PHY557_hw02_solutions

PHY557_hw02_solutions - Homework 2 1 Homework II and...

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Homework 2 1 11/1/2005 Homework II and Solutions Problems: II.1 Show that: () δψ δ ψ ⋅∂ ∂ = ×∂ θ r θθ rr r Solution: Use cylindrical coordinates and the z -axis as the rotation axis for θ : ± ± ± ± ± ± ± ± ,, LHS: RHS: z z z z z z z δδ θ ρ θρ ψψ ρρ δθ ∂∂ == = + = + + ⋅= = ⎛⎞ ⋅× = ⋅ + × + + ⎜⎟ ⎝⎠ =⋅ + + = ∂∂∂ θ zz r ρ z ρθ z θ r θ θ θ rz ρ z z r z z ±±± ± ±± ± II.2 The short range of the strong interaction. a. Show that the field ψ ( r ) = α / r e mr is a solution of the time-independent Klein-Gordon equation [ ² / r ² + 2 / r / r m 2 ] ( r ) = 0 b. Calculate the value of m for a mean range of 0.5 fm. Solution: 22 3 2 mr mr mr m e r r r e r α +− = = 2 mr me r + 2 mr r + 2 mr r + 2 2 mr e 2 mr 2 mr r 0 = The meson mass for a mean range r = 0.5 fm is: 2 2 2 2 2 2 2 1 ()4 4 4 ; 4 1 Normalization: 1 ( ) 4 4 4 2 1 Thus: 0.5fm; 197MeV fm/1fm 200MeV
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PHY557_hw02_solutions - Homework 2 1 Homework II and...

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