{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter3_13 - x is conserved Time Translations The unitary...

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_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 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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 ε − = 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!...
View Full Document

{[ snackBarMessage ]}