Final Fa00 Solutions

Introduction to the Standard Model of Particle Physics

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PHY557 Final Exam 1 Fall 2000 PHY557 – Final Examination I Short answer problems (100 points; about 6 points each) 1. Show, by explicit substitution, that a solution of the Dirac equation, ψ ( x ) = u ( p )e ipx , is also a solution of the Klein-Gordon equation 2 ( x , t )/ t 2 = ( 2 m 2 ) ( x , t ). The only t and x dependence is in the exponential function. 2 ( x , t )/ t 2 = u ( p )e ipx (– iE ) 2 =?= ( 2 m 2 ) ( x , t ) = u ( p )e ipx [( i p ) 2 m 2 ] E 2 =?= [ p 2 m 2 ], which indeed is the relativistic energy-momentum-mass relationship. 2. Starting with the Dirac Lagrangian L D = ψ ( x )[ i γ µ m ] ( x ), where 0 , derive the Dirac equation as the Euler-Lagrange equations of L D . Note: 0 50 1 2 3 01 0 1 0 2 10 0 0 1 2 0 0 ,, 0 0 , (Dirac representation); , i i i i i γγ σ          == = = σ 1 γ 1 σ 0( ) 0 () im µµ ψγ ψψ ∂∂ =∂ = ∂ + = LL , and ) 0 γψ =− = 3. Show that the invariance of the Dirac Lagrangian L D under global phase rotations of the fields leads to the existence of a conserved current of the form j ψγ . Invariance of L requires: 0 because of E-L equations ) ( ) 0 ie δδ δ δψ αψ = + ∂= + =  =+ = + =   !"""#"""$ L L L L L L 0 = where we used = e ie α ie for small variations. The final quantity in square brackets is a covariant fourvector, and is conserved: 0, with jj e e i = L 4. Prove that ( σ⋅ p ) 2 = | p | 2 1 2 , where 1 2 represents the 2 × 2 unit matrix. ( σ⋅ p ) 2 = i p i j p j = i j p i p j = ½( i j + j i ) p i p j + ½( i j j i ) p i p j = ij p i p j + i ε ijk k p i p j = p x 2 + p y 2 + p z 2 + i k ( p × p ) k = ( p x 2 + p y 2 + p z 2 ) 1 2 + 0 = | p | 2 1 2
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PHY557 Final Exam 2 Fall 2000 5. Prove that the Dirac Hamiltonian H D = α⋅ p + β m = 2 2 m m   ⋅−  p σ p 1 commutes with the helicity operator ½( σ⋅ p )/| p | 1 4 . ^^ ^ ^ 22 ^ ^ 00 11 1 2 mm m m ⋅⋅ == p p σ p σ pp p σ p σ p p 1 σ p σ p σ p , or oth- erwise stated: all 2 × 2 sub-matrices commute trivially, because the only non-commuting op- erators contained in both operators are the same: σ⋅ p . 6. Prove the property u (1,2) ( p ) u (1,2) ( p ) = 2 m for the first two Dirac solutions. u (1,2) ( p ) = N (1,2) (1,2) (1,2) 10 , with , 01 Em φ +    σ p , N = ( E + m ) ½ . u u = u γ 0 u = ( E + m )(1 [ σ⋅ p /( E + m )] 2 )| (1,2) | 2 = ( E + m )(1 p 2 /( E + m ) 2 )1 = E + m E + m = 2 m 7. Show that the averaging over initial spin states, and summing over final spin states for a Dirac current j µ (1 3) gives the tensor (1 3) 3 1 2 Tr ( Lp µν = 31 )( mp ν + 1 ) m  +  . Use the completeness relation () 1,. .,4 i i i uu p m p = =+ = m + . 13 †† ( 1 3 ) 1 3 3 1 ,, 3 3 1 ( )()() ()() ( ) ()()() ss Lu u u u u u u u p αα δ αβ α γδ γγ ≡= ∑∑ !"#"$ δα + 3 3 1 2 m p βγ + = + 1 Tr 2 += + 1 m + 8. Show that 2 (1 3) 3 1 1 3 1 3 ( ) p p p g p p =++ , assuming m 1 = m 3 .
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Final Fa00 Solutions - PHY557 Final Exam 1 Fall 2000 PHY557...

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