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Lecture_17

Lecture_17 - PHYS 342 all 2011 Fall 2011 Lecture 17 Angular...

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Unformatted text preview: PHYS 342 all 2011 Fall 2011 Lecture 17: Angular Momentum and he Hydrogen Atom the Hydrogen Atom e- , m U(r) Lecture 17 Last lecture, we discussed how n determines the allowed energies 2 2 2 2 2 2 1 1 1 1 1 3 . 6 e e e V E m m c 2 2 2 2 2 4 2 ' 1 2 integer 0; 0 1 e e o n n n e n , , , 1,2,... integer 0; 0 1 4 ( , , ) ( ) ( , ) o l m n l l m n n c r R r Y , , , , n l m n l l m Now let’s talk more about the quantum numbers ℓ and m The angular wave functions Θ ℓ ,m ( θ ) and Φ m ( φ ) satisfy the following two equations 2 2 ) ( ) d 2 2 ( ) ( ) 1 sin ( ) ( 1 ) ( ) m m m m d d d m , , 2 2 sin sin w here ( + 1) and m are separation constants m m d d Like before, in the 1D case, the angular wave functions can be used to find the probability that an lectron t me ngle r respective f electron is at some angle φ o (or θ o ) irrespective of r * ( ) 2 o m o m o o P d * , , ( ) o m o m o o P d z d θ y θ o x d φ φ o ….but there is a deeper significance First… a word about classical angular momentum 2 : / L r p r mv units m kg s J s ˆ ˆ ˆ i j k y z Note: “angular momentum in z ” = “position in x ” times “linear momentum in y ” minus “position in ˆ ˆ ˆ x y z x y z p p p p zp i zp xp j xp yp k y ” times “linear momentum in x ” ˆ ˆ ˆ z y x z y x x y z yp zp i zp xp j xp yp k L i L j L k radius r z right hand ( ) ˆ ˆ ˆ ˆ cos ˆ ˆ cos o o o o orbital velocity r r t i r s i n t j x i y j v v s i n t i v t j o L rule x y L L m e y 2 2 cos sin z e o o e o o L m v r t t m v r L momentum radius v o x Calculating the Kinetic Energy for Circular Motion 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ; 2 2 x y x y o o e e o p p p p r K r x y m m r 2 2 2 2 2 2 2 2 2 2 2...
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Lecture_17 - PHYS 342 all 2011 Fall 2011 Lecture 17 Angular...

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