Lect33_Spin - Lecture 33 Quantum Mechanical Spin Phy851...

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Lecture 33: Quantum Mechanical Spin Phy851 Fall 2009
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Intrinsic Spin Empirically, we have found that most particles have an additional internal degree of freedom, called ‘spin’ The Stern-Gerlach experiment (1922): Each type of particle has a discrete number of internal states: 2 states --> spin _ 3 states --> spin 1 – Etc….
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Interpretation It is best to think of spin as just an additional quantum number needed to specify the state of a particle. Within the Dirac formalism, this is relatively simple and requires no new physical concepts The physical meaning of spin is not well- understood Fro Dirac eq. we find that for QM to be Lorentz invariant requires particles to have both anti-particles and spin. The ‘spin’ of a particle is a form of angular momentum
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Spin Operators Spin is described by a vector operator: The components satisfy angular momentum commutation relations: This means simultaneous eigenstates of S 2 and S z exist: z z y y x x e S e S e S S r r r r + + = y x z x z y z y x S i S S S i S S S i S S h h h = = = ] , [ ] , [ ] , [ 2 2 2 2 z y x S S S S + + = s s m s s s m s S , ) 1 ( , 2 2 + = h s s z m s m m s S , , h =
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Allowed quantum numbers For any set of 3 operators satisfying the angular momentum algebra, the allowed values of the quantum numbers are: For orbital angular momentum, the allowed values were further restricted to only integer values by the requirement that the wavefunction be single-valued For spin, the quantum number, s , can only take on one value The value depends on the type of particle S =0: Higgs s =1/2: Electrons, positrons, protons, neutrons, muons,neutrinos, quarks,… s =1: Photons, W, Z, Gluon s=2: graviton { } K , , 1 , , 0 2 3 2 1 j { } j j j m j , , 1 , K + { } s s s m s , , 1 , K +
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Complete single particle basis A set of 5 commuting operators which describe the independent observables of a single particle are: Or equivalently: Some possible basis choices: When dealing with a single-particle, it is permissible to drop the s quantum number z S S R , , 2 r z z S S L L R
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