Chapter 9

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: