Chapter6_20 - If the energy is not an explicit function of...

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PHY3063 R. D. Field Department of Physics Chapter6_20.doc University of Florida The Hamiltonian and Constants of the Motion Hamiltonian: The energy operator is referred to as the “Hamiltonian” of the system with t i H E op = = h and op H i t h = and t t x i t x H op Ψ = Ψ ) , ( ) , ( h Consider an observable O with expectation value given by Ψ Ψ >= < dx t x O t x O op ) , ( ) , ( * , and look at the change of <O> with time as follows: Ψ Ψ + > < = Ψ Ψ + Ψ Ψ + Ψ Ψ = > < dx O H H O t O i dx t O t O O t i dt O d i op op op op op op op ) ( * * * * h h h Thus, > < + > < = > < ] , [ op op H O t O i dt O d i h h Energy Conservation:
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Unformatted text preview: If the energy is not an explicit function of time then energy is conserved. Proof: ] , [ &gt;= &lt; + &gt; &lt; = &gt; &lt; op op H H t H i dt H d i h h . Constants of the Motion: If the observable O does not depend explicitly on time then if O commutes with the Hamiltonian then &lt;O&gt; is a constant ( i.e. independent of time, conserved quantity). Proof: ] , [ &gt;= =&lt; &gt; &lt; op op H O dt O d i h , provided &gt;= &lt; t O and [O op , H op ] = 0....
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