Lecture 9 - Angular momentum Basic Properties Angular...

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Unformatted text preview: Angular momentum Basic Properties Angular momentum in QM Classical definition = L r p is converted into a QM operator by replacing coordinate and momentum by respective operators ; ; x y x L i y z L i z x z y x z L i x y y x = -- = -- = = -- L r p h h h All components of the angular momentum are, of course, Hermitian operators ( 29 ( 29 ( 29 ; x z y x z y z y z y z y L yp zp L yp zp p y p z p y p z yp zp =- =- =- =- =- We used here a property of hermitian conjugation: ( 29 AB B A = Commutation relations for angular momentum Components of the operator of the angular momentum in Cartesian coordinates ; ; x z y y x z z y x L yp zp L zp xp L xp yp =- =- =- The first thing we need to know about these operators is their commutators ( 29 ( 29 ( 29 ( 29 [ ] [ ] [ ] , , , , , , , x y y x z y x z x z z y z x z y z y x z x x y z y z z z y x y x z z x z z z x y y y z x zp zp yp xp zp zp x L L L L yp zp zp xp zp xp yp zp y p z p p x p z i xp i yp i L L L i L L L p yp yp zp zp y zp xp x p i L L L i L p zp- =----- =-- +- + +- =- =- = = = = h h h h h h Non-compatibility of components of angular momentum According to generalized uncertainty principle 2 2 2 2 2 2 2 2 2 2 2 2 2 , 2 4 ; 4 4 x y x z y z x y L L z L L y L L z L L L i L L = h h h Therefore, it is impossible to create such a quantum state, in which all three components of the angular momentum have a definite value. The best we can do is to have a state with definite value of one of the components, then the two others will be statistically distributed with non- zero uncertainty. Square of the angular momentum [ ] [ ] 2 2 2 2 2 2 2 2 2 2 2 2 2 , , , , , , , , 0; , x x x y x z x y x x y z x x z y x y x y y x y x y z x z...
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Lecture 9 - Angular momentum Basic Properties Angular...

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