Dirac_13 - commute with the Dirac Hamiltonian, so S r is...

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PHY4604 R. D. Field Department of Physics Dirac_13.doc University of Florida Angular Momentum and Spin Orbital Angular Momentum: The orbital angular momentum operator is defined as follows: op op op p r L r r r × = It is easy to show that the orbital angular momentum operator does not commute with the Dirac Hamiltonian, so L r is not conserved! In particular, ) ( ] , [ op op op p c i L H r r h r × = α , where 2 mc p c H op β + = r r and I used ij j i i x p δ h = ] , [ . Proof: We see that x op y z z y y z z z y y y z z z y y x x y z x op p c i p p c i p z p c p y p c zp yp p p p c zp yp mc p c L H ) ( ) ( ] , [ ] , [ )] ( ), ( [ )] ( , [ ] , [ 2 r r h h r r × = = = + + = + = The other components are done similarly. Intrinsic Spin Angular Momentum: The intrinsic angular momentum operator is defined as follows: = σ r r h r 0 0 2 op S It is easy to show that the spin angular momentum operator also does not
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Unformatted text preview: commute with the Dirac Hamiltonian, so S r is also not conserved! In particular, ) ( ] , [ op op op p c i S H r r h r × + = . Proof: We see that x op y z z y x z x z y x y x x x x z z y y x x x x op p c i p p c i S mc p S c p S c p S c S mc p p p c S mc p c S H ) ( ) ( ] , [ ] , [ ] , [ ] , [ ] , ) ( [ ] , [ ] , [ 2 2 2 r r h h r r × = − = + + + = + + + = + ⋅ = where I used the fact that ] , [ ] , [ = = x x x S S , z x y i S h − = ] , [ , and y x z i S h = ] , [ . The other components are done similarly....
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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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