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lecture11 - of both operators simultaneously e.g can find...

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5.61 Fall 2007 Lecture #11 page 1 PRINCIPLES OF QUANTUM MECHANICS (cont’d) COMMUTATORS Order counts when applying multiple operators! e.g. A ˆ = p ˆ x ˆ = ? and can we write p ˆ x ˆ = x ˆ p ˆ ? operate on function to obtain . Af ˆ x x p ˆ x ˆ x i ! d x x d d f ( ) ( ) = g ( ) ( ) f ( ) = dx ( ) f ( ) = i ! x dx f x x i ! x dx i ! x ( ) i ! f ( ) = A ˆ = i ! x d dx i ! = ( p ˆ x ˆ ) Now try B ˆ = x ˆ p ˆ Bf ˆ x x d d x i ! x x B ˆ = i ! x d ( ) = ( ) i ! dx f ( ) = dx f ( ) = x ˆ p ˆ dx x ˆ p ˆ p ˆ x ˆ Define commutator For two operators A ˆ and B ˆ , A ˆ , B ˆ = A ˆ B ˆ B ˆ A ˆ = C ˆ need not be zero! e.g. x ˆ, p ˆ = x ˆ p ˆ p ˆ x ˆ = i ! 0 !
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5.61 Fall 2007 Lecture #11 page 2 Important general statements about commutators: 1) For operators that commute A ˆ , B ˆ = 0 it is possible to find a set of wavefunctions that are eigenfunctions
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Unformatted text preview: of both operators simultaneously. e.g. can find wavefunctions ψ n such that A ˆ = a and B ˆ = b n n n n n n • This means that we can know the exact values of both observables A and B simultaneously (no uncertainty limitation). 2) For operators that do not commute ⎡ ⎣ A ˆ , B ˆ ⎤ ⎦ ≠ 0 • it is not possible to find a set of wavefunctions that are simultaneous eigenfunctions of both operators. • This means that we cannot know the exact values of both observables and simultaneously ⇒ uncertainty ! e.g. ⎡ ⎣ x ˆ, p ˆ ⎤ ⎦ = i ! ≠ 0 ⇒ Δ x Δ p ≥ ! 2...
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