Pauli Matrices

8 Key excerpts on "Pauli Matrices"

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  Applications of Group Theory to Atoms, Molecules, and Solids

  • Thomas Wolfram, Şinasi Ellialtıoğlu(Authors)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  5.1 Pauli spin matrices The spin state of an electron can be represented as a two-component spinor. The Pauli Matrices [5.3] for spin 1/2 are matrices that represent the three-dimensional spin-vector operator in terms of matrices that operate on the two-component 123 124 Electron spin and angular momentum B A Upper surface Figure 5.1 Consider a paper model of a Möbius strip. On the right-hand side of the paper, mark the upper surface with an A and the other side of the paper with a B. Starting from A, trace a line through 360 ◦ counter-clockwise on the Möbius strip, arriving at B on the other side of the paper. To return to A requires another 360 ◦ around the Möbius strip. spinors. The Pauli σ -matrices are related to the spin-vector operator by s x = (/2)σ x , s y = (/2)σ y , and s z = (/2)σ z , where σ x =  0 1 1 0  , (5.1) σ y =  0 −i i 0  , (5.2) σ z =  1 0 0 −1  . (5.3) The Pauli Matrices anticommute, meaning that σ i σ j + σ j σ i = {σ i , σ j } = 0 for i  = j . For i = j , {σ i , σ j } = 2 I , where I is the unit matrix. The eigenvectors of Pauli Matrices correspond to the spin states with the spin vector aligned parallel or antiparallel to the direction of the x , y , or z spatial axis. The spin eigenvalues are always ±(/2). This is easily verified by diagonalizing the σ -matrices. The eigenvectors and eigenvalues of s x , s y , and s z are χ + x = 1 √ 2  1 1  , eigenvalue + 1 2 , spin aligned along the + x -axis, (5.4) χ − x = 1 √ 2  1 −1  , eigenvalue − 1 2 , spin aligned along the − x -axis, (5.5) χ + y = 1 √ 2  1 i  , eigenvalue + 1 2 , spin aligned along the + y -axis, (5.6) χ − y = 1 √ 2  1 −i  , eigenvalue − 1 2 , spin aligned along the − y -axis, (5.7) χ + z = 1 √ 2  1 0  , eigenvalue + 1 2 , spin aligned along the + z -axis, (5.8) χ − z = 1 √ 2  0 1  , eigenvalue − 1 2 , spin aligned along the − z -axis. (5.9)

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  Lectures on Quantum Mechanics

  A Primer for Mathematicians

  • Philip L. Bowers(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  19 Wave Mechanics VII: Pauli’s Spinor Theory Pauli himself was trying to incorporate spin into quantum mechanics and introduced the Pauli Matrices, but he could not arrive at a complete theory. He also clearly recognized the necessity of making the theory relativistic and yet could not achieve it. He was known to be a perfectionist who severely criticized other people’s work, but he had no alternative other than to publish his unsatisfactory ad hoc theory. Now on the other hand, Dirac established his theory with totally unexpected acrobatics and solved all the problems on spin as well as making the theory relativistic . . . However, in 1934 Pauli ended up retaliating against Dirac. Pauli showed that Dirac’s argument that nature is satisfied only by the Dirac equation and not by the Klein–Gordon equation is incorrect. According to him, the Klein–Gordon equation is not in contradiction with the framework of quantum mechanics, and there is no reason that nature abhors a particle with spin 0. Sin-Itiro Tomonaga The Story of Spin, 1997 We have seen how the whole-integer representations of su(2) are used in wave mechanics to represent the operators that correspond to orbital angular momentum, and how this leads to an understanding of the ir- reducible representations of the rotation group SO(3) on the space of spherical harmonics. This was the content of Lecture 14 and was used subsequently in Lecture 15 to investigate the hydrogenic potential in quantum mechanics. Orbital angular momentum is an example of an observable that might be labeled as extrinsic, in that it arises from what we might call external degrees of freedom for the particle. It is not a property of the quantum particle itself, but rather a property of 285 286 Wave Mechanics VII: Pauli’s Spinor Theory that particle’s placement and motion within its environment and is very much dependent on that environment. For example, it is measured with respect to a center that may be chosen quite arbitrarily.

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  Quantum Electrodynamics

  • Eugene Stefanovich(Author)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  B Quantum fields of fermions Usually QFT textbooks claim that (quantum) fields are the fundamental ingredients of nature, and the main task of QFT is to apply the laws of quantum mechanics to these systems with an infinite number of degrees of freedom. We do not adhere to this point of view. We believe that matter consists of particles, and quantum fields are just ab-stract mathematical creations, whose purpose is to simplify the construction of rela-tivistically invariant and cluster-separable operators of interactions between particles. Therefore we placed our discussion of quantum fields in this appendix, and not in the main body of the book. Here we will talk about quantum fields for massive fermions with spin 1/2 (electrons, protons and their antiparticles). In the next appendix we will consider the photon quantum field. B.1 Pauli Matrices Generators of the spin-1/2 representation of the rotation group (see Table 1 -I.1) are con-veniently expressed through so-called Pauli Matrices σ i ( i = x , y , z ). We have S i = ℏ 2 σ i , (B.1) where σ x ≡ σ 1 = [ 0 1 1 0 ] , σ y ≡ σ 2 = [ 0 − i i 0 ] , σ z ≡ σ 3 = [ 1 0 0 − 1 ] . In calculations we will need the following properties of these matrices: [ σ i , σ j ] = 2 i 3 ∑ i = 1 ϵ ijk σ k , { σ i , σ j } = 2 δ ij , σ 2 i = 1 . Sometimes it is useful to define the fourth Pauli matrix σ t ≡ σ 0 = [ 1 0 0 1 ] . https://doi.org/10.1515/9783110493207-006 108 | B Quantum fields of fermions For arbitrary numerical 3-vectors a and b we have ( σ ⋅ a ) σ = a σ 0 + i [ σ × a ], (B.2) σ ( σ ⋅ a ) = a σ 0 − i [ σ × a ], (B.3) ( σ ⋅ a )( σ ⋅ b ) = ( a ⋅ b ) σ 0 + i σ ⋅ [ a × b ].

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  Quantum Computing

  From Linear Algebra to Physical Realizations

  • Mikio Nakahara, Tetsuo Ohmi(Authors)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  18 QUANTUM COMPUTING EXERCISE 1.12 Let H be a Hermitian matrix. Show that U = ( I + iH )( I − iH ) − 1 is unitary. This transformation is called the Cayley transformation . 1.7 Pauli Matrices Let us consider spin 1 / 2 particles, such as an electron or a proton. These parti-cles have an internal degree of freedom: the spin-up and spin-down states. (To be more precise, these are expressions that are relevant when the z -component of an angular momentum S z is diagonalized. If S x is diagonalized, for example, these two quantum states can be either “spin-right” or “spin-left.”) Since the spin-up and spin-down states are orthogonal, we can take their components to be | ↑ = 1 0 , | ↓ = 0 1 . (1.32) Verify that they are eigenvectors of σ z satisfying σ z | ↑ = | ↑ and σ z | ↓ = −| ↓ . In quantum information, we often use the notations | 0 = | ↑ and | 1 = | ↓ . Moreover, the states | 0 and | 1 are not necessarily associated with spins. They may represent any two mutually orthogonal states, such as horizontally and vertically polarized photons. Thus we are free from any physical system, even though the terminology of spin algebra may be employed. For electrons and protons, the spin angular momentum operator is conve-niently expressed in terms of the Pauli Matrices σ k as S k = ( / 2) σ k . We often employ natural units in which = 1. Note the tracelessness property tr σ k = 0 and the Hermiticity σ † k = σ k . ‡ In addition to the Pauli Matrices, we introduce the unit matrix I in the algebra, which amounts to expanding the Lie algebra su (2) to u (2). The Pauli Matrices satisfy the anticommutation relations { σ i , σ j } = σ i σ j + σ j σ i = 2 δ ij I. (1.33) Therefore, the eigenvalues of σ k are found to be ± 1. The commutation relations between the Pauli Matrices are [ σ i , σ j ] = σ i σ j − σ j σ i = 2 i k ε ijk σ k , (1.34) ‡ Mathematically speaking, these two properties imply that iσ k are generators of the su (2) Lie algebra associated with the Lie group SU(2).

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  Solid State and Quantum Theory for Optoelectronics

  • Michael A. Parker(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  The use of the Pauli operators will next be used to show how the wave functions produce the required average value of the spin vector. β 1 Im β 2 Re β 2 Y ψ X θ θ/2 φ Z S |1 |2 FIGURE 5.33 An intuitive map between the classical spin (average) and the quantum mechanical wave function. 314 Solid State and Quantum Theory for Optoelectronics 5.6.3 P AULI S PIN M ATRICES We can use the link between the Hilbert space and the classical spin to establish the spin operator. We look for operators ^ ~ S ¼ ~ xS ^ x þ ~ yS ^ y þ ~ zS ^ z that allow us to deduce the classical particle spin ~ S ¼ ^ ~ S D E as given in the previous section. Conventionally, the operators are de fi ned in terms of the Pauli spin operators to remove pesky factors of Planck ’ s constant. S ^ x ¼ h 2 ^ s x S ^ y ¼ h 2 ^ s y S ^ z ¼ h 2 ^ s z ! ^ ~ S ¼ h 2 ~ s ¼ h 2 ~ x ^ s x þ ~ y s y þ ~ z ^ s z ( 5 : 147 ) The objective consists of fi nding the Pauli Matrices. The operator measuring the spin along the z -direction ^ s z can be built from the eigenvalues and eigenvectors. ^ s z j 1 i¼þ 1 j 1 i and ^ s z j 2 i¼ 1 j 2 i ( 5 : 148 a) In the usual manner prescribed in linear algebra, the operator can be written in the basis vector expansion as ^ s z ¼j 1 ih 1 jj 2 ih 2 j or s z ¼ 1 0 0 1 ( 5 : 148 b) We can now demonstrate the Pauli spin matrices for the x -and y -directions. We demonstrate the Pauli y -spin matrix and leave the one for the x -direction to the problems. Using Equation 5.146a with u ¼ 90 and w ¼ 90 (i.e., the classical spin points along the y -axis in the laboratory), we want the following conditions to hold s y 1 ffiffi 2 p 1 i ¼þ 1 ffiffi 2 p 1 i s y 1 ffiffi 2 p 1 i ¼ 1 ffiffi 2 p 1 i ( 5 : 149 ) That is, we want the column vector 1 ffiffi 2 p 1 i to represent an average spin along the physical y -direction. We need to fi nd an operator, ^ s y in this case, that can be used as a type of ‘‘ indicator ’’ that tells us when a wave function produces a classical component parallel to the physical y -direction.

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  Quantum Theory

  Elements

  • D. R. Bates(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  In Table I we give a brief summary of the four most important operators of quantum mechanics; the observables in question are understood to refer to systems classically described as groups of mass points having 3 « degrees of freedom ( / = 1,2,.. .,n), subject to no external forces (total energy constant) and not requiring relativity treatment. T A B L E I 2. F U N D A M E N T A L PRINCIPLES OF Q U A N T U M MECHANICS 35 (2) Commutation rules, like (1) and (2), are often sufficient to define the operators involved without recourse to their explicit form, but the latter is usually helpful. Turning now to the matrices which are to be associated with observables, we observe that by the rule (41) of Chapter 1 each one of the operators of our list can be converted into a matrix, and this is possible in an unlimited number of ways. Any complete orthonormal set of functions satisfying Eq. (40) in Chapter 1 involving the variables which appear in the operator under consideration is available for this conversion. To give just one example of the formation of matrices from operators, let us employ the orthonormal set known as Hermite functions, with properties described in Appendix 3.1. Their form is * -' = ( i á ? r < ^ - - » K i j ? > -( í ) LxLy -LyL, = ihL, LyLg LzLy = ihL, -L,L, = ihLy Í Ly and L , may be obtained from in Table I by cyclical permutation of coordinates. Linear momentum, a differential operator, is basic in the construction of the last two entries in the table. When the operator corresponding to the linear momentum ρ of a single particle is written in the vector form — ihV, those corresponding to angular momentum and energy of this particle may be constructed according to classical formulas. Angular momentum = r χ ρ = — t 'Är χ P, and energy = (l/2w )^2 ^. y = - {n^ßm)V^ + V. These vector forms are valid in all other systems of coordinates and should be used as the basis for transformations.

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  Applications Of Unitary Symmetry And Combinatorics

  • James D Louck(Author)
  • 2011(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 1 Composite Quantum Systems 1.1 Introduction The group and angular momentum theory of composite quantum sys-tems was initiated by Weyl [80] and Wigner [82]. It is an intricate, but well-developed subject, as reviewed in Biedenharn and van Dam [7], and documented by the many references in [6]. It is synthesized further by the so-called binary coupling theory developed in great detail in [L]. It was not realized at the time that recoupling matrices, the objects that encode the full prescription for relating one coupling scheme to another, are doubly stochastic matrices. This volume develops this aspect of the theory and related topics. We review in this first chapter some of the relevant aspects of the coupling theory of angular momenta for ease of reference. Curiously, these developments relate to the symmetric group S n , which is a finite subgroup of the general unitary group and which is also considered in considerable detail in the previous volume. But here the symmetric group makes its appearance in the form of one of its simplest matrix (reducible) representations, the so-called permutation matrices. The symmetric group is one of the most important groups in physics (Wybourne [87]), as well as mathematics (Robinson [67]). In physics, this is partly because of the Pauli exclusion principle, which ex-presses a collective property of the many entities that constitute a com-posite system; in mathematics, it is partly because every finite group is isomorphic to a symmetric group of some order. While the symmetric group is one of the most studied of all groups, many of its properties that relate to doubly stochastic matrices, and other matrices of physi-cal importance, seem not to have been developed. This review chapter provides the background and motivation for this continued study. 1

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  Pauli And The Spin-statistics Theorem

  • Ian Duck, E C George Sudarshan(Authors)
  • 1998(Publication Date)
  • World Scientific

    (Publisher)

  . . §3. Guide to Pauli's Proof. We now look at the details of Pauli's proof which requires only very simple and general properties of van der Waerden spinors [14.8]. Recall the 348 basic spinor-vector connection V aP = yn a aP = (yO + - . y)a£ (^ and where Lorentz indices /i = 0,1,2,3, spinor indices a,/?=l,2, and the Pauli Matrices a^ — X.a. Spinor indices are raised and lowered with the alternating symbol e°^ = —e^*, e a p = -ep a with e 12 = e 2 = 1, etc. The inverse relation for the vector is V = -. (3) More generally, spinors V^Q'„ can be characterized by two positive in-tegers (r, q) expressed in terms of angular momentum quantum numbers (j, k) with r — 2j + 1 and g = 2A; + 1 and (jf, k) integral or half-integral. The number of lower, undotted indices is 2j, and of upper, dotted indices is 2k. The number of independent components is rq = (2j + l)(2k + 1). For example (j, A;) = (0,0) is a scalar, ( | , ) is a 4-vector, (1,0) is self-dual anti-symmetric tensor, (1,1) is a traceless symmetric tensor, and so on. A Dirac four-spinor is a combination of two irreducible spinors (|,0) and (0,^). A product of two irreducible representations ?7i(ji, fci)C/2(j2» ^2) decom-poses into a sum of irreducible representations U(j, k) with 3 = Ji + h,3i + Jf2 -1 , • • ■ > |ji - J2I, fc = fci + k 2 ,h + k 2 -l,---,|fci -k 2 . (4) Under 27r-space rotations, representations undergo a sign change if one of (j, k) is half-integral and the other integral, and no sign change if both of (j, k) are integral or both half-integral. Thus (1,0), (1,1), ( | , | ) , etc are single-valued under rotations; (|,0), ( | , 1 ) , etc are double-valued. The product of two single-valued or two double-valued representations is a single-valued representation; the product of one single-valued and one double-valued representation is double-valued. 349 Consider first single-valued representations which all have j + k integral.

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