Chapter6_26 - α 2 (g 2 ) op +i α [f op ,g op ]| ψ > =...

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PHY3063 R. D. Field Department of Physics Chapter6_26.doc University of Florida Eigenstates and the Uncertainty Principle Simultaneous Eigenstates: Suppose that the state ψ is an eigenstate of the operator A op with eigenvalue a and also an eigenstate of the operator B op with eigenvalue b as follows: A op | ψ > = a| ψ > and B op | ψ > = b| ψ > . Then [A,B] = 0. Proof: We see that [A,B]| ψ > = AB| ψ > - BA| ψ > = (ab-ba)| ψ > = 0 . The Uncertainty Relations: Define ( A) 2 <(A - <A>) 2 > = <A 2 > - (<A>) 2 ( B) 2 <(B - <B>) 2 > = <B 2 > - (<B>) 2 Then 2 4 1 2 2 ) ] , [ ( ) ( ) ( > < B A i B A One can simultaneously know the precise values of commuting observables, but there is a lower limit on how well one can simultaneously know the values of non-commuting observables. Proof: Let f op = A op - <A> and g op = B op - <B> where A op and B op are hermitian operators (hence f op and g op are hermitian) and let |f α > =( f op +i α g op )| ψ > where α is a (real) constant. Note that F( α ) = <f α |f α > 0 for any value of α . Thus, F( α ) = <f α |f α > = < ψ |( f op -i α g op ) ( f op +i α g op )| ψ > = < ψ |(f 2 ) op +
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Unformatted text preview: α 2 (g 2 ) op +i α [f op ,g op ]| ψ > = < ψ |(f 2 ) op | ψ > + α 2 < ψ | (g 2 ) op | ψ > +i α < ψ |[f op ,g op ]| ψ > = <f 2 > + α 2 <g 2 > + α <i[f op ,g op ]> ≥ 0. Now we find the value of α that minimizes of F( α ) which occurs when dF/d α = 0 . Hence 2 α min <g 2 > +<i[f op ,g op ]> = 0 and α min = -<i[f op ,g op ]>/(2<g 2 >) = 0 . Thus, 4 ] , [ 2 ] , [ 4 2 2 2 2 2 2 2 2 2 min ≥ > < > < > < − > < > < + > >< < = g g g f i g g f i g f F op op op op which implies that <f 2 ><g 2 > ≥ <i[f op ,g op ]> 2 /4 . Now we use < f 2 > = <(A-<A>) 2 > = ( ∆ A) 2 and < g 2 > = <(B-<B>) 2 > = ( ∆ B) 2 and [f op ,g op ] = [A op ,B op ] to arrive at 2 4 1 2 2 ) ] , [ ( ) ( ) ( > < ≥ ∆ ∆ B A i B A | ] , [ | ) )( ( 2 1 > < ≥ ∆ ∆ B A i B A Find maximal set of commutation operators and label the states according to the quantum numbers of these operators!...
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This note was uploaded on 05/29/2011 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.

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