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3D Schr. Eqn.:Radial Eqn. • For V funtion of radius only. Look at radial equation r 2 dr dr 1 d r 2 dR ( ) 2 2M [V (r ) E ]R l (l 1) R r2 • often rewritten as u (r ) rR (r ) 2M 2 dr 2 d 2u [V 2M 2 ] u Eu l ( l 1) r2 • note l(l+1) term. Angular momentum. Acts like repulsive potential (ala classical mechanics) • energy eigenvalues typically depend on 2 quantum numbers (n and l). Only 1/r potentials depend only on n (and true for hydrogen atom only in first order. After adding perturbations due to spin and relativity, depends on n and j=l+s). P460 - 3D S.E. II 1 Particle in spherical Box • Griffiths Example 4.1. E&R section 15-8. Good V (r ) r a first model for nuclei V (r ) 0 ra • plugdr into radial equation r 2M 2 2M [ V u 2 d2 2 • look first at l=0 ] u Eu u ( r ) rR ( r ) l ( l 1) 2 d 2u dr 2 k 2u with k 2 ME u A sin( kr) B cos( kr) • boundary conditions. R=u/r and must be finite at r=0. Gives B=0. For continuity, must have R=u=0 at r=a. gives sin(ka)=0 and En 0 n 2 2 2 2 Ma 2 n 00 n 1,2,3.... sin(nr / a ) 1 r 2a • note plane wave solution are e ik r r P460 - 3D S.E. II 2 Particle in spherical Box • For l>0 solutions are Bessel functions (see Griffiths). Often arises in scattering off spherically symmetric potentials (like nuclei…..) • energy will depend on both quantum numbers Enl E10 E11 E12 E20 E21 E22 ..... • and so 1s 1p 1d 2s 2p 2d 3s 3d …………….and ordering (except higher E for higher n,l) depending on details • gives what nucleii (what Z or N) have filled (sub)shells being different than what atoms have filled electronic shells. In atoms: Z 2 4 10 ( He Be Ne) 1S 2S 2 P • in nuclei (with j subshells) Z 2 6 8 14 16 ( He C O Si S ) 1s 1 p 3 1 p 1 1d 5 2s 1 2 2 2 2 P460 - 3D S.E. II 3