Spin and Addition of Angular Momentum Type Operators
We have seen in Sect. 7.4 that representations of the angular momentum Lie algebra ( 7.51 ) are labeled by a quantum number ℓ which can take half-integer or integer values. However, we have also seen in Sect. 7.5 that ℓ is limited to integer values when
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Spin and Addition of Angular Momentum Type Operators
We have seen in Sect. 7.4 that representations of the angular momentum Lie algebra (7.51) are labeled by a quantum number which can take half-integer or integer values. However, we have also seen in Sect. 7.5 that is limited to integer values when the operators M actually refer to angular momentum, because the wave functions1 x|n, , m or x|k, , m for angular momentum eigenstates must be single valued. It was therefore very surprising when Stern, Gerlach, Goudsmit, Uhlenbeck and Pauli in the 1920s discovered that half-integer values of are also realized in nature, although in that case cannot be related to an angular momentum any more. Half-integer values of arise in nature because leptons and quarks carry a representation of the “covering group” SU(2) of the proper rotation group SO(3), where SU(2) stands for the group which can be represented by special unitary 2 × 2 matrices.2 The designation “special” refers to the fact that the matrices are also required to have determinant 1. The generators of the groups SU(2) and SO(3) satisfy the same Lie algebra (7.51), but for every rotation matrix ˆ there are two unitary 2 × 2 matrices U (ϕ) = − U (ϕ + 2π ϕ). ˆ R(ϕ) = R(ϕ + 2π ϕ) In that sense SU(2) provides a double cover of SO(3). We will use the notations l and M for angular momenta, and s or S for spins.
1 We
denote the magnetic quantum number with m in this chapter because m will denote the mass of a particle. 2 Ultimately, all particles carry representations of the covering group SL(2,C) of the group SO(1,3) of proper orthochronous Lorentz transformations, see Appendices B and H. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_8
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8 Spin and Addition of Angular Momentum Type Operators
8.1 Spin and Magnetic Dipole Interactions A particle of charge q and mass m which moves with angular momentum l through a constant magnetic field B has its energy levels shifted through a Zeeman term in the Hamiltonian, HZ = −
q l · B. 2m
(8.1)
We will explore the origin of this term in Chap. 15, see Problem 15.2, but for now we can think of it as a magnetic dipole term with a dipole moment μl =
q l. 2m
(8.2)
The relation between μl and l can be motivated from electrodynamics, but is actually a consequence of the coupling to magnetic vector potentials in the Schrödinger equation. The quantization , m |lz |, m = hm ¯ for angular momentum components in a fixed direction yields a Zeeman shift E = −
q h¯ Bm , 2m
− ≤ m ≤ ,
(8.3)
of the energy levels of a charged particle in a magnetic field. For orbital momentum the resulting number 2 + 1 of energy levels is odd. However, the observation of motion of Ag atoms through an inhomogeneous field by Stern and Gerlach in 1921 revealed a split of energy levels of these atoms into two levels in a magnetic field. This com
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