Download Quantum Mechanics of Hydrogen Atom: Separation of Variables and Radial Functions - Prof. T and more Study notes Physical Chemistry in PDF only on Docsity! 9-1 H atom made up of proton and electron Schrödinger equation is ( ) 2 2 0 04 4 e e V r r rπε πε = − = − r r ( ) ( ) ( ) ( ) ( ) 2 22 2 2 2 2 2 0 , ,1 2 , , , ,1 1 sin sin sin , , , , 4 r r r r r r r r r e r E r r ψ θ φ µ ψ θ φ ψ θ φ θ θ θ θ θ φ ψ θ φ ψ θ φ πε ∂ ∂ + ∂ ∂ − ∂ ∂ ∂ + ∂ ∂ ∂ − = h 9-2 Because V depends only on r, can achieve separation of variables. are spherical harmonic functions. Only function not known is R(r). ( ) ( ) ( ) ( ), ,r R rψ θ φ θ φ= Θ Φ ( ) ( )θ φΘ Φ ( ) ( ) ( ) 2 2 2 2 2 2 0 ( 1) 2 2 4 d R rd l l e r R r E R r r d r d r r rµ µ πε + − + − = h h Effective potential 9-5 Need 3 QN to describe state of H 1, 2, 3, 4,..... 0,1, 2, 3, ..., 1 0, 1, 2, 3, ...l n l n m l = = − = ± ± ± ± ( ) ( ) ( ) 0 0 0 3 / 2 / 10 0 3 / 2 /2 20 0 0 3 / 2 / 2 21 0 0 1 1, 0 2 1 1 2, 0 2 8 1 1 2, 1 24 r a r a r a n l R r e a r n l R r e a a r n l R r e a a − − − = = = = = = − = = = Radial functions 9-6 Complete normalized total energy eigenfunctions include spherical harmonics ( ) ( ) ( ) ( ) 0 0 0 0 3 / 2 / 100 0 3 /2 / 2 200 0 0 3 / 2 / 2 210 0 0 3 /2 / 2 21 1 0 0 1 1 1, 0, 0 1 1 2, 0, 0 2 32 1 1 2, 1, 0 , , cos 32 1 1 2, 1, 1 , , sin 8 r a l r a l r a l r a i l n l m r e a r n l m r e a a r n l m r e a a r n l m r e e a a φ ψ π ψ π ψ θ φ θ π ψ θ φ θ π − − − − ± ± = = = = = = = = − = = = = = = = ± = Note that EF real only if ml = 0! 9-7 Convenient to combine orbital functions with their complex conjugate to create real functions ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 3 / 2 / 2 1 1 2 1 1 0 0 3 / 2 / 2 1 1 2 1 1 0 0 3 / 2 / 2 0 2 1 0 0 1 1 , , sin cos 128 1 1 , , sin sin 128 1 1 , , cos 32 x y z r a p r a p r a p r r e Y Y a a r r e Y Y a a r r e Y a a ψ θ φ θ φ π ψ θ φ θ φ π ψ θ φ θ π − − − − − = + = − = 9-10 9-11 9-12 9-15 Calculating the probability of finding the e- in H within dV 9-16 9-17 Total energy EF have n-l-1 radial and l angular nodes Example Problem 9.3 Locate the nodal surfaces in The angular part, cosθ, is zero for θ=π/2. In 3D space, this corresponds to the plane z = 0. ( ) 0 3 / 21 /2 2 / 3 310 2 0 0 0 1 2 1 , , 6 cos 81 r ar rr e a a a ψ θ φ θ π − = − 9-20 Define radial probability distribution function P(r)dr ( ) ( )( ) ( )( ) 2 2 0 0 22 sin 4 P r d r d d R r dr r R r d r π π φ θ θ π = = ∫ ∫ rpd gives the probability of finding the electron in a spherical shell of radius r and thickness dr. 9-21 rpd example for 1s orbital. Note that rpd function goes to zero as r goes to zero
Radial Distribution Function
1s 2p 2s
yyy
44
mo, “eee
5
10 15 20
Distance (ao)
9-22
