them all { just jump right to the inner products. • The system is now in state jai. Further measurements of A^ will give as a result the eigenvalue a, should we make those measurements without allowing our state to evolve in time. Whether this all seems obvious or intimidating, practice { make sure it sticks. Practice Measurement: Spinors Given the spinor χ0 below represented terms of the z basis states χ+ and χ−, what is the probability of obtaining −~=2 for a measurement of the component of spin in the x-direction? χ0 = A [(1 + i)χ+ + (3 − i)χ−] 3.
Load more

Recommended publications
  • Package 'Einsum'
    Package ‘einsum’ May 15, 2021 Type Package Title Einstein Summation Version 0.1.0 Description The summation notation suggested by Einstein (1916) <doi:10.1002/andp.19163540702> is a concise mathematical notation that implicitly sums over repeated indices of n- dimensional arrays. Many ordinary matrix operations (e.g. transpose, matrix multiplication, scalar product, 'diag()', trace etc.) can be written using Einstein notation. The notation is particularly convenient for expressing operations on arrays with more than two dimensions because the respective operators ('tensor products') might not have a standardized name. License MIT + file LICENSE Encoding UTF-8 SystemRequirements C++11 Suggests testthat, covr RdMacros mathjaxr RoxygenNote 7.1.1 LinkingTo Rcpp Imports Rcpp, glue, mathjaxr R topics documented: einsum . .2 einsum_package . .3 Index 5 1 2 einsum einsum Einstein Summation Description Einstein summation is a convenient and concise notation for operations on n-dimensional arrays. Usage einsum(equation_string, ...) einsum_generator(equation_string, compile_function = TRUE) Arguments equation_string a string in Einstein notation where arrays are separated by ’,’ and the result is separated by ’->’. For example "ij,jk->ik" corresponds to a standard matrix multiplication. Whitespace inside the equation_string is ignored. Unlike the equivalent functions in Python, einsum() only supports the explicit mode. This means that the equation_string must contain ’->’. ... the arrays that are combined. All arguments are converted to arrays with
    [Show full text]
  • Notes on Manifolds
    Notes on Manifolds Justin H. Le Department of Electrical & Computer Engineering University of Nevada, Las Vegas [email protected] August 3, 2016 1 Multilinear maps A tensor T of order r can be expressed as the tensor product of r vectors: T = u1 ⊗ u2 ⊗ ::: ⊗ ur (1) We herein fix r = 3 whenever it eases exposition. Recall that a vector u 2 U can be expressed as the combination of the basis vectors of U. Transform these basis vectors with a matrix A, and if the resulting vector u0 is equivalent to uA, then the components of u are said to be covariant. If u0 = A−1u, i.e., the vector changes inversely with the change of basis, then the components of u are contravariant. By Einstein notation, we index the covariant components of a tensor in subscript and the contravariant components in superscript. Just as the components of a vector u can be indexed by an integer i (as in ui), tensor components can be indexed as Tijk. Additionally, as we can view a matrix to be a linear map M : U ! V from one finite-dimensional vector space to another, we can consider a tensor to be multilinear map T : V ∗r × V s ! R, where V s denotes the s-th-order Cartesian product of vector space V with itself and likewise for its algebraic dual space V ∗. In this sense, a tensor maps an ordered sequence of vectors to one of its (scalar) components. Just as a linear map satisfies M(a1u1 + a2u2) = a1M(u1) + a2M(u2), we call an r-th-order tensor multilinear if it satisfies T (u1; : : : ; a1v1 + a2v2; : : : ; ur) = a1T (u1; : : : ; v1; : : : ; ur) + a2T (u1; : : : ; v2; : : : ; ur); (2) for scalars a1 and a2.
    [Show full text]
  • The Mechanics of the Fermionic and Bosonic Fields: an Introduction to the Standard Model and Particle Physics
    The Mechanics of the Fermionic and Bosonic Fields: An Introduction to the Standard Model and Particle Physics Evan McCarthy Phys. 460: Seminar in Physics, Spring 2014 Aug. 27,! 2014 1.Introduction 2.The Standard Model of Particle Physics 2.1.The Standard Model Lagrangian 2.2.Gauge Invariance 3.Mechanics of the Fermionic Field 3.1.Fermi-Dirac Statistics 3.2.Fermion Spinor Field 4.Mechanics of the Bosonic Field 4.1.Spin-Statistics Theorem 4.2.Bose Einstein Statistics !5.Conclusion ! 1. Introduction While Quantum Field Theory (QFT) is a remarkably successful tool of quantum particle physics, it is not used as a strictly predictive model. Rather, it is used as a framework within which predictive models - such as the Standard Model of particle physics (SM) - may operate. The overarching success of QFT lends it the ability to mathematically unify three of the four forces of nature, namely, the strong and weak nuclear forces, and electromagnetism. Recently substantiated further by the prediction and discovery of the Higgs boson, the SM has proven to be an extraordinarily proficient predictive model for all the subatomic particles and forces. The question remains, what is to be done with gravity - the fourth force of nature? Within the framework of QFT theoreticians have predicted the existence of yet another boson called the graviton. For this reason QFT has a very attractive allure, despite its limitations. According to !1 QFT the gravitational force is attributed to the interaction between two gravitons, however when applying the equations of General Relativity (GR) the force between two gravitons becomes infinite! Results like this are nonsensical and must be resolved for the theory to stand.
    [Show full text]
  • Multilinear Algebra and Applications July 15, 2014
    Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3.
    [Show full text]
  • Abstract Tensor Systems As Monoidal Categories
    Abstract Tensor Systems as Monoidal Categories Aleks Kissinger Dedicated to Joachim Lambek on the occasion of his 90th birthday October 31, 2018 Abstract The primary contribution of this paper is to give a formal, categorical treatment to Penrose’s abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for traced symmetric monoidal categories. 1 Introduction This paper formalises the connection between monoidal categories and the ab- stract index notation developed by Penrose in the 1970s, which has been used by physicists directly, and category theorists implicitly, via the diagrammatic languages for traced symmetric monoidal and compact closed categories. This connection is given as a representation theorem for the free traced symmet- ric monoidal category as a syntactically-defined strict monoidal category whose morphisms are equivalence classes of certain kinds of terms called Einstein ex- pressions. Representation theorems of this kind form a rich history of coherence results for monoidal categories originating in the 1960s [17, 6]. Lambek’s con- arXiv:1308.3586v1 [math.CT] 16 Aug 2013 tribution [15, 16] plays an essential role in this history, providing some of the earliest examples of syntactically-constructed free categories and most of the key ingredients in Kelly and Mac Lane’s proof of the coherence theorem for closed monoidal categories [11].
    [Show full text]
  • Quantum Chromodynamics (QCD) QCD Is the Theory That Describes the Action of the Strong Force
    Quantum chromodynamics (QCD) QCD is the theory that describes the action of the strong force. QCD was constructed in analogy to quantum electrodynamics (QED), the quantum field theory of the electromagnetic force. In QED the electromagnetic interactions of charged particles are described through the emission and subsequent absorption of massless photons (force carriers of QED); such interactions are not possible between uncharged, electrically neutral particles. By analogy with QED, quantum chromodynamics predicts the existence of gluons, which transmit the strong force between particles of matter that carry color, a strong charge. The color charge was introduced in 1964 by Greenberg to resolve spin-statistics contradictions in hadron spectroscopy. In 1965 Nambu and Han introduced the octet of gluons. In 1972, Gell-Mann and Fritzsch, coined the term quantum chromodynamics as the gauge theory of the strong interaction. In particular, they employed the general field theory developed in the 1950s by Yang and Mills, in which the carrier particles of a force can themselves radiate further carrier particles. (This is different from QED, where the photons that carry the electromagnetic force do not radiate further photons.) First, QED Lagrangian… µ ! # 1 µν LQED = ψeiγ "∂µ +ieAµ $ψe − meψeψe − Fµν F 4 µν µ ν ν µ Einstein notation: • F =∂ A −∂ A EM field tensor when an index variable µ • A four potential of the photon field appears twice in a single term, it implies summation µ •γ Dirac 4x4 matrices of that term over all the values of the index
    [Show full text]
  • About Matrices, Tensors and Various Abbrevia- Tions
    About matrices, tensors and various abbrevia- tions The objects we deal with in this course are rather complicated. We therefore use a simplifying notation where at every step as much as possible of this complexity is hidden. Basically this means that objects that are made out of many elements or components. are written without indices as much as possible. We also have several different types of indices present. The ones that show up in this course are: • Dirac-Indices a; b; c • Lorentz indices, both upper and lower µ, ν; ρ • SU(2)L indices i; j; k • SU(3)c indices α; β; γ Towards the right I have written the type of symbols used in this note to denote a particular type of index. The course contains even more, three-vector indices or various others denoting sums over types of quarks and/or leptons. An object is called scalar or singlet if it has no index of a particular type of index, a vector if it has one, a matrix if it has two and a tensor if it has two or more. A vector can also be called a column or row matrix. Examples are b with elements bi: 0 1 b1 B C B b2 C b = (b1; b2; : : : ; bn) b = (b1 b2 ··· bn) b = B . C (1) B . C @ . A bn In the first case is a vector, the second a row matrix and the last a column vector. Objects with two indices are called a matrix or sometimes a tensor and de- noted by 0 1 c11 c12 ··· c1n B C B c21 c22 ··· c2n C c = (cij) = B .
    [Show full text]
  • Tensor Algebra
    TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC Introduction to Tensors Tensor Algebra 2 Introduction SCALAR , , ... v VECTOR vf, , ... MATRIX σε,,... ? C,... 3 Concept of Tensor A TENSOR is an algebraic entity with various components which generalizes the concepts of scalar, vector and matrix. Many physical quantities are mathematically represented as tensors. Tensors are independent of any reference system but, by need, are commonly represented in one by means of their “component matrices”. The components of a tensor will depend on the reference system chosen and will vary with it. 4 Order of a Tensor The order of a tensor is given by the number of indexes needed to specify without ambiguity a component of a tensor. a Scalar: zero dimension 3.14 1.2 v 0.3 a , a Vector: 1 dimension i 0.8 0.1 0 1.3 2nd order: 2 dimensions A, A E 02.40.5 ij rd A , A 3 order: 3 dimensions 1.3 0.5 5.8 A , A 4th order … 5 Cartesian Coordinate System Given an orthonormal basis formed by three mutually perpendicular unit vectors: eeˆˆ12,, ee ˆˆ 23 ee ˆˆ 31 Where: eeeˆˆˆ1231, 1, 1 Note that 1 if ij eeˆˆi j ij 0 if ij 6 Cylindrical Coordinate System x3 xr1 cos x(,rz , ) xr2 sin xz3 eeeˆˆˆr cosθθ 12 sin eeeˆˆˆsinθθ cos x2 12 eeˆˆz 3 x1 7 Spherical Coordinate System x3 xr1 sin cos xrxr, , 2 sin sin xr3 cos ˆˆˆˆ x2 eeeer sinθφ sin 123sin θ cos φ cos θ eeeˆˆˆ cosφφ 12sin x1 eeeeˆˆˆˆφ cosθφ sin 123cos θ cos φ sin θ 8 Indicial or (Index) Notation Tensor Algebra 9 Tensor Bases – VECTOR A vector v can be written as a unique linear combination of the three vector basis eˆ for i 1, 2, 3 .
    [Show full text]
  • Introduction to Quantum Mechanics Unit 0. Review of Linear Algebra
    Introduction to Quantum Mechanics Unit 0. Review of Linear Algebra A. Linear Space 1. Definition: A linear space is a collection S of vectors |a>, |b>, |c>, …. (usually infinite number of them) on which vector addition + is defined so that (i) (S,+) forms an Abelien group, i,e,, S is closed under + + is associate: (|a>+|b>)+|c> = |a>+(|b>+|c>) + is commutative: |a>+|b>=|b>+|a> Existence of identity: |a> + |0> = |a> for all |a> Existence of inverse: |a> + (|-a>) = |0> and also scalar (usually complex number) * is defined so that (ii) S is closed under * (iii) * and complex number multiplication is associative: α(β*|a>)=(αβ)*|a> (iv) 1|a>=|a> (and hence 0|a>=|0> and |a>=-|a>) (v) * and + is distributive: (α+β)|a>=α|a>+β|a> and α(|a>+|b>)=α|a>+α|b> 2. Basis and dimension (i) Definition of a basis: A basis is a collection of vectors B={|a1>, |a2>,|a3>,….|aN>} such that any vector in S can be written as a linear combination of a1>, |a2>,|a3>,….|aN>: a = α1 a1 + α 2 a 2 + α3 a 3 +Lα N a N where α1 ,α 2 ,Kα N are complex numbers Furthermore, ={|a1>, |a2>,|a3>,….|aN> are independent of each other, i.e. 0 = α1 a1 + α 2 a 2 + α 3 a 3 +Lα N a N if and only if α1 = α 2 = K = α N = 0 (ii) Choice of a basis is not unique. (iii) However, all basises have the same number of elements. This number (say, N) depends only on the linear space S, and it is known as the dimension of the linear space S.
    [Show full text]
  • Lost in the Tensors: Einstein's Struggles with Covariance Principles 1912-1916"
    JOHN EARMAN and CLARK GL YMOUR LOST IN THE TENSORS: EINSTEIN'S STRUGGLES WITH COVARIANCE PRINCIPLES 1912-1916" Introduction IN 1912 Einstein began to devote a major portion of his time and energy to an attempt to construct a relativistic theory of gravitation. A strong intimation of the struggle that lay ahead is contained in a letter to Arnold Sommerfeld dated October 29, 1912: At the moment I am working solely on the problem of gravitation and believe 1 will be able to overcome all difficulties with the help of a local, friendly mathemat- ician. But one thing is certain, that I have never worked so hard in my life, and that I have been injected with a great awe of mathematics, which in my naivet~ until now I only viewed as a pure luxury in its subtler forms! Compared to this problem the original theory of relativity is mere child's play.' Einstein's letter contained only a perfunctory reply to a query from Sommerfeld about the Debye-Born theory of specific heats. Obviously disappointed, Som- merfeld wrote to Hilbert: 'My letter to Einstein was in vain . Einstein is evidently so deeply mired in gravitation that he is deaf to everything else? Sommerfeld's words were more prophetic than he could possibly have known; the next three years were to see Einstein deeply mired in gravitation, sometimes seemingly hopelessly so. In large measure, Einstein's struggle resulted from his use and his misuse, his understanding and his misunderstanding of the nature and implications of covariance principles. In brief, considerations of general covariance were bound up with Einstein's motive for seeking a 'generalized' theory of relativity; mis- understandings about the meaning and implementation of this motivation threatened to wreck the search; and in the end, the desire for general covariance helped to bring Einstein back onto the track which led to what we now recognize *Present address c/o Department of Philosophy, University of Minnesota, Minneapolis, Minn, U.S.A.
    [Show full text]
  • Indicial Notation
    Appendix A Indicial Notation A.1 Indicial Notation for Vector and Matrix Operations In tensor analysis, an extensive use of indicial notation is made. Operations using Cartesian components of vectors and matrices can be expressed efficiently using this notation. A.1.1 Elements of a Matrix A set of n variables x1, x2,...,xn is denoted using indicial notation such as as xi , i = 1, 2,...,n. Let us consider now the equation that describes a plane in a three dimensional space x1, x2, x3, a1x1 + a2x2 + a3x3 = p, where ai and p are constants. This equation can be expressed alternatively as 3 ai xi = p. i=1 At this point, we can use the convention usually referred to as Einstein notation or Einstein summation convention, which implies summation over a set of indexed terms in a formula when an index appears twice in a single term along the variational rank of the indexes (1 to 3 in the above equation). Therefore, ai xi = p In the same spirit, © Springer International Publishing AG, part of Springer Nature 2018 227 E. Cueto and D. González, An Introduction to Structural Mechanics for Architects, Structural Integrity 4, https://doi.org/10.1007/978-3-319-72935-0 228 Appendix A: Indicial Notation • Vectorial scalar product: u · v = ui vi • Norm of a vector: √ √ ||u|| = u · u = ui ui • Derivative of a function: n ∂ f ∂ f df = dx = dx = f, dx ∂x i ∂x i i i i=1 i i 1 Let us define the Kronecker Delta δij as a symbol whose values can be, = δ = 1ifi j ij 0ifi = j In the same way, let us define the permutation symbol εijk as, ⎧ ⎨ 0ifi = jj= ki= k ε = , , ijk ⎩ 1ifi j k for an even number permutation −1ifi, j, k form an odd permutation A.1.
    [Show full text]
  • 1 Einstein Notation
    PH 108 : Electricity & Magnetism Einstein Notation Rwitaban Goswami 1 Einstein Notation 2.6. The gradient operator r behaves like a vector in \ some sense". For example, divergence of a curl (r:r × A~ = 0) for any A~, may suggest that it is just like A:~ B~ × C~ being zero if any two vectors are equal. Prove that r × r × F~ = r(r:F~ ) − r2F~ . To what extent does this look like the well known expansion of A~ × B~ × C~ ? Solution: We will be using Einstein summation convention. According to this convention: 1. The whole term is summed over any index which appears twice in that term 2. Any index appearing only once in the term in not summed over and must be present in both LHS and RHS Don't be confused by this. The convention only is to simply omit the P symbol. You can mentally insert the P symbol to make sense of what the summation will look like. Now let us go over some common symbols that are used extensively in Einstein's notation: • δij is the Kronecker-Delta symbol. It is defined as ( 1 if i = j δ = ij 0 if i 6= j • ijk is the Levi-Civita symbol. It is defined as 81 if ijk is a cyclic permutation of xyz, eg xyz, yzx, zxy <> ijk = −1 if ijk is an anticyclic permuation of xyz, eg xzy, zyx, yxz :>0 if in ijk any index is repeating, eg xxy, xyy etc Now, you can verify: ~ ~ A · B = AiBi ~ ~ A × B = ijkAjBk e^i wheree ^i is the unit vector in i direction In the cross product formula, when the P symbol is reinserted, this simply means X X X ijkAjBke^i i=x;y;z j=x;y;z k=x;y;z A few things to note: P • δii = 3.
    [Show full text]