Math 2.0

Quantum theory Definition

Spin

Builds on Angular momentum operatorSU(2)Irreducible representationTensor product of Hilbert spacesPauli matrices

Definition

Let s∈{0,12,1,32,…}s \in \set{0, \frac12, 1, \frac32, \ldots} and ℏ>0\hbar > 0. Let S=(S1,S2,S3)S = (S_1, S_2, S_3) be a triple of Hermitian operators on ℂ2s+1\C^{2s+1} that acts irreducibly. Assume it is an angular momentum operator: [S1,S2]=iℏS3[S_1, S_2] = i\hbar S_3, and the same for cyclic permutations. Then it comes from the irreducible representation of SU(2) of dimension 2s+12s + 1. It satisfies S12+S22+S32=ℏ2s(s+1)IS_1^2 + S_2^2 + S_3^2 = \hbar^2 s(s+1) I. A particle of spin ss has the state space L2(ℝ3)⊗ℂ2s+1L^2(\R^3) \otimes \C^{2s+1}. This is the tensor product of the space of wave functions with the space of internal states. The triple SS, acting on the second factor, is its spin operator. Its total angular momentum is J=L⊗I+I⊗SJ = L \otimes I + I \otimes S, where LL is the orbital angular momentum.

Remarks

The eigenvalues of S3S_3 are ℏm\hbar m for m=−s,…,sm = -s, \ldots, s. A rotation by angle θ\theta about a unit axis nn acts on internal states by e−iθ n⋅S/ℏe^{-i\theta\, n \cdot S/\hbar}. For θ=2π\theta = 2\pi this is (−1)2sI(-1)^{2s} I, so for half-integer ss the internal states form a representation of SU(2) that is only a projective representation of SO(3).

Example: for s=12s = \frac12, S=ℏ2σS = \frac{\hbar}{2}\sigma with the Pauli matrices σ=(σ1,σ2,σ3)\sigma = (\sigma_1, \sigma_2, \sigma_3). Since σk2=I\sigma_k^2 = I, one gets S2=3ℏ24I=ℏ2⋅12⋅32 IS^2 = \frac{3\hbar^2}{4} I = \hbar^2 \cdot \frac12 \cdot \frac32 \, I, as required, and S3S_3 has eigenvalues ±ℏ2\pm\frac{\hbar}{2}.

Non-example: the orbital angular momentum LL on L2(ℝ3)L^2(\R^3) is an angular momentum operator, but it is not a spin. The space L2(ℝ3)L^2(\R^3) is infinite-dimensional, L12+L22+L32L_1^2 + L_2^2 + L_3^2 is not a multiple of the identity, and only integer values ℓ\ell occur.

Reading path

To understand this page from the ground up, read these 39 pages in order. The foundations come first.

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Angular momentum operator, concept, no evidence levelAngular momentum oper…conceptCross product, concept, no evidence levelCross productconceptDeterminant, concept, no evidence levelDeterminantconceptGroup, concept, no evidence levelGroupconceptHermitian matrix, concept, no evidence levelHermitian matrixconceptHilbert space, concept, no evidence levelHilbert spaceconceptIdentity matrix, object, no evidence levelIdentity matrixobjectIrreducible representation, concept, no evidence levelIrreducible represent…conceptLie algebra, concept, no evidence levelLie algebraconceptLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionOrthonormal basis, concept, no evidence levelOrthonormal basisconceptReal projective space, object, no evidence levelReal projective spaceobjectTensor product of vector spaces, concept, no evidence levelTensor product of vec…conceptTrace of a matrix, concept, no evidence levelTrace of a matrixconceptUnitary matrix, concept, no evidence levelUnitary matrixconceptVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionTensor product of Hilbert spaces, definition, no evidence levelTensor product of Hil…definitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleDeterminant, definition, no evidence levelDeterminantdefinitionUnitary group, object, no evidence levelUnitary groupobjectSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleRotation group SO(3), example, no evidence levelRotation group SO(3)exampleSU(2), example, no evidence levelSU(2)examplePauli matrices, object, no evidence levelPauli matricesobjectSpin, definition, no evidence levelSpindefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Angular momentum operator concept
  2. Cross product concept
  3. Determinant concept
  4. Group concept
  5. Hermitian matrix concept
  6. Hilbert space concept
  7. Identity matrix object
  8. Irreducible representation concept
  9. Lie algebra concept
  10. Lie group concept
  11. Matrix definition
  12. Orthonormal basis concept
  13. Real projective space object
  14. Tensor product of vector spaces concept
  15. Trace of a matrix concept
  16. Unitary matrix concept
  17. Vector space definition
  18. Ring definition
  19. Tensor product of Hilbert spaces definition
  20. Square matrix definition
  21. Transpose definition
  22. Linear map definition
  23. Unit of a ring definition
  24. Conjugate transpose definition
  25. Multilinear map definition
  26. Orthogonal matrix definition
  27. Division ring definition
  28. Alternating multilinear map definition
  29. Orthogonal group object
  30. Unitary matrix definition
  31. The quaternions example
  32. Determinant definition
  33. Unitary group object
  34. Special orthogonal group object
  35. Special unitary group SU(n) example
  36. Rotation group SO(3) example
  37. SU(2) example
  38. Pauli matrices object
  39. Spin definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Spin_(physics)
About this page
TagHJ36MXZFJ6
URIm2://HJ36MXZFJ6
Kinddefinition
WritingW0 stub
MSC81R05, 81R25, 81V72
DefinesSpin
UsesAngular momentum operator, SU(2), Irreducible representation, Tensor product of Hilbert spaces, Pauli matrices
Changed3 November 2026

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