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Spin in Quantum Mechanics 120 5.11TheSphericalBasis...............................106 5.11.1 Transformation of a spherical vector under rotation of the coordi- natesystem ...............................107 5.11.2 Rotationofquantummechanicalstates . 109 5.12 Special Galilei Transformations: Determination of the Unitary Operator Representing the Transformation . 111 5.13 Invariance Under Special Galilei Transformations . .112 5.14 GeneralConsiderations. 115 5.14.1 ExponentialFormoftheRepresentation . 115 5.14.2 CasimirOperators . .. .. .116 5.15 ProjectionOperators . .. .. 117 6 Spin in Quantum Mechanics 120 6.1 SpinorsandTheirProperties. .120 6.1.1 Geometric Visualization of Spinors . 121 6.1.2 RotationsofSpinors ..........................122 6.2 Pauli Matrices and Their Eigenvectors . 123 6.3 Finite Rotations and Pauli Matrices . 125 6.3.1 RotationsAboutthe3-Axis . .125 6.3.2 RotationsAboutthe2-Axis . .125 1 6.3.3 Spinor Nature of Spin- 2 Rotations ..................126 1 6.3.4 General Euler-Angle Rotations for Spin- 2 ...............126 6.4 SpinSpace....................................127 iv 6.4.1 Spinor Space and Its Matrix Representation . 127 6.4.2 Angular Momentum Operators in Spinor Representation . 128 6.4.3 IncludingSpinSpace ..........................128 6.5 Single Particle States with Spin . 131 6.5.1 Coordinate-Space-Spin Representation . 131 6.5.2 Helicity Representation . 131 6.6 Isospin......................................133 6.6.1 Nucleon-Nucleon (NN) Scattering . 135 7 Combining Angular Momentum Eigenstates 137 7.1 AdditionofTwoAngularMomenta . .137 7.2 ConstructionoftheEigenstates . 141 7.3 SpecialCases ..................................143 7.3.1 Spin-OrbitInteraction . .143 1 7.3.2 Coupling of Two Spin- 2 Particles ...................145 7.4 PropertiesofClebsch-GordanCoefficients . 146 7.5 Wigner’s 3 − j Coefficients...........................149 7.6 CartesianandSphericalTensors . 152 7.7 TensorOperatorsinQuantumMechanics . 156 7.8 Wigner-EckartTheorem . .. .. .158 7.9 Applications...................................161 8 Symmetries III: Time Reversal 167 v Chapter 6 Spin in Quantum Mechanics 6.1 Spinors and Their Properties Spinors are mathematical entities, which are useful when describing half-integer spins in the context of rotations of physical systems. Definition of Spinors: A mathematical entity S is a spinor if it satisfies the requirement that it changes sign under a 2π rotation: S(θ +2π) = −S(θ) . (6.1) Part of this requirement is the assumption that S can have a sign associated with it. 1 As it turns out, spin- 2 objects (primarily electrons and subatomic particles) have wave functions that change sign, if the object is rotated through 2π and which can, therefore, be represented by spinors. 120 6.1.1 Geometric Visualization of Spinors A unit circle in the x − z plane is centered at the origin O and intercepting the positive z-axis at P . Consider a ray OC making an angle θ with the z-axis, so that θ is the usual polar angle. Now, bisect the angle so that the point S bisects PC, since OP = OC. Thus OS(θ) = cos(θ/2) ; PS(θ) = sin(θ/2) . (6.2) OS and PS are taken to be signed quantities, OS is negative for π <θ < 2π, as is PS for 2π <θ </4π. As the point C goes once around the unit circle, the point S goes from P to O. A second 1 revolution of C brings S back to P . The path of S is a circle of radius 2 . This path is traced out in a period of 4π of θ. Considering (6.2), OS and PS change sign when θ −→ θ +π. Therefore, according to our rule, the pair (OS,PS) forms a spinor associated with C. In the two-dimensional representation shown in Fig. 6.1, one may view C as marking the −→ tip of the vector OC, which has unit magnitude and θ as polar angle. Then OS and PS −→ form the components of a spinor associated with OC. 121 6.1.2 Rotations of Spinors The properties under rotations of the spinor (OS,PS) associated with C can also be worked out from Fig. 6.1. If C is rotated through an angle β from its position at θ, then one has the following transformations: θ −→ θ + β OS −→ OSβ = cos(β/2) OS − sin(β/2) PS PS −→ PSβ = sin(β/2) OS + cos(β/2) PS, (6.3) which follow from the trigonometric identities for the sum of two angles. The last two equations can be written as OS(θ + β) cos(β/2) − sin(β/2) OS(θ) = (6.4) PS(θ + β) sin(β/2) cos(β/2) PS(θ) or cos(β/2) − sin(β/2) S(θ + β) = S(θ) , (6.5) sin(β/2) cos(β/2) where S(θ) represents the spinor. In this matrix representation, it is clear that the basic spinor property, S(θ +2π)= −S(θ), is fulfilled. The matrix in (6.5) is the 2 × 2 matrix 1 that transforms any spin- 2 angular momentum eigenstate under rotation by β about the y-axis. If we want to define rotations of spinors around the z-axis, then we should require that those spinors are eigenfunctions of the angular momentum operator component for this axis, ∂S J S(θ) = −i = ±S(θ) (6.6) 3 ∂θ where we have used the definition (4.24) for a generalized angular momentum J~. The linear combination of spinors that satisfy this equation are complex, namely S(θ) = e±iθ/2 . (6.7) The matrix form of rotations around the z-axis, which also satisfy (6.6) are e+iα/2 0 S(θ + α) = S(θ) . (6.8) 0 e−iα/2 Thus the spinor S(θ) is complex valued. 122 6.2 Pauli Matrices and Their Eigenvectors Pauli matrices are here introduced through their algebraic properties. They describe 1 the simplest, non-trivial spin system, namely spin 2 . In cartesian coordinates the Pauli matrices are given by 0 1 0 −i 1 0 σ = σ = σ = (6.9) 1 1 0 2 i 0 3 0 −1 and they are collectively denoted by ~σ = (σ1, σ2, σ3). In terms of Paul matrices, the ”spin” matrices S~ are given as 1 S~ = ~σ. (6.10) 2 Properties of Pauli Matrices are: 2 σj = 1 (6.11) for j =1, 2, 3. They anticommute [σi, σk]+ = σiσk + σkσi = 0 ; k =6 i . (6.12) 1 For the quantum mechanical description of a spin- 2 particle, one needs obviously the ~ 1 representation of the angular momentum operators J with j = 2 . The most general form 1 of spin- 2 state would be given by 1 1 1 1 α | , + i + β | , − i (6.13) 2 2 2 2 or written as column vector α . (6.14) β 1 1 The two basis vectors are then | 2 , ± 2 i or 1 0 and (6.15) 0 1 Let us consider 1 S = σ , (6.16) 3 2 3 123 the component of the spin along an arbitrarily chosen 3-axis. We need to find eigenvalues and eigenvectors of σ3. The eigenvalue equation for σ3 reads a a σ + = λ + (6.17) 3 a ± a − − where λ± are the eigenvalues. The solutions for the eigenvectors are given by 1 0 χ = ; χ = (6.18) + 0 − 1 with the corresponding eigenvalues λ+ = +1 and λ− = −1. The eigenvectors χ± are orthonormal, as required for the eigenvectors of a hermitian operator. The eigenstates χ± are usually referred to as Pauli spinors and χ+ represents a ”spin-up,” χ− a ”spin-down” state. According to our general considerations in Chapter 3, we introduce the ladder operators 0 1 0 0 S ≡ and S = (6.19) + 0 0 − 1 0 and it can be easily verified that S+χ+ = | 0i S+χ− = χ+ S−χ+ = χ− S−χ− = | 0i (6.20) Furthermore 1 1 S = (S + S ) = σ 1 2 + − 2 1 1 1 S = (S − S ) = σ (6.21) 2 2i + − 2 2 which is exactly the definition given in (6.10). One could have also introduced σ± = (σ1 ± iσ2) (6.22) and then defined the ladder operators as 0 0 0 2 σ = ; σ = σ , σ = , (6.23) −1 2 0 0 3 +1 0 0 which is often referred to as ’spherical representation.’ Here we have 2 2 σ± = 0 and σ0 = 1 . (6.24) 124 From this follows that higher powers of these spherical-basis matrix elements must be 1 zero. Furthermore, the basis vectors χ± span the spin-space for spin- 2 particles. The completeness relation is given by 1 0 χ χ† + χ χ† = = 1 . (6.25) + + − − 0 1 6.3 Finite Rotations and Pauli Matrices As discussed in the previous chapter, J3 describes infinitesimal rotations and the Ji are the generators of a Lie algebra. From the generators all finite rotations can be constructed. 1 For spin- 2 states the rotation operator has the following form (cp. 5.95 g) 1 α β γ D 2 (αβγ) = e−i 2 σ3 e−i 2 σ2 e−i 2 σ3 . (6.26) Here the description of rotations via Euler angles is used. 6.3.1 Rotations About the 3-Axis Expanding the expression e−iασ3/2 and using the properties of the Pauli matrices, namely 2m 2m+1 that for n even, n =2m, σ3 = 1, and m odd, n =2m +1, σ3 = σ3, one obtains −iγσ3/2 e = cos(γ/2) 1 − i sin(γ/2) σ3 . (6.27) By explicitly inserting σ3 this leads to e−iγ/2 0 e−iγσ3/2 = . (6.28) 0 e+iγ/2 Thus, matrix elements of rotations around the 3-axis are given by −imγ e δm′m . (6.29) 6.3.2 Rotations About the 2-Axis Since σ2 is not diagonal, the rotation matrix must have a more complicated form. The rotation matrix for a rotation of a spinor was already heuristically derived in 6.1.2 and is 125 given by cos(β/2) − sin(β/2) e−iβσ2/2 = = d1/2(β) .
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