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c120a_lecture30_s07 - Chem 120A Spring 2007 Spin Algebra...

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Chem 120A Spin Algebra and Statistics 04/06/07 Spring 2007 Lecture 30 Reading: Ratner/Schatz Chs. 7, 8 Atkins/Friedman Ch. 7.8-7.11 This lecture contains details of the derivation of the eigenvalues of general angular momentum, in addition to the material discussed in class. This is additional material for those interested to see where the eigenvalues of S 2 , S z , L 2 and L z come from. 1 Recap and more about Spin Elementary particles and composite particles carry an intrinsic angular momentum called “spin” = S , which is related to the intrinsic magnetic moment by μ = - ge 2 m S . Here g (the g-factor) is a unitless factor. For electrons, g 2. Let’s briefly look at the similarities and differences between spin angular momentum and orbital angular momentum. For orbital angular momentum: μ = - e 2 m L For spin angular momentum: μ = - ge 2 m S Orbital angular momentum has a classical analog where the magnetic moment μ comes from the electron spinning about some axis (see Figure 1) Figure 1: A charge moving, q , moving at a velocity, v in a loop of radius, r , produces a magnetic moment, μ . The right-hand rule determines the direction of the magnetic moment The intrinsic angular momentum of an electron, on the other hand, has nothing to do with its orbital motion, but it does lead to an intrinsic μ . This is a relativistic effect that can be derived from relativistic quantum mechanics Again, and we can’t stress this enough, electron spin is not orbital angular momentum in the classical sense. Experiments tell us, however, that we can take most general properties we derive for the QM operator L = r × p = ˆ L x i + ˆ L y j + ˆ L z k and we can simply apply them to the operator S = S x i + ˜ S y j + ˜ S z k . Finally the orbital angular momentum of an electron is described by physical coordinates, ψ ( r , θ , φ ) . The wavefunction for spin is represented only by a single quantum number. As we will see shortly ψ ( m s ) = ± 1 2 for electrons From last time we saw that to understand the behavior of an electron’s intrinsic magnetic moment μ (which is an observable we can measure), then we must understand the behavior of its intrinsic angular momentum = S . Mathmatically, spin angular momentum, S , can be described in terms as orbital angular momentum, L . Here we will give a unified treatment of angular momentum which holds for both L and S , thus we will use the symbol for general angular momentum, J Classically, angular momentum is J = r × p = ˆ J x i + ˆ J y j + ˆ J z k where i , j , k are the usual cartesian unit vectors. To understand angular momentum in QM, we turn the classical observables into operators and study the Chem 120A, Spring 2007, Lecture 30 1
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“algebra” of J = r × p in QM. The four important operators associated with angular momentum are: ˆ J x , ˆ J y , ˆ J z , and J 2 = ˆ J x 2 + ˆ J y 2 + ˆ J z 2 .
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c120a_lecture30_s07 - Chem 120A Spring 2007 Spin Algebra...

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