Summary2.1 The Schrodinger equation andits solution for one-electron atoms2.1.1 The Schrodinger equationTheHamiltonianOperator of one-electron atomsHatom, Het and Li2+RhH=T,+T+V.+V心Vn2n-e8元8元mmn12
2.1 The Schrödinger equation and its solution for one-electron atoms 2.1.1 The Schrödinger equation The Hamiltonian Operator of one-electron atoms H atom, He+ and Li2+ e n e e n n n e n e V m h m h H T T V ˆ 8 8 ˆ ˆ ˆ ˆ 2 2 2 2 2 2 N e Summary
Considerthat the electron approximately surrounds the atomicnucleus,theHamilton operator canbe simplifiedashH=T+V.n-6n-e08元mVhQ++2n-eayaz8元oxmeZe2十n-4元%TheSchrodingerequationHy=Ey12Separationofvariables
e n e e e n e V m h H T V ˆ 8 ˆ ˆ ˆ 2 2 2 r Ze Vn e 0 2 4 ˆ Consider that the electron approximately surrounds the atomic nucleus, the Hamilton operator can be simplified as r e 2 2 2 r x y z 2 n-e 2 2 2 2 2 e 2 2 Vˆ ) x y z ( 8 m H - ˆ h Separation of variables ? The Schrödinger equation H ˆ E
h?Ze?H=T+V62m4元grSphericalpolar coordinates8元aCayTeWmsine(E+=0Ora0sin0a0sin?adOr4元y(r,0,g)= R(r)0(0)(gSeparation of variables!1 d'Φ1im,dΦ(m, = 0,±1,±2...m0m2元Φdbasin0(0+βsino=m(sin6.(0), β=l(l+1)a0a00(0) dR8元Zem2E+=β一Rn.(r)+h?R drdr4元12Quantum numbers----n,lm
r Ze m H T V e e e n e 0 2 2 2 4 ˆ 2 ˆ ˆ ˆ 2 2 2 2 2 2 2 2 2 2 0 1 1 1 8 ( ) (sin ) ( ) 0 sin sin 4 me Ze r E r r r r r h r q q q q q f 2 2 2 1 ml d d F F f r Ze E h m r dr dR r dr d R e 0 2 2 2 2 2 4 1 8 (r,q,f) R(r)(q )F(f) Separation of variables! Quantum numbers- n，l, ml Spherical polar coordinates ( 0, 1, 2.) 2 1 F ( ) l i m m e l m l f f ( ) ( ) , l l 1 ml l q ， ( ) , R r n l 2 2 m q q q q q q q ) sin ( ) (sin ( ) sin
·ForH andH-likeions.their one-electron wavefunction店(atomicorbital) canbe expressed as(0).Φ(r,0,Φ)=R(r). mn.l.mmiThese are the eigenfunctions ofHamiltonian, following theeigenequationHyn,l.m (r,,0)=Enyn,l.m (r,0,0)with the eigenvalue depending solely on the principalquantum number n, i.e.,Z-h?2Zh-REn222?8元*mnan8元nmaoR= 13.6 eV ---Rydberg constant12
R= 13.6 eV - Rydberg constant • For H and H-like ions, their one-electron wavefunction (atomic orbital) can be expressed as, 2 2 2 2 2 0 2 2 2 0 2 2 2 2 ) 8 ( 8 n Z R n Z m a h m n a Z h E e e n (r ) R (r) n l m n l l , , , ,q ,f ( ,q ,f) ( ,q ,f) ˆ , , , , H r E r l ml n l m n n l These are the eigenfunctions of Hamiltonian, following the eigenequation, with the eigenvalue depending solely on the principal quantum number n, i.e., (q) (f) ml ml l, F
2.2 The physical meaning ofquantum numbers(n, , m)MagneticPrincipalAngularMomentumQuantum NumberQuantum NumberQuantumNumber12
2.2 The physical meaning of quantum numbers (n, l, ml ) Principal Quantum Number Angular Momentum Quantum Number Magnetic Quantum Number