Chapter3_13 - x is conserved! Time Translations: The...

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PHY4604 R. D. Field Department of Physics Chapter3_13.doc University of Florida Displacement Operators Spatial Translations: The unitary operator D op (a) that generates finite spatial translation is given by ) ( ) ( ) ( x a D a x op ψ = + , where h / ) ( ) ( op x p ia op e a D = ( D op is unitary since (p x ) op is hermitian). The momentum operator is the generator of spatial translations. Suppose that H op | ψ >=E| ψ > and hence < ψ |H op | ψ >=E with < ψ | ψ >=1 . The translated state D(a)| ψ > = |D a ψ > is also an eigenstate of H op with energy E provided [p x ,H op ]=0. Proof: <D a ψ |H op |D a ψ > = < ψ |D (a)H op D(a)| ψ > = < ψ H op | ψ > = E, where I used op p ia op p ia op op op H e H e a D H a D op x op x = = h h / ) ( / ) ( ) ( ) ( provided 0 ] , ) [( = op op x H p . If H op is invariant under the space translation x x +a then [p x ,H op ] = 0 and p
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Unformatted text preview: x is conserved! Time Translations: The unitary operator T op (a) that generates finite time translation is given by ) ( ) ( ) ( t a T a t op = + , where t a iaE op e e a T op = = h / ) ( ( T op is unitary since E op is hermitian) Infinitesimal Time Translations: The energy operator is the generator of time translations op op E i T h = 1 ) ( , ( ) > + < = > < >= =< > < ) ( ) ( lim ] , [ 1 t H t H i t H i H E dt E d i op op h h h If H op is invariant under the time translation t t +a then [E op ,H op ] = 0 and energy is conserved!...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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