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Chapter 9

Chapter 9. Spin. 6.B.2 6.C.1. Angular momentum Let us recall key results for the angular momentum. 6.C.2. Angular momentum Let us recall key results for the angular momentum. 6.C.3. Angular momentum Let us recall key results for the angular momentum. 6.C.3. Angular momentum

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Chapter 9

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  1. Chapter 9 Spin

  2. 6.B.2 6.C.1 Angular momentum • Let us recall key results for the angular momentum

  3. 6.C.2 Angular momentum • Let us recall key results for the angular momentum

  4. 6.C.3 Angular momentum • Let us recall key results for the angular momentum

  5. 6.C.3 Angular momentum • Matrices corresponding to subspaces E (k,j) depend on the value of j, which is determined by the specificity of the studied system • When N = 1, then j = ½ and the dimensionality of the matrices is (2j + 1) × (2j + 1) = 2 × 2

  6. 6.C.3 Angular momentum • Using the expression for matrix elements below:

  7. 6.C.3 Angular momentum • Using the expression for matrix elements below:

  8. 6.C.3 Angular momentum • Using the expression for matrix elements below: • These matrices represent three components of an angular momentum operator in a basis for which the J2 operator is diagonalizedwith eigenvalues 3ћ2/2and the Jz operator is diagonalized with eigenvalues ±ћ/2

  9. 4.A.2 9.A.2 Spin angular momentum • The matrices that we just introduced represent one of the examples of a spin vector operator • It turns out that such observable does indeed exist in nature • Moreover, it is one of the most fundamental properties of an electron and other elementary particles

  10. 4.A.2 9.A.2 Spin angular momentum • Spin can be measured experimentally, and it gives rise to many macroscopic phenomena (such as, e.g., magnetism) • If one imagines that a particle with a spin has a certain spatial extension, then a rotation around its axis would give rise to an intrinsic angular momentum • However, if it were the case, the value of j would necessarily be integral, not half-integral

  11. 9.A.2 Spin angular momentum • Therefore, the spin angular momentum has nothing to do with motion in space and cannot be described by any function of the position variables • Spin has no classical analogue! • Here we will introduce spin variables satisfying the following postulates: • 1) The spin operator S is an angular momentum: • 2) It acts in a spin state spaceEs, where S2 and Sz constitute a CSCO

  12. 9.A.2 Spin angular momentum • The space Es is spanned by the set of eigenstates common to S2 and Sz: • Spin quantum number s can take both integer and half-integer values • Every elementary particle has a specific and immutable value of s

  13. 9.A.2 Spin angular momentum • Pi-meson: s = 0 • Electron, proton, neutron: s = 1/2 • Photon: s = 1 • Delta-particle: s = 3/2 • Graviton: s = 2 • Etc. • Every elementary particle has a specific and immutable value of s

  14. 9.A.2 9.B Spin angular momentum • 3) All spin observables commute with all orbital observables • Therefore the state space E of a given system is: • Let us restrict ourselves to the case of the particles with spin 1/2 • In this case the space Es is 2D • In this space we will consider an orthonormal basis of eigenstates common to S2 and Sz:

  15. 9.B Spin 1/2 • The eigenproblem: • The basis: • Recall: • Thus: • In this space we will consider an orthonormal basis of eigenstates common to S2 and Sz:

  16. 9.B Spin 1/2 • Any spin state in Es can be represented by an arbitrary vector:

  17. 9.B Spin 1/2 • Any operator acting in Es can be represented by a 2×2 matrix in basis • E.g.: • σ’s are called Pauli matrices: • Their properties: Wolfgang Ernst Pauli (1900 – 1958)

  18. 9.B Spin 1/2 • Any operator acting in Es can be represented by a 2×2 matrix in basis • E.g.: • Therefore: • Their properties: Wolfgang Ernst Pauli (1900 – 1958)

  19. 9.C.1 Observables and state vectors • Since • A CSCO in E can be obtained through juxtaposition of a CSCO in Er and a CSCO in Es • E.g.: • The basis used will be: • Then:

  20. 9.C.1 Observables and state vectors • This basis is orthonormal and complete: • Any state in E can be expanded as: • Where: • I.e.: • This can be written in a spinor form:

  21. 9.C.1 Observables and state vectors • This basis is orthonormal and complete: • An associated bra can be expanded as: • Where: • I.e.: • This can be written in a spinor form:

  22. 9.C.1 Observables and state vectors • A scalar product can be written as: • Normalization:

  23. 9.C.1 Observables and state vectors • It may happen that some state vector can be factored as: • Then:

  24. 9.C.1 Observables and state vectors • Consider the operator equation: • In the 2 × 2 matrix representation: • For example:

  25. 9.C.1 Observables and state vectors • Consider the operator equation: • In the 2 × 2 matrix representation: • For example:

  26. 9.C.1 Observables and state vectors • Consider the operator equation: • In the 2 × 2 matrix representation: • For example:

  27. 9.C.1 Observables and state vectors • Consider the operator equation: • Then:

  28. 9.C.1 Observables and state vectors • Consider the operator equation: • Then:

  29. 9.C.1 Observables and state vectors • Similarly we can obtain expressions for “mixed” operators in the 2 × 2 matrix representation, e.g.:

  30. 9.C.2 Measurements • There exist only one state vector that corresponds to specific values of particle’s position and spin z-component (since X, Y, Z and Sz are members of a CSCO): • The probability of finding this particle in a volume dxdydz with a spin parallel to the z-axis is: • The probability of finding this particle in a volume dxdydz with a spin antiparallel to the z-axis is:

  31. 9.C.2 Measurements • The probability of finding this particle in a volume dxdydz and not measuring the spin is: • The probability of finding this particle with a spin parallel to the z-axis is: • What about measurements of Sx?

  32. 9.C.2 Measurements • We need to find eigenvalues and eigenspinors of Sx • The probabilities are:

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