Two nucleons of reduced mass interact through an idealised finite spherical square…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Central potentials and canonical three-dimensional systems

The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem

Problem

Two nucleons of reduced mass μc2=469.5 MeV\mu c^2 = 469.5\,\mathrm{MeV} interact through an idealised finite spherical square well V(r)=−V0V(r)=-V_0 for r<ar<a and V(r)=0V(r)=0 for r>ar>a, with range a=3.0 fma=3.0\,\mathrm{fm} and V0>0V_0>0. Take ℏc=197.3 MeV fm\hbar c = 197.3\,\mathrm{MeV\,fm}. ((a)) Use the general threshold condition jl−1(k0a)=0j_{l-1}(k_0a)=0, with k0=2μV0/ℏk_0=\sqrt{2\mu V_0}/\hbar, to find the threshold value of V0a2V_0a^2 for binding a first l=2l=2 state, in MeV fm2\mathrm{MeV\,fm^2}. You will need j1(x)=sin⁡x/x2−cos⁡x/xj_1(x)=\sin x/x^2-\cos x/x. ((b)) Hence find the minimum depth V0,min⁡V_{0,\min} (in MeV) for which this well binds an l=2l=2 state. ((c)) A well of this range has depth V0=60 MeVV_0=60\,\mathrm{MeV}. Which of l=0l=0, l=1l=1 and l=2l=2 does it bind? Justify with numbers. ((d)) Each bound state forms a (2l+1)(2l+1)-fold multiplet under rotations alone. State briefly, contrasting with the isotropic harmonic oscillator, why this square well never shows any degeneracy beyond that 2l+12l+1.

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