# ph135_hw4 - ˆ S 12 operator on the two-nucleon angular...

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Physics 135c Homework 4 1.) The central part of the nucleon-nucleon potential can be written as a sum of four terms: V ( r ) = - V 0 [ W ( r ) + B ( r ) ˆ P σ + M ( r ) ˆ P x + H ( r ) ˆ P x ˆ P σ ] where ˆ P σ is the spin exchange operator and ˆ P x is the space coordinate exchange operator. Using the symmetry properties of the spin-singlet and triplet S and P states, determine the relation between the above interactions ( W,B,M,H ) and the four interactions: V 1 S ( r ) ,V 3 S ( r ) ,V 1 P ( r ) ,V 3 P ( r ); corresponding to the nucleon-nucleon potentials with the nucleons in 1 S , 3 S , 1 P , 3 P states respec- tively. 2.) Supplemental Problem 2 (SP2). 3.) Supplemental Problem 3 (SP3). 4.) Show that a N-N potential for the deuteron containing a tensor term of the form ˆ S 12 = ( 3 r 2 )(ˆ σ 1 · ˆ r )(ˆ σ 2 · ˆ r ) - ˆ σ 1 · ˆ σ 2 can produce a mix of S - and D - states by calculating the eFect of the
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Unformatted text preview: ˆ S 12 operator on the two-nucleon angular momentum states with L = 0 and L = 2 for J = 1 and S = 1 ±xed (i.e. 3 S 1 , 3 D 1 ). [Hint: Show that ˆ S 12 | 3 S 1 > = α | 3 D 1 > + β | 3 S 1 >, and ˆ S 12 | 3 D 1 > = γ | 3 D 1 > + δ | 3 S 1 > by determining the coe²cients α,β,γ,δ .] 5.) ³or a non-local potential ˆ V ( r , r ′ ), the potential energy operator ˆ V acting on the wave function ψ is i ˆ V ( r , r ′ ) ψ ( r ′ ) d 3 r ′ . Show that such a non-local potential is equivalent to a momentum (and hence velocity-) dependent potential. [Hint: consider momentum space transforms.]...
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