Chapter 9 Spin

Total angular momentum Let us recall key results for the total angular momentum 6.B.2 6.C.1

Total angular momentum Let us recall key results for the total angular momentum 6.C.2

Total angular momentum Let us recall key results for the total angular momentum 6.C.3 E (k,j) E (k’,j) E (k’,j’) … E (k,j) matrix (2j + 1) × (2j + 1) 00 0 E (k’,j) 0 matrix (2j + 1) × (2j + 1) 0 0 E (k’,j’) 00 matrix (2j’ + 1) × (2j’+ 1) 0 …000

Total 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.C.3

Total angular momentum Using the expression for matrix elements below: 6.C.3

Total angular momentum Using the expression for matrix elements below: 6.C.3

Spin angular momentum Previously we introduced a matrix corresponding to a spin vector operator: Such observable does indeed exist in nature 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 4.A.2 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 space E s, where S 2 and S z constitute a CSCO 9.A.2

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

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 E s is 2D In this space we will consider an orthonormal basis of eigenstates common to S 2 and S z : 9.A.2 9.B

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

Spin 1/2 Any spin state in E s can be represented by an arbitrary vector: 9.B

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

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

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

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

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

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

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

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

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

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

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

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 S z 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: 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 S x ? 9.C.2

Measurements We need to find eigenvalues and eigenspinors of S x The probabilities are: 9.C.2