Electron Spin

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2Uhlenbeck and Goudsmit Postulate Electron Spin, 1925In 1925, Uhlenbeck and Goudsmit proposed an explanation for the Stern-Gerlach experiment. They postulated thatthe electron possesses an intrinsic angular momentum, referred to as "spin". This intrinsic angular momentum givesrise to a magnetic moment in the electron that interacts with magnetic fields. The electron spin and the relatedmagnetic moment are quantized such that there are only two possible discrete values. This quantization of themagnetic moment is what leads to the deflection of the beam of silver atoms either up or down in theinhomogeneous magnetic field.It is important to note that the term "electron spin" is a bit of a misnomer; the electron cannot be thought of as a tinyball of charge spinning about its axis. The surface of the electron would have to be rotating at greater than the speedof light in order to produce a magnetic moment the size of that measured for the electron. Electron spin is anintrinsic property of the particle, like charge or mass; however, in the case of spin, the property is purely quantummechanical in nature.Dirac Equation, 1928While Uhlenbeck and Goudsmit’s postulate of the existence of an intrinsic angular momentum for the electronprovided an explanation of the Stern-Gerlach experiment, there was no solid theoretical framework for theirpostulate at the time. However, in 1928 Paul Dirac (Figure 3) developed a relativistic version of quantummechanics.Figure 3. P. A. M. Dirac (from http://discover.positron.edu.au/popups/paul-dirac).Using his relativistic equation, Dirac was able to derive the existence of electron spin and show that the properties ofelectron spin were in accord with Uhlenbeck and Goudsmit’s postulate and the results of the Stern-Gerlachexperiment.
3Electron Spin as an Intrinsic Angular MomentumBecause spin corresponds to an intrinsic angular momentum of the electron, it behaves in very much the same wayas orbital angular momentum. That is, electron spin is a vector quantity that possesses similar eigenvalue equations,commutators, and measurement relations. For example, there are two eigenvalue equations that have the formˆ S 2 ψ spin = 2 s s +1 ( ) ψ spinˆ S z ψ spin = m s ψ spin .€Here, s corresponds to the spin quantum number (analogous to the orbital angular momentum quantum number )and m s corresponds to the magnetic€spin quantum number (analogous to the magnetic quantum number m ). Themajor difference between the electron spin quantum numbers and the orbital angular momentum quantum numbersis that s is a half integer instead of an integer,€€s = 1 2m s = ± s = ± 1 2 .Because there are only two possible values of the magnetic spin quantum number corresponding to the twodeflections observed in the Stern-Gerlach € experiment, there are two electron spin eigenfunctions, ψ spin . One spineigenfunction corresponds to the "spin up" state, with quantum numbers s = 1 , m 2 s = 1 2, and is usually labeled α.The other spin eigenfunction corresponds to the "spin down" state, with quantum numbers s = 1 2 , m s = − 1 2 , and isusually labeled β. The eigenvalue equations for the spin up and spin down states may be € written as€S ˆ 2 α = 3 4 2 α S ˆ 2 β = 3 4 2 βˆ€S z α = 1 2 αS ˆz β = − 1 2 β .The spin up and spin down eigenfunctions are normalized and orthogonal,€∫ α * α dσ = ∫ β * β dσ = 1∫ α * β dσ = ∫ β * α dσ = 0 .Here, the volume element dσ is used to denote integration over the electron spin variable.€The commutator relationships for the spin operators also are similar to those for the orbital angular momentum.Each component € operator commutes with the spin squared; however, the components do not commute amongthemselves.S ˆ[ 2 , S ˆx ] = S ˆ[ 2 , S ˆy ] = ˆS ˆ[ , S ˆx y ] = iS ˆz ; S z[ S 2 , S ˆz ] = 0S ˆ , ˆ S y x ;[ ] = i ˆS ˆ[ , S ˆz x ] = iS ˆy .€
Page 1: Chemistry 362Spring 2013Dr. Jean M.

Electron Spin

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