hole h vs. e configurations: l l for N > 2 l + 1 J = H as example of localization, delocalization, tunneling ikx k

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1 Infinite 1-D Lattice CTDL, pages LAST TIME: ( ) ( ) + N + 1 N hoe h vs. e configurations: for N > + 1 e rij unchanged ζ( NLS) ζ( NLS) [ ζn unchanged ] Hund s 3rd Rue (Lowest L - S term of Zeeman Effect Wigner-Ecart Theorem used to define g J E Zeeman = µ M g B J J z ony) N < +1 E for J = L S reguar N = +1 MIN ( + 1)+ 1 N > +1 E for J = L+ S inverted MIN ( ) + ( + ) ( + ) JJ+ 1 SS 1 LL 1 g J = 1 + JJ ( + 1) Confirm by H Zeeman in Sater determinanta basis TDAY: S N + 1 J = S state: no fine structure eua spacings g J as L,S,J diagnostic next ecture 1. H as exampe of ocaization, deocaization, tunneing +. secuar euation for simpified 1- D attice 3. eigenvectors by eua probabiity tric 4. restrict to <π : 1st Briouin Zone ( ) = ( ) 5. EE Acos a of the aowed states? ix 6. Boch functions ψ ( x) = e u ( x) 7. wavepacets,motion, group veocity 8. transitions energy bands and intensity profies 9. conductivity updated September 19, :

2 37 - Start with H +, a attice with ony euivaent sites. uaitative picture: atomic energy eves tunneing between identica ocaized states sow behind big barrier (sma spitting) fast behind sma barrier (arge spitting) eves bands, of width reated to tunneing rate H + R + R x ( ) E n +1 ( ) E n atoms states R >> a E owest = R and douby degenerate 1 for exact degeneracy, can choose any inear combination () Locaized basis set ψ = ψ or ψ Deocaized basis set ocaized deocaized () eft right () () [ eft right] 1 / ψ = ψ ± ψ updated September 19, :

3 If initiay in ocaized state, tunneing rate depends on ( ) () * height reative to E of barrier * width of barrier n () * size of overap between exponentia tais of ψ and ψ eft () right 37-3 cear that tunneing rate (i.e. spitting) increases * as n at constant R (internucear separation) * as R at constant n E R E 1 doube degeneracy at R is tunneing spitting gets arger as R N ATMS ALNG A STRAIGHT LINE wide narrow N atoms, N states each eectronic state of isoated atom becomes band of states for attice. Energy width of each band increases as the principa.n. increases because atomic states reuire more room: r n a n. Tunneing gets faster. Greater sensitivity to word outside one atom. updated September 19, :

4 37-4 Simpified Mode for 1 Dimensiona Lattice: basis for uaitative insights and eary time predictions. 1. Each ion, caed, has one bound state, ν at E = ν Hν [diagona eement of H] (actuay spin-orbitas). permit orbitas ony on adjacent ions to interact [simpifying assumption] ie Hüce theory. 3. symmetry: a ions are euay spaced, x +1 x =, and a adjacent-orbita interaction matrix eements are identica so ν Hν +1 Α [off-diagona eements of H] (IAI woud increase as ) [reasons for A sign choice ater.] E A H = A E A A tridiagona infinite matrix since this is infinite, need a tric to diagonaize it. genera variationa function ϕ = c = ν superposition of A s at each site get reuirements on c by pugging this into Schrödinger euation H ϕ = E ϕ eft mutipy by ν updated September 19, :

5 37-5 th position LHS ( 1 ) pics out -th row of H ν H ϕ = E ν ϕ [ ]= [ ] RHS E ν E c ϕ ( AE A ) c M M c + = Ac + E c Ac = c [ E E] c A c A comes from the assumed simpe form of mode TRICK: probabiity of finding e on each attice site shoud be the same for a sites (compex ampitudes might differ but probabiities wi be constant) et c = e i c = 1 for a This choice of c is a good guess that is consistent with expectation of eua probabiities on each attice site. is distance between adjacent atoms is integer is the coordinate of the -th site: oos ie e ix pane wave is of dimension 1 probem reduces to finding aowed vaues of. periodicity of attice provides the important resut that if is repaced by, where = + π, the wavefunction does not change (transationa symmetry) updated September 19, :

6 37-6 π i + i i i iπ c = e = e = e e{ = c = 1 Since a distinguishabe ϕ may be generated by choosing in the interva π < π, restrict to this range: caed First Briouin zone. Return to uestion about what happens when is not in 1st Briouin Zone next time [get another part of the band structure using uaitative perturbation theory rather than a matrix diagonaization cacuation]. Pug i c = e into Schrödinger Euation = c ( E E) A c + c divide by e i and rearrange [ ] E = E A1e 4+ e4 3 E = E Acos ( ) ( ( ) ( ) ) = e ( E E) A e + e i i 1 i 1 ii This is the condition on E, that must be satisfied for a eigenfunctions of cos the Schrödinger euation updated September 19, :

7 37-7 E varies continuousy over finite interva E ± A E() E E A E + A π +π The choice ν H ν + 1 = Aeads to minimum E at =. Are these a of the aowed energy eves that arise from a singe orbita at each attice site? Apparenty not see next time. ny haf of the states. [ne orbita per atom two spin-orbitas per atom. Antisymmetrization gives another separate band.] Coud repeat cacuation for a higher energy state at each site. Woud get a broader band centered at higher energy. updated September 19, :

8 37-8 coser oo at spatia form of ϕ ( x) x ϕ goa is to repace infinite sum by singe term: + i ϕ( x) = x ϕ = e x ν 13 = ν ( x) show that: ϕ This is caed a Boch function ( )~ ( ) ix x e u x pane wave (Free partice) periodicity of attice begin by reuiring that ϕ i ( x) = e ν ( x) = Transationa symmetry imposes a reationship between ν (x) and ν each ν (x) is ocaized at site. ν ( x) = ν x i ϕ ( x) = e ν x ( ) = ( ) ( ) i ϕ( x+ ) = e ν x = = ν ( x ( 1) ) ( ( ) ) i i ( 1) = e e ν x 1 shift x by to get from site to site updated September 19, :

9 37-9 re-index sum (repace 1 by ) ϕ i ( x+ ) = e ϕ ( x) transation by! This form of φ has a of the symmetry properties we wi need. This form is sufficient to satisfy the symmetry reuirements (boundary conditions). This means, instead of writing ϕ ( x) as sum over atom - ocaized ν ( x) s, it is possibe to write ϕ ( x) as product of factors ϕ ix x = e u x ( ) ( ) 1st factor conveys transationa symmetry of a pane wave with wavevector, nd factor buids in transationa symmetry of attice with spacing. This is a more genera expression that incorporates a of the properties of the origina definition of ϕ (x) as a sum over ocaized orbitas. note that u x+ u ( x) ϕ ( ) = ( ) = + ix i iix x+ e e u x e e u ( x) ( ) = [ ] = e i ϕ ( x) as reuired. Note aso that Iϕ (x + n)i = Iϕ (x)i impies that, as reuired, e has eua probabiity of being found on each site. updated September 19, :