Lectures on Atomic Physics, Schemes and Mind Maps of Physics

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Lectures on Atomic Physics

Walter R. Johnson

Department of Physics, University of Notre Dame

Notre Dame, Indiana 46556, U.S.A.

January 4, 2006

Contents

• 1 Angular Momentum Preface xi
  • 1.1 Orbital Angular Momentum - Spherical Harmonics
    • 1.1.1 Quantum Mechanics of Angular Momentum
    • 1.1.2 Spherical Coordinates - Spherical Harmonics
  • 1.2 Spin Angular Momentum
    • 1.2.1 Spin 1/2 and Spinors
    • 1.2.2 Infinitesimal Rotations of Vector Fields
    • 1.2.3 Spin 1 and Vectors
  • 1.3 Clebsch-Gordan Coefficients
    • 1.3.1 Three-j symbols
    • 1.3.2 Irreducible Tensor Operators
  • 1.4 Graphical Representation - Basic rules
  • 1.5 Spinor and Vector Spherical Harmonics
    • 1.5.1 Spherical Spinors
    • 1.5.2 Vector Spherical Harmonics
• 2 Central-Field Schr¨odinger Equation
  • 2.1 Radial Schr¨odinger Equation
  • 2.2 Coulomb Wave Functions
  • 2.3 Numerical Solution to the Radial Equation
    • 2.3.1 Adams Method (adams)
    • 2.3.2 Starting the Outward Integration (outsch)
    • 2.3.3 Starting the Inward Integration (insch)
    • 2.3.4 Eigenvalue Problem (master)
  • 2.4 Quadrature Rules (rint)
  • 2.5 Potential Models
    • 2.5.1 Parametric Potentials
    • 2.5.2 Thomas-Fermi Potential
  • 2.6 Separation of Variables for Dirac Equation
  • 2.7 Radial Dirac Equation for a Coulomb Field
  • 2.8 Numerical Solution to Dirac Equation
    • 2.8.1 Outward and Inward Integrations (adams, outdir, indir)
    • 2.8.2 Eigenvalue Problem for Dirac Equation (master) ii CONTENTS
    • 2.8.3 Examples using Parametric Potentials
• 3 Self-Consistent Fields
  • 3.1 Two-Electron Systems
  • 3.2 HF Equations for Closed-Shell Atoms
  • 3.3 Numerical Solution to the HF Equations
    • 3.3.1 Starting Approximation (hart)
    • 3.3.2 Refining the Solution (nrhf)
  • 3.4 Atoms with One Valence Electron
  • 3.5 Dirac-Fock Equations
• 4 Atomic Multiplets
  • 4.1 Second-Quantization
  • 4.2 6-j Symbols
  • 4.3 Two-Electron Atoms
  • 4.4 Atoms with One or Two Valence Electrons
  • 4.5 Particle-Hole Excited States
  • 4.6 9-j Symbols
  • 4.7 Relativity and Fine Structure
    • 4.7.1 He-like ions
    • 4.7.2 Atoms with Two Valence Electrons
    • 4.7.3 Particle-Hole States
• 5 Hyperfine Interaction & Isotope Shift
  • 5.1 Hyperfine Structure
  • 5.2 Atoms with One Valence Electron
    • 5.2.1 Pauli Approximation
  • 5.3 Isotope Shift
    • 5.3.1 Normal and Specific Mass Shifts
  • 5.4 Calculations of the SMS
    • 5.4.1 Angular Decomposition
    • 5.4.2 Application to One-Electron Atom
  • 5.5 Field Shift
• 6 Radiative Transitions
  • 6.1 Review of Classical Electromagnetism
    • 6.1.1 Electromagnetic Potentials
    • 6.1.2 Electromagnetic Plane Waves
  • 6.2 Quantized Electromagnetic Field
    • 6.2.1 Eigenstates of Ni
    • 6.2.2 Interaction Hamiltonian
    • 6.2.3 Time-Dependent Perturbation Theory
    • 6.2.4 Transition Matrix Elements
    • 6.2.5 Gauge Invariance
    • 6.2.6 Electric Dipole Transitions
    • 6.2.7 Magnetic Dipole and Electric Quadrupole Transitions CONTENTS iii
    • 6.2.8 Nonrelativistic Many-Body Amplitudes
  • 6.3 Theory of Multipole Transitions
• 7 Introduction to MBPT
  • 7.1 Closed-Shell Atoms
    • 7.1.1 Angular Momentum Reduction
    • 7.1.2 Example: 2nd-order Energy in Helium
  • 7.2 B-Spline Basis Sets
    • 7.2.1 Hartree-Fock Equation and B-splines
    • 7.2.2 B-spline Basis for the Dirac Equation
    • 7.2.3 Application: Helium Correlation Energy
  • 7.3 Atoms with One Valence Electron
    • 7.3.1 Second-Order Energy
    • 7.3.2 Angular Momentum Decomposition
    • 7.3.3 Quasi-Particle Equation and Brueckner-Orbitals
    • 7.3.4 Monovalent Negative Ions
  • 7.4 Relativistic Calculations
    • 7.4.1 Breit Interaction
    • 7.4.2 Angular Reduction of the Breit Interaction
    • 7.4.3 Coulomb-Breit Many-Electron Hamiltonian
    • 7.4.4 Closed-Shell Energies
    • 7.4.5 One Valence Electron
  • 7.5 CI Calculations
    • 7.5.1 Relativistic CI Calculations
• 8 MBPT for Matrix Elements
  • 8.1 Second-Order Corrections
    • 8.1.1 Angular Reduction
  • 8.2 Random-Phase Approximation
    • 8.2.1 Gauge Independence of RPA
    • 8.2.2 RPA for hyperfine constants
  • 8.3 Third-Order Matrix Elements
  • 8.4 Matrix Elements of Two-particle Operators
    • 8.4.1 Two-Particle Operators: Closed-Shell Atoms
    • 8.4.2 Two-Particle Operators: One Valence Electron Atoms
  • 8.5 CI Calculations for Two-Electron Atoms
    • 8.5.1 E1 Transitions in He
• A Exercises
  • A.1 Chapter
  • A.2 Chapter
  • A.3 Chapter
  • A.4 Chapter
  • A.5 Chapter
  • A.6 Chapter

iv CONTENTS

A.7 Chapter 7............................... 243 A.8 Chapter 8............................... 244

vi LIST OF TABLES

4.2 Comparison of V (^) HFN^ − 1 energies of (3s 2 p) and (3p 2 p) particle-hole excited states of neon and neonlike ions with measurements.... 114 4.3 First-order relativistic calculations of the (1s 2 s) and (1s 2 p) states of heliumlike neon (Z = 10), illustrating the fine-structure of the (^3) P multiplet.............................. 120

5.1 Nonrelativistic HF calculations of the magnetic dipole hyperfine constants a (MHz) for ground states of alkali-metal atoms com- pared with measurements....................... 134 5.2 Lowest-order matrix elements of the specific-mass-shift operator T for valence states of Li and Na.................. 139

6.1 Reduced oscillator strengths for transitions in hydrogen..... 159 6.2 Hartree-Fock calculations of transition rates Aif [s−^1 ] and life- times τ [s] for levels in lithium. Numbers in parentheses represent powers of ten.............................. 160 6.3 Wavelengths and oscillator strengths for transitions in heliumlike ions calculated in a central potential v 0 (1s, r). Wavelengths are determined using first-order energies................. 170

7.1 Eigenvalues of the generalized eigenvalue problem for the B-spline approximation of the radial Schr¨odinger equation with l = 0 in a Coulomb potential with Z = 2. Cavity radius is R = 30 a.u. We use 40 splines with k = 7....................... 189 7.2 Comparison of the HF eigenvalues from numerical integration of the HF equations (HF) with those found by solving the HF eigenvalue equation in a cavity using B-splines (spline). Sodium, Z = 11, in a cavity of radius R = 40 a.u............... 192 7.3 Contributions to the second-order correlation energy for helium. 193 7.4 Hartree-Fock eigenvalues ≤v with second-order energy corrections E(2) v are compared with experimental binding energies for a few low-lying states in atoms with one valence electron......... 196 7.5 Expansion coefficients cn, n = 1 · ·20 of the 5s state of Pd−^ in a basis of HF orbitals for neutral Pd confined to a cavity of radius R = 50 a.u................................ 199 7.6 Contributions to the ground-state energy for He-like ions..... 207 7.7 MBPT Coulomb (E(n)), Breit (B(n)) and reduced mass – mass polarization (RM/MP) contributions to the energies of 2s and 2p states of lithiumlike Ne (Johnson et al., 1988b).......... 208 7.8 Contribution δEl of (nlml) configurations to the CI energy of the helium ground state. The dominant contributions are from the l = 0 nsms configurations. Contributions of configurations with l ≥ 7 are estimated by extrapolation................. 210 7.9 Energies (a.u.) of S, P, and D states of heliumlike iron (FeXXV) obtained from a relativistic CI calculation compared with ob- served values (Obs.) from the NIST website............ 212

LIST OF TABLES vii

8.1 First-order reduced matrix elements of the electric dipole opera- tor in lithium and sodium in length L and velocity V forms.... 214 8.2 Second-order reduced matrix elements and sums of first- and second-order reduced matrix elements for E 1 transitions in lithium and sodium in length and velocity forms. RPA values given in the last column are identical in length and velocity form. 216 8.3 Comparison of HF and RPA calculations of hyperfine constants A(MHz) for states in Na (μI = 2.2176, I = 3/2) with experimen- tal data................................. 220 8.4 Contributions to the reduced matrix element of the electric dipole transition operator ωr in length-form and in velocity form for the 3s − 3 p 1 / 2 transition in Na. Individual contributions to T (3) from Brueckner orbital (B.O.), structural radiation (S.R.), nor- malization (Norm.) and derivative terms (Deriv.) are given. (ω 0 = 0.072542 a.u. and ω(2)^ = 0.004077 a.u.).......... 222 8.5 Third-Order MBPT calculation of hyperfine constants A(MHz) for states in Na (μI = 2.2176, I = 3/2) compared with experi- mental data............................... 223 8.6 Matrix elements of two-particle operators Breit operator for 4s and 4p states in copper (Z=29) Numbers in brackets represent powers of 10; a[−b] ≡ a × 10 −b^................... 225 8.7 Contributions to specific-mass isotope shift constants (GHz amu) for 3s and 3p states of Na...................... 226 8.8 Contributions to field-shifts constants F (MHz/fm^2 ) for 3s and 3 p states in Na............................ 226 8.9 Isotope shifts δν^22 ,^23 (MHz) for 3s and 3p states of Na...... 227 8.10 Relativistic CI calculations of wavelengths λ(˚A), transition rates A(s−^1 ), oscillator strengths f , and line strengths S(a.u.) for 2P → 1S & 2S states in helium. Numbers in brackets represent powers of 10.............................. 229

List of Figures

1.1 Transformation from rectangular to spherical coordinates..... 4

2.1 Hydrogenic Coulomb wave functions for states with n = 1, 2 and

3.................................... 30 2.2 The radial wave function for a Coulomb potential with Z = 2 is shown at several steps in the iteration procedure leading to the 4 p eigenstate............................. 40 2.3 Electron interaction potentials from Eqs.(2.99) and (2.100) with parameters a = 0.2683 and b = 0.4072 chosen to fit the first four sodium energy levels......................... 46 2.4 Thomas-Fermi functions for the sodium ion, Z = 11, N = 10. Upper panel: the Thomas-Fermi function φ(r). Center panel: N (r), the number of electrons inside a sphere of radius r. Lower panel: U (r), the electron contribution to the potential...... 50 2.5 Radial Dirac Coulomb wave functions for the n = 2 states of hy- drogenlike helium, Z = 2. The solid lines represent the large com- ponents P 2 κ(r) and the dashed lines represent the scaled small components, Q 2 κ(r)/αZ....................... 56

3.1 Relative change in energy (E(n)^ − E(n−1))/E(n)^ as a function of the iteration step number n in the iterative solution of the HF equation for helium, Z = 2...................... 68 3.2 Solutions to the HF equation for helium, Z = 2. The radial HF wave function P 1 s(r) is plotted in the solid curve and electron potential v 0 (1s, r) is plotted in the dashed curve......... 69 3.3 Radial HF wave functions for neon and argon........... 85 3.4 Radial HF densities for beryllium, neon, argon and krypton... 85

4.1 Energy level diagram for helium................... 105 4.2 Variation with nuclear charge of the energies of 1s 2 p states in he- liumlike ions. At low Z the states are LS-coupled states, while at high Z, they become jj-coupled states. Solid circles 1 P 1 ; Hollow circles 3 P 0 ; Hollow squares 3 P 1 ; Hollow diamonds 3 P 2....... 121

ix

x LIST OF FIGURES

5.1 Comparison of the nuclear potential |V (r)| and field-shift fac- tor F (r) calculated assuming a uniform distribution (given by solid lines) with values calculated assuming a Fermi distribution (dashed lines)............................. 142

6.1 Detailed balance for radiative transitions between two levels... 153 6.2 The propagation vector ˆk is along the z′^ axis, ˆ≤ 1 is along the x′ axis and ˆ≤ 2 is along the y′^ axis. The photon angular integration variables are φ and θ......................... 165 6.3 Oscillator strengths for transitions in heliumlike ions........ 172

7.1 We show the n = 30 B-splines of order k = 6 used to cover the interval 0 to 40 on an “atomic” grid. Note that the splines sum to 1 at each point........................... 188 7.2 B-spline components of the 2s state in a Coulomb field with Z = 2 obtained using n = 30 B-splines of order k = 6. The dashed curve is the resulting 2s wave function.................. 190 7.3 The radial charge density ρv for the 3s state in sodium is shown together with 10×δρv , where δρv is the second-order Brueckner correction to ρv............................ 198 7.4 Lower panel: radial density of neutral Pd (Z=46). The peaks corresponding to closed n = 1, 2, · · · shells are labeled. Upper panel: radial density of the 5s ground-state orbital of Pd−. The 5 s orbital is obtained by solving the quasi-particle equation... 200 7.5 MBPT contributions for He-lile ions................ 207

8.1 First- and second-order Breit corrections to the ground-stste en- ergies of neonlike ions shown along with the second-order corre- lation energy. The first-order Breit energy B(1)^ grows roughly as Z^3 , B(2)^ grows roughly as Z^2 and the second-order Coulomb energy E(2)^ is nearly constant.................... 224

xii PREFACE

The final section of the material began with a discussion of second- quantization. This approach was used to study a number of structure prob- lems in first-order perturbation theory, including excited states of two-electron atoms, excited states of atoms with one or two electrons beyond closed shells and particle-hole states. Relativistic fine-structure effects were considered using the “no-pair” Hamiltonian. A rather complete discussion of the magnetic-dipole and electric quadrupole hyperfine structure from the relativistic point of view was given, and nonrelativistic limiting forms were worked out in the Pauli ap- proximation. Fortran subroutines to solve the radial Schr´odinger equation and the Hartree-Fock equations were handed out to be used in connection with weekly homework assignments. Some of these assigned exercises required the student to write or use fortran codes to determine atomic energy levels or wave func- tions. Other exercises required the student to write maple routines to generate formulas for wave functions or matrix elements. Additionally, more standard “pencil and paper” exercises on Atomic Physics were assigned. I was disappointed in not being able to cover more material in the course. At the beginning of the semester, I had envisioned being able to cover second- and higher-order MBPT methods and CI calculations and to discuss radiative transitions as well. Perhaps next year! Finally, I owe a debt of gratitude to the students in this class for their patience and understanding while this material was being assembled, and for helping read through and point out mistakes in the text.

South Bend, May, 1994

The second time that this course was taught, the material in Chap. 5 on electromagnetic transitions was included and Chap. 6 on many-body methods was started. Again, I was dissapointed at the slow pace of the course.

South Bend, May, 1995

The third time through, additional sections of Chap. 6 were added.

South Bend, December, 2000

The fourth time that this course was taught, the section in Chap. 4 on hyper- fine structure was moved to a separate chapter, Chap. 5. Material on the isotope shift was also included in Chap. 5. The material on electromagnetic transitions (now Chap. 6) remains unchanged. The chapter on many-body methods (now Chap. 7) was considerably extended and Chap. 8 on many-body methods for matrix elements was added. In order to squeeze all of this material into a one semester three credit hour course, it was necessary to skip most of the material on numerical methods. However, I am confident that the entire book could be covered in a four credit hour course.

South Bend, December, 2005

Chapter 1

Angular Momentum

Understanding the quantum mechanics of angular momentum is fundamental in theoretical studies of atomic structure and atomic transitions. Atomic energy levels are classified according to angular momentum and selection rules for ra- diative transitions between levels are governed by angular-momentum addition rules. Therefore, in this first chapter, we review angular-momentum commu- tation relations, angular-momentum eigenstates, and the rules for combining two angular-momentum eigenstates to find a third. We make use of angular- momentum diagrams as useful mnemonic aids in practical atomic structure cal- culations. A more detailed version of much of the material in this chapter can be found in Edmonds (1974).

1.1 Orbital Angular Momentum - Spherical

Harmonics

Classically, the angular momentum of a particle is the cross product of its po- sition vector r = (x, y, z) and its momentum vector p = (px, py , pz ):

L = r × p.

The quantum mechanical orbital angular momentum operator is defined in the same way with p replaced by the momentum operator p → −i¯h∇. Thus, the Cartesian components of L are

Lx = ¯h i

y (^) ∂z∂ − z (^) ∂y∂

, Ly = ¯h i

z (^) ∂x∂ − x (^) ∂z∂

, Lz = ¯h i

x (^) ∂y∂ − y (^) ∂x∂

With the aid of the commutation relations between p and r:

[px, x] = −i¯h, [py , y] = −i¯h, [pz , z] = −i¯h, (1.2)

1.1. ORBITAL ANGULAR MOMENTUM - SPHERICAL HARMONICS 3

Since J+ raises the eigenvalue m by one unit, and J− lowers it by one unit, these operators are referred to as raising and lowering operators, respectively. Furthermore, since J (^) x^2 + J y^2 is a positive definite hermitian operator, it follows that λ ≥ m^2.

By repeated application of J− to eigenstates of Jz , one can obtain states of ar- bitrarily small eigenvalue m, violating this bound, unless for some state |λ, m 1 〉,

J−|λ, m 1 〉 = 0.

Similarly, repeated application of J+ leads to arbitrarily large values of m, unless for some state |λ, m 2 〉 J+|λ, m 2 〉 = 0.

Since m^2 is bounded, we infer the existence of the two states |λ, m 1 〉 and |λ, m 2 〉. Starting from the state |λ, m 1 〉 and applying the operator J+ repeatedly, one must eventually reach the state |λ, m 2 〉; otherwise the value of m would increase indefinitely. It follows that m 2 − m 1 = k, (1.14)

where k ≥ 0 is the number of times that J+ must be applied to the state |λ, m 1 〉 in order to reach the state |λ, m 2 〉. One finds from Eqs.(1.8,1.9) that

λ|λ, m 1 〉 = (m^21 − m 1 )|λ, m 1 〉, λ|λ, m 2 〉 = (m^22 + m 2 )|λ, m 2 〉,

leading to the identities

λ = m^21 − m 1 = m^22 + m 2 , (1.15)

which can be rewritten

(m 2 − m 1 + 1)(m 2 + m 1 ) = 0. (1.16)

Since the first term on the left of Eq.(1.16) is positive definite, it follows that m 1 = −m 2. The upper bound m 2 can be rewritten in terms of the integer k in Eq.(1.14) as m 2 = k/2 = j.

The value of j is either integer or half integer, depending on whether k is even or odd:

j = 0,

It follows from Eq.(1.15) that the eigenvalue of J 2 is

λ = j(j + 1). (1.17)

The number of possible m eigenvalues for a given value of j is k + 1 = 2j + 1. The possible values of m are

m = j, j − 1 , j − 2 , · · · , −j.

4 CHAPTER 1. ANGULAR MOMENTUM

x

y

z

θ r

Figure 1.1: Transformation from rectangular to spherical coordinates.

Since J− = J +†, it follows that

J+|λ, m〉 = η|λ, m + 1〉, 〈λ, m|J− = η∗〈λ, m + 1|.

Evaluating the expectation of J 2 = J−J+ + J z^2 + Jz in the state |λ, m〉, one finds

|η|^2 = j(j + 1) − m(m + 1).

Choosing the phase of η to be real and positive, leads to the relations

J+|λ, m〉 =

(j + m + 1)(j − m) |λ, m + 1〉, (1.18) J−|λ, m〉 =

(j − m + 1)(j + m) |λ, m − 1 〉. (1.19)

1.1.2 Spherical Coordinates - Spherical Harmonics

Let us apply the general results derived in Section 1.1.1 to the orbital angular momentum operator L. For this purpose, it is most convenient to transform Eqs.(1.1) to spherical coordinates (Fig. 1.1):

x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, r =

x^2 + y^2 + z^2 , θ = arccos z/r, φ = arctan y/x.

In spherical coordinates, the components of L are

Lx = i¯h

sin φ

∂θ

• cos φ cot θ

∂φ

Ly = i¯h

− cos φ

∂θ

• sin φ cot θ

∂φ

Lz = −i¯h

∂φ

6 CHAPTER 1. ANGULAR MOMENTUM

one obtains

Θl,−l(θ) =

2 ll!

(2l + 1)! 2

sinl^ θ. (1.30)

Applying Ll ++ mto Yl,−l(θ, φ), leads to the result

Θl,m(θ) =

(−1)l+m 2 ll!

(2l + 1)(l − m)! 2(l + m)!

sinm^ θ

dl+m d cos θl+m^

sin^2 l^ θ. (1.31)

For m = 0, this equation reduces to

Θl, 0 (θ) =

(−1)l 2 ll!

2 l + 1 2

dl d cos θl^ sin^2 l^ θ. (1.32)

This equation may be conveniently written in terms of Legendre polynomials Pl(cos θ) as

Θl, 0 (θ) =

2 l + 1 2

Pl(cos θ). (1.33)

Here the Legendre polynomial Pl(x) is defined by Rodrigues’ formula

Pl(x) =

2 ll!

dl dxl^

(x^2 − 1)l^. (1.34)

For m = l, Eq.(1.31) gives

Θl,l(θ) =

(−1)l 2 ll!

(2l + 1)! 2 sinl^ θ. (1.35)

Starting with this equation and stepping backward l − m times leads to an alternate expression for Θl,m(θ):

Θl,m(θ) =

(−1)l 2 ll!

(2l + 1)(l + m)! 2(l − m)!

sin−m^ θ

dl−m d cos θl−m^

sin^2 l^ θ. (1.36)

Comparing Eq.(1.36) with Eq.(1.31), one finds

Θl,−m(θ) = (−1)mΘl,m(θ). (1.37)

We can restrict our attention to Θl,m(θ) with m ≥ 0 and use (1.37) to obtain Θl,m(θ) for m < 0. For positive values of m, Eq.(1.31) can be written

Θl,m(θ) = (−1)m

(2l + 1)(l − m)! 2(l + m)!

P (^) lm (cos θ) , (1.38)

where P (^) lm (x) is an associated Legendre functions of the first kind, given in Abramowitz and Stegun (1964, chap. 8), with a different sign convention, defined by

P (^) lm (x) = (1 − x^2 )m/^2

dm dxm^

Pl(x). (1.39)

1.2. SPIN ANGULAR MOMENTUM 7

The general orthonormality relations 〈l, m|l′, m′〉 = δll′^ δmm′^ for angular mo- mentum eigenstates takes the specific form ∫ (^) π

0

∫ (^2) π

0

sin θdθdφ Y (^) l,m∗ (θ, φ)Yl′^ ,m′^ (θ, φ) = δll′^ δmm′^ , (1.40)

for spherical harmonics. Comparing Eq.(1.31) and Eq.(1.36) leads to the relation

Yl,−m(θ, φ) = (−1)mY (^) l,m∗ (θ, φ). (1.41)

The first few spherical harmonics are:

Y 00 =

1 4 π

Y 10 =

3 4 π cos^ θ^ Y^1 ,±^1 =^ ∓

3 8 π sin^ θ e

±iφ

Y 20 =

5 16 π (3 cos

(^2) θ − 1) Y 2 ,± 1 = ∓

15 8 π sin^ θ^ cos^ θ e

±iφ

Y 2 ,± 2 =

15 32 π sin

(^2) θ e± 2 iφ

Y 30 =

7 16 π cos^ θ^ (5 cos

(^2) θ − 3) Y 3 ,± 1 = ∓

21 64 π sin^ θ^ (5 cos

(^2) θ − 1) e±iφ

Y 3 ,± 2 =

105 32 π cos^ θ^ sin

(^2) θ e± 2 iφ (^) Y 3 ,± 3 =^ ∓

35 64 π sin

(^3) θ e± 3 iφ

1.2 Spin Angular Momentum

1.2.1 Spin 1 / 2 and Spinors

The internal angular momentum of a particle in quantum mechanics is called spin angular momentum and designated by S. Cartesian components of S satisfy angular momentum commutation rules (1.4). The eigenvalue of S^2 is ¯h^2 s(s + 1) and the 2s + 1 eigenvalues of Sz are ¯hm with m = −s, −s + 1, · · · , s. Let us consider the case s = 1/2 which describes the spin of the electron. We designate the eigenstates of S^2 and Sz by two-component vectors χμ, μ = ± 1 /2:

χ 1 / 2 =

, χ− 1 / 2 =

These two-component spin eigenfunctions are called spinors. The spinors χμ satisfy the orthonormality relations

χ† μχν = δμν. (1.43)

The eigenvalue equations for S^2 and Sz are

S^2 χμ = 34 ¯h^2 χμ, Sz χμ = μ¯hχμ.