Question

1. Give the Schrodinger equation for free electrons. Explain all terms: Hamiltonian, potential. 2. Solve the Schrodinger equation and find the wavefunction and the energy. 3. Draw the energy dispersion.

Answer 1

Schroding or eques written as - Ĥ 4 = E4 where. Å & ft Hamiltonian operator E energy of the system Î = 2 → Kinetic energos perrt. potential energy part îcit momentum operator in position representation 62 am <? a ti D2 2 m 2m :: Schrodinger equ t 2 m 74 +v4 = E 41 for a free Dartide, V=0 So. Sehrotinges laws of → potential vanishes. Free eletron, a 2 pe 4 E 4 0 am E 124 + 2 => Answer take, am K² 42 we so from ejus. Ci . get. 724 +K²4 =0 cii) Let, 4 ) = 4(2,4, +) x(x) Yly) Z (2) using the substitution from eqin chil we reget, gex 2²2 1 ay - + Kºco dya now, k²-kt ky +k2 + + Dx2 Z a z² take the depart only deg tka a x dae a dad + ka x = 0

X = A e-ikaa xrormalisation, nofieiton) (actory, de of TT Jet x* x da "Ox=d. =Y A = att A 12 - ikxx e 1 VOT Varm Z = e é ikyt 30, similarly, y ikzz Van .. 4 (9) -i (kg x + ky 4 + kz z). e (2) 342 + time independent > 40). eilano) (2)/2 have fo. if eve take time dependent sof energy I=tw / كه So ware of 4t) = 4 (rot) - 4(%) et CEGA e-ilk-sónar) (27) 3/2 time dependent wifi And from the assumption we regel, Some per te

tdk? 7 E am Broglie Lajpother lexis again from de b 스 2 IT d ay p=tk. d tek 212 pe am Lu Answer) 2 m energy energy dispaansion grapher, E k t² k² gives a parabolar 2m