p31318notes05.pdf - 2 The drawing below shows an angular...

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2 The drawing below shows an angular momentum vector of magnitude ) 46 . 3 ( 12 = and the possible orientations it can have in order that its z -component should be one of the seven values 3 , 2 , 1 , 0 , 1 , 2 , 3 + + + - - - .
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1 P313 Notes 05 The Eigenvalues of the Eigenfunctions We shall need to do two things with these functions: 1. We want to see how the functions are related to the possible values of angular momentum and its z- component. (These Notes) 2. We would like to know what the eigenfunctions look like. (Notes 06). It is easy to see that > lmn | is an eigenfunction to the operator L z with corresponding eigenvalues h m . Thus (if we agree to express angular momentum in units of h rather than J s) it is easy to see that > = > lmn m lmn | | z L . And, since we already know that > = > lmn E lmn | | H , it follows that L z commutes with the Hamiltonian. We already know that L 2 commutes with L z (and hence also with the Hamiltonian), so we know that > lmn | must also be an eigenfunction to the operator L 2 . It needs a
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