Chapter3_14 - 2 (g 2 ) op +i [f op ,g op ]| > =...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
PHY4604 R. D. Field Department of Physics Chapter3_14.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 +
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 (g 2 ) op +i [f op ,g op ]| &gt; = &lt; |(f 2 ) op | &gt; + 2 &lt; | (g 2 ) op | &gt; +i &lt; |[f op ,g op ]| &gt; = &lt;f 2 &gt; + 2 &lt;g 2 &gt; + &lt;i[f op ,g op ]&gt; 0. Now we find the value of that minimizes of F( ) which occurs when dF/d = 0 . Hence 2 min &lt;g 2 &gt; +&lt;i[f op ,g op ]&gt; = 0 and min = -&lt;i[f op ,g op ]&gt;/(2&lt;g 2 &gt;) = 0 . Thus, 4 ] , [ 2 ] , [ 4 2 2 2 2 2 2 2 2 2 min &gt; &lt; &gt; &lt; &gt; &lt; &gt; &lt; &gt; &lt; + &gt; &gt;&lt; &lt; = g g g f i g g f i g f F op op op op which implies that &lt;f 2 &gt;&lt;g 2 &gt; &lt;i[f op ,g op ]&gt; 2 /4 . Now we use &lt; f 2 &gt; = &lt;(A-&lt;A&gt;) 2 &gt; = ( A) 2 and &lt; g 2 &gt; = &lt;(B-&lt;B&gt;) 2 &gt; = ( B) 2 and [f op ,g op ] = [A op ,B op ] to arrive at 2 4 1 2 2 ) ] , [ ( ) ( ) ( &gt; &lt; B A i B A | ] , [ | ) )( ( 2 1 &gt; &lt; B A i B A Find maximal set of commutation operators and label the states according to the quantum numbers of these operators!...
View Full Document

Ask a homework question - tutors are online