Chapter3_12 - ( 1 ) ( ) ( x U x p i x x x op op x = + = + =...

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PHY4604 R. D. Field Department of Physics Chapter3_12.doc University of Florida Symmetries and Conserved Quantities In quantum mechanics there is a relationship between symmetries and conservation laws. The time derivative of the expectation value of the observable operator O op is given by > < + > < = > < ] , [ op op H O t O i dt O d i h h and if O op does not depend explicitly on time then ] , [ op op H O dt O d i =< > < h . Linear Momentum: It is easy to show that dx x df i x f p op x ) ( )] ( , ) [( h = and hence () > + < = > < >= =< > < ) ( ) ( lim ] , ) [( 1 0 x H x H i x H i H p dt p d i op op x x ε h h h Thus, if the Hamiltonial is invariant ( i.e. does not change) under the infinitesimal spatial transformation x x + ε , then p x is conserved! Infinitesimal Generator of Spatial Translations: We see that ) ( ) ( ) ( )
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Unformatted text preview: ( 1 ) ( ) ( x U x p i x x x op op x = + = + = + h , where op op x op ih p i U + = + = 1 ) ( 1 ) ( h The hermitian operator h op is said to be the generator of the transformation. Finite Spatial Translations: The operator U op (a) that generates a finite spatial translation is given by ) ( ) ( ) ( x a U a x op = + , where h / ) ( ) ( op x op p ia iah op e e a U = = ( U op is unitary since h op is hermitian) Proof: = = = + ) ( ! 1 ) ( ) ( n n dx d a x dx d a n x e a x ( Taylor Series Expansion )...
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