1. Chapter 9
Spin

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

3. Angular momentum Let us recall key results for the angular momentum

4. Angular momentum Let us recall key results for the angular momentum
E (k,j)
E (k’,j)
E (k’,j’) …
matrix
(2j + 1) × (2j + 1)
(2j’ + 1) × (2j’+ 1)

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. Angular momentum Using the expression for matrix elements below:
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 diagonalized with eigenvalues 3ћ2/2 and 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 space Es, 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. 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. 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. 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. 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. 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. Observables and state vectors
A scalar product can be written as:
Normalization:

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

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

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

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

27. Observables and state vectors
Consider the operator equation:
Then:

28. Observables and state vectors
Consider the operator equation:
Then:

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