Schrodinger Equation in 3-d

Transcription

1 Schrodingr Equation in 3-d ψ( xyz,, ) ψ( xyz,, ) ψ( xyz,, ) Vxyz (,, ) ψ( xyz,, ) = Eψ( xyz,, ) m x y z p p p x y + + z m m m + V = E p m + V = E E + k V = E

2 Infinit Wll in 3-d V = x > L, y > L, z > L3 x, y, z < 0 V = 0 for 0 < x < L, 0 < y < L, 0 < z < L. for or. 3 ψ( xyz,, ) = ψ ( x) ψ ( y) ψ ( z) 3 ψ ( xyz,, ) = Asin( kx) sin( ky) sin( kz) 3 Boundary Conditions rquir that. E k nπ nπ n3π =, k =, k3 = L L L Enrgy ignvalus ar. nnn 3 If th wll is cubical,, E ( ) m k k k n n π = = + + m L L L = L = L = L 3 π = ml n n n + + ( ) thn and. nnn 3 So this ignvalu has a thr-fold dgnracy. 3 E = E = E 3 n L 3 3

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4 Schrodingr Equation in Sphrical Coordinats Why? Som potntial nrgy functions ar radial (.g. Coulomb potntial). z = x = y = rcosθ rsinθcosφ rsinθsinφ dτ = dv = dxdydz dτ = ( rsin θ dφ)( r dθ)( dr) dτ = r sinθ dr dθ dφ

5 For a potntial nrgy function that dpnds only on r: + + = m r r r ψ θ ψ ψ sin V() rψ r m r sinθ θ θ sin θ φ Eψ This looks nasty, but it had bn wll studid vn bfor Schrodingr wrot it down, so solutions ar known for various potntial nrgy functions. Assum that th solutions ar sparabl (turns out to b a good assumption): ψ ( r, θ, φ) = R( r) f ( θ) g( φ) Substituting this back into Schrodingr s quation givs: d [ ] Rr dr r dr () r mr d df ( θ) d g( φ) + E V() r = sinθ () dr f ( θ)sinθ dθ dθ g( φ)sin θ dφ Sinc th lft sid is only a function of r and th right sid is only a function of angls, both sids must qual a constant, which w will call ( + )..

6 d [ ] Rr dr r dr () r mr + E V() r = ll ( + ) () dr h f d df ( θ) d g( φ) sinθ = ll ( + ) ( θ)sinθ dθ dθ g( φ)sin θ dφ Sinc th potntial nrgy function is only in th radial quation, th angular part of th wav function is th sam for all potntials (just lik th tim part arlir). A radial potntial nrgy function affcts only th radial part of th wav function. Rarranging th angular quation to sparat thta and phi givs: g( φ) d g ( φ) dφ = ll ( + )sin θ sinθ d df ( θ) sinθ f ( θ) dθ dθ Again, sinc only phi appars on th lft and only thta on th right, both sids must qual a constant which w will call -m.

7 g( φ) d g ( φ) dφ = m ll ( + )sin θ sinθ d df ( θ) sinθ = m f ( θ) dθ dθ Th solutions to th angular part ar wll known and ar calld Sphrical Harmonics. g f m lm, ( φ) = imφ m (sin θ) d ( θ) = l l! d(cos θ) l+ m Y f g l lm = lm m ( cos θ ) Boundary conditions rquir that l = 03,,,,... and m = 0, ±, ±,... ± l. ( θ, φ) ( θ) ( φ)

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9 Angular Momntum ψ ψ ψ Vψ = Eψ p + m x y z m V = E + + = m r r r ψ θ ψ ψ sin V() rψ r m r sinθ θ θ sin θ φ p m pm V p L r t r + + = + + V = m mr E Eψ whr L is th angular momntum. So th oprators for th angular momntum squard and th z-componnt of momntum ar ( L ) op ( L ) z op = sinθ ψ + sinθ θ θ sin θ φ = i φ ψ Our radial wav functions (sphrical harmonics) ar ignfunctions of ths oprators, so w can asily valuat th xpctation valus for L and L z. It turns out that both ar quantizd!

10 ( L ) L opψ = ψ ( L ) op R( r) Y m ( θφ, ) = R( r)( L ) op Y m ( θφ, ) ( L ) Y ( θφ, ) = L Y ( θφ, ) op m m ( L ) opym ( θφ, ) = ( + ) Y m ( θφ, ) L = ( + ) So angular momntum is quantizd using th intgr, which w will now call th angular momntum quantum numbr or orbital quantum numbr. Allowd valus ar = 03,,,,... ( L ) Lψ ( L ) R( r) f ( θ) g( φ) = R( r) f ( θ)( L ) g( φ) ( L ) g( φ) = L g( φ) imφ i = m φ L z ψ = z op z z op z op z op z = m whr m = 0, ±, ±,... ±. imφ

11 Angular Momntum is spac quantizd!

12 + + = m r r r ψ θ ψ ψ sin V() rψ r m r sinθ θ θ sin θ φ Eψ d + + m r dr r dr () r ( ) dr m r Rr () + V() rrr () = ERr () Now w can mov on to th radial part of th wav function. Th solutions to this part dpnd on th potntial nrgy function. For an lctron in a bound orbit around a nuclus of charg +Z, th potntial nrgy function is th Coulomb potntial, V() r = kz r.

13 d = m r dr r dr () r kz ( ) Rr () ERr () dr r m r E n = Z n 4 k m = Z n E

14 Th solutions to this quation hav th form: R nl () r = with th rstriction that < n. a0 r L ran / r 0 nl a 0

15 For hydrogn and hydrogn-lik atoms, th nrgy is dtrmind only by n, th principal quantum numbr. Th angular momntum is dtrmind by, th angular momntum or orbital quantum numbr. Th diffrnt valus of hav bn givn cod lttrs: = 0 S ( Sharp) = P (Pr incipal) = D ( Diffus) = 3 F ( Fundamntal) = 4 G Slction ruls for lctric dipol transitions (incl. optical): = ± m = 0 or ±

16 Th full spatial wav functions for hydrogn-lik (singl lctron) atoms ar ψ (, r θ, φ) = C R () r f () θ g () φ nlm nlm nl lm m Th constant C is th normalization constant for ach stat and is found using π π ψ * ψ dτ = ψ * ψ r sinθ dφ dθ dr = Howvr, it is convnint to normaliz ach sparabl function: ( )( ) R *( r ) R ( r ) r π π dr Y *( θφ, ) Y ( θφ, )sin θdθdφ = Notic th normalization constants for Y and R as listd in Tabl 7- and 7-. W can dtrmin th probability of finding th lctron btwn r and r+dr as Prdr () = R*() rrrrdr () and th probability of finding th lctron btwn r and r is r r Prdr () = R*() rrrrdr () r r

17 Th wav function squard and th radial probability distribution for th hydrogn ground stat:

18 Th radial probability function for th first xcitd stat of hydrogn: Notic that th largst angular momntum corrsponds to smallr xpctation valu for r. Also notic that th lowr angular momntum stat spnds mor tim closr and mor tim farthr from th nuclus than th largr angular momntum stat.

19 Plantary orbits: Ths orbits all hav th sam nrgy. Th mor lliptical orbits hav lowr angular momntum but spnd mor tim both closr and farthr from th sun than th high angular momntum orbits.

20 Hydrogn atom wav functions

21 Orbital Angular Momntum L = ( + ) = 03,,,,...,n m L z = m = 0, ±, ±,... ± Magntic Dipol Momnt µ = = + = + µ m L ( ) ( ) B m µ z z µ B m L m = = = m m Magntic momnt is also quantizd.

22 Elctron Spin Elctrons hav an intrinsic angular momntum (protons and nutrons, too). It s as if thy wr small sphrs spinning vry quickly. S = s( s + ) = 3 4 Sz = ms s = m s = ± Spin Magntic Dipol Momnt µ = = µ m gs 3 g s 4 s B Th g-factor is not on if th charg distribution is not th sam as th mass distribution. µ = z gs s z mg s sµ B gsµ B m = = ± g s = Hydrogn Wav Functions: ψ nmms Hydrogn Ground Stat: ψ = ψ + ψ

23 Strn-Grlach Exprimnt F z = µ ( db/ dz) z Silvr atoms ar known to hav only on unpaird lctron, which is in an stat. = 0 So Strn and Grlach xpctd to s on lin. Instad thy saw two lins (c) bcaus of th two spin stats of th lctron.

24 Total Angular Momntum of ach lctron is mad up of two contributions, orbital and spin. j J = + = j( j + ) s or j = s J = L + S J m z = mj = j, j +,..., j, j j

25 Spctroscopic Notation For singl lctrons: n = K Shll Principal quantum numbrs corrspond to shlls n = L Shll n = 3 M Shll Sobr Physicists Don't Find Giraffs Hiding In Kitchns Lik Min = 0 S ( Sharp) = P (Pr incipal) Th diffrnt valus of hav bn givn cod lttrs: = D ( Diffus) = 3 F ( Fundamntal) = 4 G

26 Spctroscopic Notation For dscribing th stat of a particular lvl in an atom which may contain mor than on lctron: n s+ whr w substitut th appropriat cod lttr for P is its an S or D or tc... P j So, th ground stat of hydrogn is S / and th ground stat of hlium is S. o Slction ruls for lctric dipol transitions (incl. optical): = ± m = 0 or ± j = 0 or ±

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28 Spin Orbit Coupling Bcaus th lctron is orbiting th nuclus, or from th rfrnc fram of th lctron, th nuclus is orbiting th lctron, th lctron is in a magntic fild. Th intraction of th lctron s magntic momnt (orbital and spin) with th magntic fild contributs to its nrgy. U = µ B U V For hydrogn, U 0 3 V for sodium

29 Two or mor idntical particls Th wav function for th systm bcoms ψ ( r, r ) opψ = ψ and th Schrodingr quation bcoms H ( r, r ) E ( r, r ) For on dimnsion: ψ( x, x) ψ( x, x) + V( x, x) ψ( x, x) = Eψ( x, x) m x m x V( x, x ) = V ( x ) + V ( x ) + V ( x, x ) whr. V( x, x) = V( x) + V( x) thn th wav functions will b ψ ( x, x ) = ψ ( x ) ψ ( x ) If th two particls ar non-intracting thn ψ ( x) and ψ ( x) nm n m whr ar two of th on-particls solutions for this potntial. n m To normaliz th wav function of th systm: = ψnm = ψn ψm ( x, x ) dx dx ( x ) dx ( x ) dx

30 Th probability of finding particl btwn x and x + dx and finding particl btwn x and x + dx is probability = ψ ( x, x ) dx dx nm Sinc th particls ar idntical, th probability should b th sam if w xchang thm: ψ( x, x ) = ψ( x, x ) ψ ( x, x) = + ψ ( x, x) ψ ( x, x ) = ψ ( x, x ) Which mans that ithr Symmtric or Anti-symmtric So th gnral solutions for th systm hav th forms [ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )] ψ = Cψ x ψ x + ψ x ψ x S n m n m ψ = Cψ x ψ x ψ x ψ x A n m n m Symmtric Anti-symmtric But if n=m, thn th anti-symmtric wavfunction is zro and not a good solution. Elctrons, protons and nutrons ar all frmions and must hav anti-symmtric total wav functions. Thrfor, no two lctrons in on systm can b in th sam stat. Thy must all hav distinct quantum numbrs. This is calld th Pauli Exclusion Principl.

31 Ground Stats of Multi-lctron Atoms For any atom othr than hydrogn, th nutral atom has mor than on lctron. Th lctrons intract to chang th solutions to Schrodingr s quation and th nrgy lvls. W hav to start with th on lctron solutions and us approximations for th ffcts of th othr lctrons. Gnrally, th ffct of intractions btwn lctrons is to incras th nrgy lvls sinc th forc is rpulsiv and th potntial nrgy is thrfor positiv. Also, th largr th valu of n and l, th largr th ffct of intractions with othr lctrons. For th sam n valu, stats with lowr l will spnd mor tim clos to th nuclus and thrfor b bound tightr.

32 Hlium (Z=) Both lctrons ar in a s stat (nlm=00) but hav diffrnt spin quantum numbrs, m s. If thr wr no intraction btwn th lctrons, th nrgy of th systm would b E Z E0 Z E0 E0 E0 = = = 8E0 = 088. n n Th wav function for th ground stat of hlium would b ψ ψ ( r, θ, φ ) ψ ( r, θ, φ ) = Th potntial nrgy du to th forc btwn th two lctrons is. V. int = k r V r Using this intraction potntial and th wav function solution for th non-intracting cas, w can calculat an approximat valu for Vint = +34 V, so E 09 V + 34 V = 75 V. Z E0 E0 E = = = 4E0 = 544. n Th nrgy for th H+ ion is. So th ionization nrgy (th nrgy rquird to rmov on lctron from nutral hlium) should b ~ V. It s actually 4.5 V sinc th nrgy of th nutral atom is ~-79 V. Th ground stat configuration is writtn as s. Ionization nrgis and ground stat lctron configurations ar givn in Appndix C. V.

33 Lithium has two lctrons in a s stat (K shll) but must hav th third lctron in th L shll (n=). Sinc th s stat is lowr in nrgy than th p, th third lctron of Lithium is in th s lvl so th lctron configuration is s s. Lithium has an angular momntum of sinc it has zro orbital angular momntum but has on unpaird spin. / Boron to non (Z=5 to Z=0) fill th p orbitals. Sodium and magnsium (Z= and Z=) fill th 3s stats. Aluminum to argon (Z=3 to Z=8) fill th 3p stats. Potassium and calcium (Z=9 and Z=0) fill th 4s stats rathr than th 3d. Scandium to Zinc (Z= to Z=30) fill th 3d stats and ar calld transition mtals. Notic that filld shlls man larg ionization nrgis and small atomic radii.