Lecture27Notes

Lecture27Notes - Chem 120A Spring 2006 Spin...

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Chem 120A Spin Statistics/Approximation Methods 03/24/06 Spring 2006 Lecture 27 Reading: Engel Chapters 10 and 11 1 Spin Last time we looked at the algebra of angular momentum in quantum mechanics and then specialized to spin angular momentum. Remember that every fundamental particle has a particular value of “ s ”. That value of s then determines the possible values of m s . We then gave the fundamental theorem of spin statistics: If s is an integer = 1, 2, 3,. ... then the particle is a boson. Identical bosons are symmetric under pairwise exchange: ψ ( 1 , 2 ) = ψ ( 2 , 1 ) . If s is a half-integer = 1/2, 3/2, 5/2. .. then the particle is a fermion. Identical fermions are antisymmetric under pairwise exchange: ψ ( 1 , 2 ) = - ψ ( 2 , 1 ) . Let’s look at two identical fermions, for instance two electrons in an He atom. Electrons have s = 1 / 2, therefore m s has possible values of + 1 / 2 and - 1 / 2. The total wavefunction has a spatial part (see Lectures 20,21,24) and a spin part. Ψ ( r 1 , r 2 , s 1 , s 2 ) = ψ ( r 1 , r 2 ) φ ( s 1 , s 2 ) (1) where ψ ( r 1 , r 2 ) describes the spatial part of the two electrons and φ ( s 1 , s 2 ) describes the spin angular mo- mentum of the two electrons. Above we stated that the wavefunction describing the two particles must be antisymmetric overall. If the spatial part of the wavefuntion is symmetric, then the spin part must be antisymmetric. If the spin part is symmetric, then the spatial part must be antisymmetric (remember that the product of a symmetric and antisymmetric function is overall antisymmetric). We will use a + subscript for symmetric wavefunctions and a - subscript for antisymmetric wavefunctions. Ψ - = ψ + ( r 1 , r 2 ) φ - ( s 1 , s 2 ) (2) or Ψ - = ψ - ( r 1 , r 2 ) φ + ( s 1 , s 2 ) (3) We can equally describe the spin wavefunction with its values of m s rather than s . φ
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Lecture27Notes - Chem 120A Spring 2006 Spin...

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