Question

2. A particle is found in a three-dimensional potential well so that U = U. for r > R and U = 0 for r < R. Find the wavefunctions for stationary states. Tip: Start with the Schrödinger equation (1.106 in the lecture notes) and use the Laplacian for spherical coordinate system. Also note that in the problem considered U(r,0,0) = U(r). (3p)

Answer 1

given that paulide is found Yn 3-elimensional potential and the potential is defined aj - vo 4 ZR Ular 0 4 <R They by using schyondigay wave eq qutions - - t2tre + U14) il de dt 2m 0 820 + 2m t2 * E- U14)) ° - Ô 02 yo Roym of sphewical cay dinate system – d 42. d ) 42 de + I Sinoy2 ( sténo. d + al d² 4zsin 20 do? do Then ly eq 0 ey2. dy T I d e12 de d25 + Id I sino de easino do do | sino b) *eXinzo de + du m. (F- v.) 4 ť? 414,0,0) = R(4) 10) (0) Then -

010) al (** 42 VR + d .42 d4 R$ esino do 1 sino do do) + R © d20 aquino a 62 am -(EN) Rot t? diuided ly RO¢ multipy - dyin 20 Then sinzo e 42. dR sino (sino doo + im R dy X LE do 52 erino | d2 & do2 2- Then I 029 + m2 e dQ2 Then: sinzo d R du l կ և K sino o sino do do + 2m (E-U) xz 3 in %0 =12 d'i do 72 fo - d ello dR (4 amy? + (E-v) llet) R da ħ 2 and ma sinzo d sio Odo sino do) llet) soln dro do? + med=0 Belmo

wane funution should be noymalued $11- Then Mid m= 0,Il-.. 4 e Ter similially o coudinale XCOLO dx -sino -sino C Clo do -sino do do dx sino d dr - sin 20 do dx Then - d and sin sino do) - (- sinzo de do du de CC-rey + Lu llet) – m2 Je 1-re soln 0 - Cem Pem coro S6 solu OF Radial

I d 2m 42. dr me) + muo ellet1) 42 P20 12 du h2 e 2 it 3.0 consider the value of potential 4neo.s -2me B? 2 d + ame 2 ellt') SI- Ro 42 dl [ 4. del) + (-p? un&o.tar 12. Then son of reis f = 28.9 amey E 320²&o k2n2 دام and RIP) = FLP) Ae (Fles is The arcuitory function)