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  Lecture 23Angular Momentum Eigenstates November 29, 2010 Lecture 23  Angular Momentum Operators Generators of rotations must satisfy: [ ˆ J  i ,  ˆ J  j ] =  i ¯ h ijk  ˆ J  k ˆ J  2 = ˆ J  2 x  + ˆ J  2 y  + ˆ J  2 z  [ ˆ J  2 ,  ˆ J  i ] = 0 ⇒  can find simultaneous eigenstates of   ˆ J  2 and  ˆ J  z ˆ J  2 |  j,m   =  α (  j ) |  j,m   ˆ J  z |  j,m   =  m ¯ h |  j,m  Define:  ˆ J  ±  = ˆ J  x  ±  i  ˆ J  y [ ˆ J  z ,  ˆ J  ± ] =  ± ¯ h  ˆ J  ±  [ ˆ J  2 ,  ˆ J  ± ] = 0 ⇒  ˆ J  ± |  j,m   =  C  ± (  j,m ) |  j,m ± 1  ˆ J  +  raises and  ˆ J  −  lowers the eigenstate to onewith the same  ˆ J  2 eigenvalue but with the  ˆ J  z eigenvalue higher or lower by one unit of   ¯ h . Lecture 23 1  Angular Momentum Eigenvalues ˆ J  2 z |  j,m   = ˆ J  2 |  j,m  −  ˆ J  2 x |  j,m  −  ˆ J  2 y |  j,m ⇒  m 2 ¯ h 2 ≤  α (  j ) The raising (lowering) doesn’t just keep goingbut for a given  j  there is a maximum(minimum) value of   m . ˆ J  + |  j,m max   = 0 ˆ J  − |  j,m min   = 0 Lecture 23 2  Constraints on Values of   m ˆ J  ∓  ˆ J  ±  = ( ˆ J  x ∓ i  ˆ J  y )( ˆ J  x ± i  ˆ J  y )= ˆ J  2 x  + ˆ J  2 y  ±  i [ ˆ J  x ,  ˆ J  y ] = ˆ J  2 −  ˆ J  2 z  ∓  ¯ h  ˆ J  z ˆ J  2 = ˆ J  2 z  ±  ¯ h  ˆ J  z  + ˆ J  ∓  ˆ J  ± ˆ J  −  ˆ J  + |  j,m max   = 0 ⇒  ˆ J  2 |  j,m max   =  m max ( m max  + 1)¯ h 2 |  j,m max  ˆ J  +  ˆ J  − |  j,m min   = 0 ⇒  ˆ J  2 |  j,m min   =  m min ( m min  + 1)¯ h 2 |  j,m min  m min  =  − m max  j  ≡  m max ⇒  α (  j ) =  j (  j  + 1)¯ h 2 ⇒  ˆ J  2 |  j,m   =  j (  j  + 1)¯ h 2 |  j,m  Lecture 23 3
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