c120a_lecture31_sp2007 - Chem 120A Spring 2007 Spin...

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Chem 120A Spin Statistics/Approximation Methods 04/09/07 Spring 2007 Lecture 31 Reading: Ratner Schatz Chapter 9 1 Spin our notation later on. integer or half integer. we can label our kets as b . (The letters j and m are standard notation, but a and b are not.) We will use the standard ± ± j , m ² notation from now on. For a given value of j , m can take on values of - j , - j + 1 , - j + 2 ... j - 2 , j - 1 , j . A few examples are shown in Figure 3 specialize the above arguments to orbital angular momentum, we would simply replace all J ’s with L ’s and all the j ’s with l ’s. For spin angular momentum, we replace all J ’s with S ’s, all the j ’s with s ’s, and all the m ’s with m s ’s as shown below. 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
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This note was uploaded on 02/19/2010 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at University of California, Berkeley.

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c120a_lecture31_sp2007 - Chem 120A Spring 2007 Spin...

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