PHYS
p31318notes03.pdf

p31318notes03.pdf - 7 L x ih y z y z L y ih z x z x L z ih...

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7 . 2 ) ( ) ( ) ( 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + - + + + + + - - - - - - - y x xy x z zx z y yz z y x y x z x z y x y y x i z x x z i y z z y i h h h h 2 z y x L L L L Now convert these to spherical coordinates. . csc cot sin cot cos cos cot sin 2 2 2 2 2 2 φ θ + θ + θ θ - φ - φ φ θ + θ φ - φ φ θ + θ φ + h h h h 2 z y x L L L L i i i Commutators: z x y y x y z x x z x y z z y z z y y x x i i i L L L L L L L L L L L L L L L L L L L L L L L L L L L h h h = - = - = - = - = - = - 0 2 2 2 2 2 2 Also of interest: φ θ + θ ± = ± = φ ± ± cot i e i i y x h L L L This seems very abstract for the moment. In a little time, I’m going to ask you if any of these operators commute with the Hamiltonian.
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6 . 2 dt d dx d t i x i χ ψ = ψχ - h h h . ) ( 2 dx d dt d x i t i ψ χ = ψχ - h h h Thus, as expected, these two operators, which have a common eigenfunction, do indeed commute But you can well imagine that not all pairs of operators commute, and you cannot necessarily find a function that is simultaneously an eigenfunction to two arbitrary operators. For example, consider the position operator x (which just means “multiply by x ”) and the linear momentum operator p x (which is x i - h ). While , x x i ψ - = ψ h x xp . ψ + ψ - = ψ - = ψ x x i x x i h h x p x Thus the position and linear momentum operators do not commute, and no function can ever be found that is simultaneously an eigenfunction to each. There is no function that can tell you simultaneously the precise position and momentum of any particle. This is absolutely certain, and should be called the certainty principle. Now, the angular momentum operators. Angular momentum is defined by p r L × = and the angular momentum operator is defined by p r L × = That is to say:
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5 Ψ = Ψ - Ψ E m t V 2 2 2 ) , ( h r Let us look for solutions of the form . constant ) ( t i e ω - × = Ψ r k . Then . Ψ ω - = Ψ i & Now, recalling that ω = h E , we can write this last equation as Ψ - = Ψ h & E i , or as Ψ = Ψ E i & h Seen in this light we see that Ψ is an eigenfunction to the operator , t i h with eigenvalue E.
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