Duality S. Eq Hydrogen Outlines of 1 Wave-Particle Duality 2 The Schrödinger Equation 3 The Hydrogen Atom Schrödinger Eq. of the Hydrogen Atom Noninteracting Particles and Separation of Variables The One-Particle Central Force Problem Energy levels of the Hydrogenlike atom Hydrogenlike-Atom Wave Functions
Schrödinger Eq. of the Hydrogen Atom f½a fµ N r e H = 2 2 2M N 2 2 N 2m e Ze 2 e r r N r e Ù r = r e r N e 2 = e 2 /4πɛ 0
Noninteracting Particles and Separation of Variables Ĥ = Ĥ1 + Ĥ2, (Ĥ1 + Ĥ2)Ψ(q 1, q 2 ) = EΨ(q 1, q 2 ) Ù Ĥ 1 Úâf1k' Ĥ 2 Úâf2k' lcþ{ bψ(q 1, q 2 ) = G 1 (q 1 )G 2 (q 2 ): Ĥ 1 G 1 (q 1 )G 2 (q 2 ) + Ĥ 2 G 1 (q 1 )G 2 (q 2 ) = EG 1 (q 1 )G 2 (q 2 ) G 2 (q 2 )Ĥ 1 G 1 (q 1 ) + G 1 (q 1 )Ĥ 2 G 2 (q 2 ) = EG 1 (q 1 )G 2 (q 2 ) Ĥ 1 G 1 (q 1 ) G 1 (q 1 ) Left part must be a constant, + Ĥ2G 2 (q 2 ) G 2 (q 2 ) = E Ĥ 1 G 1 (q 1 ) = E 1 G 1 (q 1 ), Ĥ 2 G 2 (q 2 ) = E 2 G 2 (q 2 ) E 1 + E 2 = E
Noninteracting Particles and Separation of Variables Œí2n Õá âf Kµ Ĥ = Ĥ 1 + Ĥ 2 + + Ĥ n Ψ(q 1, q 2,, q n ) = G 1 (q 1 )G 2 (q 2 ) G n (q n ) E = E 1 + E 2 + + E n Ĥ i G i = E i G i, i = 1, 2,, n
Reduction of the Two-Particle Problem to Two One-Particle Problem Ÿ% IX {µ r N N r r e e R(X, Y, Z) = m 1 r 1 + m 2 r 2 m 1 + m 2 r(x, y, z) = r 1 r 2 M = m 1 + m 2 ; µ = m 1m 2 m 1 + m 2 2 R = 2 X 2 + 2 Y 2 + 2 Z 2 2 r = 2 x 2 + 2 y 2 + 2 z 2 = 1 m 1 2 1 + 1 m 2 2 2 = 1 M 2 R + 1 µ 2 r
KS.eqŒz µ lcþ{ [ 2 2M 2 R 2 2µ 2 r + V (r)]ψ = E total Ψ Ψ( R, r) = Ψ c ( R)Ψ e ( r) 2 2M 2 RΨ c ( R) = E c Ψ c ( R) [ 2 2µ 2 + V (r)]ψ e ( r) = EΨ e ( r) E total = E c + E = ) Ÿ%²ÄÚSÜ$Ä" ù { ( Ã{í2õN K"
n %å 2 = 2 r 2 ˆL 2 = 2 1 [ sin θ ˆL z = i 2 = 1 r + 2 r r + 1 1 r [ 2 sin θ φ, [ 2 2µ 2 + V (r)]ψ( r) = EΨ( r) θ (sin θ θ ) + 1 sin 2 θ 2 r 2 r 1 r 2 2 ˆL 2 θ (sin θ θ ) + 1 2 φ 2 ], sin 2 θ 2 φ 2 ], Ĥ = 2 2 2µr r r + 1 ˆL 2 2 2µr + V (r) 2
lcþ{ ) [ 2 2µr Œ± ) µ 2 r 2 r + V (r) + 1 2µr 2 ˆL 2 ]Ψ(r, θ, φ) = EΨ(r, θ, φ) Ψ = R(r)Y (θ, φ) ˆL 2 Y (θ, φ) = β 2 Y (θ, φ)» [ 1 r d 2 dr 2 r + 2 µ 2 (E V ) β r 2 ]R l (r) = 0
¼ê Ù) Y lm (θ, φ) = ( 1) m+ m 2 2π 0 ˆL 2 Y (θ, φ) = β 2 Y (θ, φ) (2l + 1)(l m )! 4π(l + m )! P m l ˆL 2 Y lm (θ, φ) = l(l + 1) 2 Y lm (θ, φ) dφ π 0 dθy lmy l m sin θ = δ ll δ mm 1 (cosθ) e imφ 2π
Radial Function and Energy levels 2 2µ (R l + 2 r R l ) + V (r)r l + l(l+1) 2 2µr 2 R l = ER l (r) %å Ù)µ R = R nl (r), 0 R nl R n lr 2 dr = δ nn E = E nl Uþ 2l + 1 { mþfêã'" éua lfv (r) = Ze 2 /r Ù) R nl (r) = N nl e Zr 2Z na ( na r)l L 2l+1 a 2 /µe 2 n+l (2Z na r) e 2 = e 2 /4πɛ 0
Energy levels of hydrogenlike atom Ψ nlm = R nl (r)y lm (θ, φ) HΨ nlm = E n Ψ nlm E n = Z 2 n 2 µe 4 2 2 L 2 Ψ nlm = l(l + 1) 2 Ψ nlm L z Ψ nlm = m l Ψ nlm l = 0, 1, 2, 3, n 1 m = 0, ±1, ±2,, ±l = hcr Z 2 M n 2 l= 0(sharp), 1(principle), 2(diffuse), 3(fundamental), hcr (e2 /4πɛ 0) 2 m e 2 2 13.6eV, a 0 2 m ee 2 /4πɛ 0 0.529Å R =109 737.315 685 39(55) cm 1, R M = R M m e+m
Hydrogenlike-Atom Wave Functions Ψ nlm = R nl (r)y lm (θ, φ) R nl (r) = ( 2Z (n l 1)! )3 na 2n[(n + l)!] Y lm (θ, φ) = ( 1) m+ m 2 where a 4πɛ 0 2 /µe 2 3 e Zr 2Z na ( na r)l L 2l+1 (2l + 1)(l m )! P m 4π(l + m )! n+l (2Z na r) 1 l (cos θ) e imφ 2π
Radial Hydrogenic Wavefunction R nl R 1s = ( Z a )3/2 2e ρ R 2s = ( Z 2a )3/2 2(1 ρ)e ρ R 2p = ( Z 2a )3/2 2 3 ρe ρ R 3s = ( Z 3a )3/2 2(1 2ρ + 2 3 ρ2 )e ρ R 3p = ( Z 4 2 3a )3/2 ρ(1 ρ/2)e ρ 3 R 3d = ( Z 2 2 3a )3/2 3 5 ρ2 e ρ l = 0ž 3r = 0 :?µ Ψ n,l=0 (0) 2 = 1 π ( Z na )3 a 4πɛ 0 2 /µe 2 ρ = Zr na 1 r = 1 r R2 nlr 2 dr 0 = 1 n 2 (Z a ) 1 r 3 = 1 l(l + 1/2)(l + 1) ( Z na )3
R 1s & R 2s & R 2p Electron in the Nucleus?!
R 3s & R 3p & R 3d
Radial Distribution Function R 2 nl r 2
Real Hydrogenlike Functions combinations of Ψ n,l,m and Ψ n,l, m : 1 Ψ n,l,m = R nl (r)s lm (θ) e imφ 2π 1 Ψ n,l, m = R nl (r)s l, m (θ) { 2π cos m φ sin m φ Ψ 2pz = Ψ 2p0 = 1 π ( Z 2a )5/2 e Zr/2a z Ψ 2px = 1 2 (Ψ 2p1 + Ψ 2p 1 ) = 1 π ( Z 2a )5/2 e Zr/2a x Ψ 2py = Ψ 3dxy = 1 i 2 (Ψ 2p 1 Ψ 2p 1 ) = 1 ( Z π 2a )5/2 e Zr/2a y 1 i 2 (Ψ 2 3d 2 Ψ 3d 2 ) = 81 2π (Z a )7/2 e Zr/3a xy
Graphs of Real Hydrogenlike Orbitals Angular wave function, Y lm (θ, φ) 2s Probability density, Ψ 2, Ψ (@y = 0) 2pz 3pz p z 3d z 2 3d xz 3d x 2 y 2 A Java viewer from: http://www.falstad.com/qmatom/
Graphs of Real Hydrogenlike Orbitals 3p z (m = 0) 3d xz (m = ±1) 3d xy (m = ±2) 3d z 2 (m = 0) 3d yz (m = ±1) 3d x 2 y 2 (m = ±2)
Graphs of Real Hydrogenlike Orbitals 4f xz 2 (m = ±1) 4f xyz (m = ±2) 4f x(x 2 3y 2 ) (m = ±3) 4f z 3 (m = 0) 4f yz 2 (m = ±1) 4f z(x 2 y 2 ) (m = ±2) 4f y(3x 2 y 2 ) (m = ±3) http://en.wikipedia.org/wiki/atomic orbital
õ>f f>f;u? }B APenetrating Effect >fkbls >fùø E(ns) < E(np) < E(nd) < E(nf ); U? " f;u?ã
ësnjj SK SN ëö SK få½ =Lv> 6.1 => 17 õ ÕáâfNX =Lv> 6.2-6.3 üâfn %å =Lv> 6.1 =Q> 6.1 =Lv>6.2 a fu? =Lv> 6.5 => 17 =Lv>6.16 a få¼ê =Lv> 6.6 =Lv>6.17,6.18 => 17 =>3-12