Question

(VI) Hydrogen atom A What is the probability that an electron in the ground state of hydrogen will be found inside the nucleus? Find the expression for the probability, in which Rc denotes the the radius of nucleus. Hints: Rc IT 127 i) Integration in spherical coordinate system (r, 0, 0)|r2 sin Ododedr Jo Jo Jo 2.c 20 e Jo a 2 B Construct the wavefunction for an electron in the state defined by the three quantum numbers: principal n = 4, azimuthal (defining the orbital angular momentum) l = 3, magnetic (projection of the orbital along the z axis) m = 3. Use the table below. 110.9= Ro60)=2(8" expl-ria) 2,091-(* 2-r1a,)eype=r120) 1,(0) - team exper/20) RM=2() "2" 3 1 16) perp=r13a) 7:40.5) - v2. since * 30.09-11coso 7,0.9) -- v sinoc 7,0.9) - sin och Rodos - rasbasmi" (3) ext=rika)

Answer 1

The expression for the ground state state wavefunction for hydrogen electron is written and the probabilty is then determined. please see attched solution with step by step solution.

312 For a hydrogen atom in ground-state, n=1,1=0.me=0 we have, L100 (2,0,0) = R1,0 (4) x Yo (0,0) = 2(992 exp(-a)* Tam* exul Now, Probability (P), For de eRe (radius of nucleus) - Rc T 21 - Plr = Rc) = ( 14 (2,0,0)1 4² sino do do de 0 0 0 9T for 4,00 O - (Note that I ainode = foco - 2) (Note . . do that Rc P = 40 ( 2e-2r10o as T193) then u (x²e 2x dx as given in Now, Let 2 = -2 the hint. & Reab b we have, 2br 24 - ao и/a> - Sales open ola

Thus, .-93) he P (EL Rc) = P(reb) of Lo3 / NO - 4 = Re I Prab) = g_e +1 92 x This is the expression for Probability to find election inside the nucleus Note: a = Boht's radius to 0.5810lom so the p would be very small, almost zeso. quantum numbers : with for an election n=u, l=3 and m=3 Sime, the yg - 4reme - Rmiela). Yeah C 0,60) 44, 3, 3 = R4,3 (r) .Y (0,0) ; Using the Table en 25" (143* ( ) - ( in ciobe) 768 35 1 44,33 = *** 3 -1 6144 * 43** (4) csillo,