pset4sol - 22.02 Problem Set 4 Solution 1 Krane problem 4.4...

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22.02 Problem Set 4 Solution 1. Krane, problem 4.4. What fraction of the time do the neutron and proton in the deuteron spend beyond the range of their nuclear force? Solution: In center of mass spherical coordinate: ) ( 2 ˆ 2 2 r V H NUC r + = µ h The energy eigenvalue problem is written as: ) , , ( ) , , ( )] ( 2 [ 2 2 ϕ θ ψ ϕ θ ψ µ r E r r V NUC r = + h ) , , ( ) , , ( ) ( 2 ) , , ( ˆ )) , , ( ( 1 2 2 2 2 2 2 ϕ θ ψ ϕ θ ψ µ ϕ θ ψ ϕ θ ψ µ r E r r V r r L r r r r NUC = + + h Separate variables within ψ as ψ = and notice . ) ( ) , ( ) , , ( r Y r m l φ ϕ θ ϕ θ m l m l Y l l Y L ) 1 ( ˆ 2 2 + = h We get: φ φ φ µ φ µ E V r l l r r r NUC = + + + 2 2 2 2 2 2 ) 1 ( ) ( 1 2 h h (1-1) Let u (1-2) and ) ( ) ( r r r φ = 2 2 2 ) 1 ( r l l V NUC eff µ + + = h V (1-1) becomes 0 ) ( ) ( ) ( 2 2 2 2 = + r u E V r r u eff µ h For ground state, l=0, V eff =V NUC 0 ) ( ) ( ) ( 2 2 2 2 = + r u E V r r u NUC µ h (1-3) (1-3) has exactly the same form as one dimensional finite potential well problem. Solve (1-3), we get: r k B r k A r u R r 1 1 cos sin ) ( + = < r k r k De Ce r u R r 2 2 ) ( + = Use boundary conditions, at , u 0 = r 0 0 ) ( ) ( = = = B r r r φ at , u +∞ = r 0 ) ( = +∞ < D r r k A r u R r 1 sin ) ( = < 2 1 ) ( 2 h NUC V E k + = µ (1-4) r k Ce r u R r 2 ) ( = 2 2 2 h E k µ = (1-5) From (1-2), (1-4) and (1-5) r r k A r R r 1 sin ) ( = < φ 2 1 ) ( 2 h NUC V E k + = µ (1-6) r Ce r R r r k 2 ) ( = φ 2 2 2 h E k µ = (1-7)
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22.02 Problem Set 4 Solution
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