Question

where a = V(2m/h2) (U – E) and k= V2mE/hº is the wavenumber of oscillation in the interior. (b) Show that the energy condition found in (a) can be rewritten as 20. Consider a particle with energy E bound to a finite square well of height U and width 2L situated on -LS x 5 +L. Because the potential energy is symmetric about the midpoint of the well, the stationary state waves will be either symmetric or antisymmetric about this point. (a) Show that for E< U, the conditions for smooth joining of the interior and exterior waves lead to the following equation for the allowed energies of the symmetric waves: k tan kL = a (symmetric case) k sec kL = V2mU Apply the result in this form to an electron trapped at a defect site in a crystal, modeling the defect as a square well of height 5 eV and width 0.2 nm. Solve the equa- tion numerically to find the ground-state energy for the electron, accurate to £0.001 eV.

Answer 1

Potential welll NO 13 VW lo perca For bound state Elvo 1D Schrodinger coam ean den + am | E-vei]4= o 1 1 Exla, il y a exc-4/ -a ah do [E-ve]yto= (D2 - sm (v. - El ] 4 (n20 D² for 102_q²) PW = 0 , 22 a { vo-E] -0 so d² X 2=0 D=74 Apak 4, (n)= Bear x2-0/2 for p.) 2 0/2 i-e ap 24 (0% ole + spony = 0 (02+ B²) 4th=0 .8² mt - D2, B²0 D=+iB Nok our potential is eymmebr to soln will be either even as odd. (4,0%) - Bear -9, Yuna (12 = 31 (2) = ceas (B4) - M Cuch (Beni= A e du 24

Todd(n - 1 4, (n) = - Bedx 2 12 = Dsin(Baj 24th = Akde axa a 24 cal nah - - 2012 - ③ Al -20% -9 Trand continuity condition 4, 1-2/2) = 4, 1-) & Y₂ (92) = 431/2) y (C/2) = 8 (92) PY 1 an) = Y3 (2) for even soll 4 (0%) = Y (2) ACCOM/B) = A é Cx-ß sin(89) = - LA e a ean al this is called B fan Ba = a cendental eyn. (a) ten (B) = (40) - 6 let donleave me vete & R² ma2 ro - 00 lang sepi (vo -2) set meneve me on 972 (20) 2 - R²_2 o eyn let R © antandan) = J pe t y = datan (4h) - 6 & y = JR? <n2 R2- an ² - y2 y²+ 2n² = R² 0 (ean of circle

dotanden) + in Sp? - -- RR soln so 0420277 (one bound state) Lan L34 (Two bound 2h Can LSEL steite) = A 2 (3 boynd eleise) where in called number of bound stare. n2a2 ma²vo 42 In2a2+2 -n=1, 2, 3 .ro [vo= a/n222 +2 ) |- na 1, 2, 3 .. gma2 * an = (net) , no 1, 2, 3 .... (Moof bourd skates 2² = 1n-1) ²2 mazen - 10-) 2q2 En =2(n-1) ²2 22 mar En 2n-1) 292721 ama2

obire som Reblaue a 22 enery where in because you qurehen have widh 22 canto Tranecer derful ear p2 Jun 2(Ba) = 20 a to , for B2 tun? (B1) = 22 +82 11+ tein (BL)] = 22 +82 B2 sec 2 (BL) = 9m [vo - te] B² ece2(BL) = em r = 18 sec / Pl_Imre energy set n=1 in ean (12), with a (3114) 2 x 65X)6342 2 x g. 181031 x 10.11092 - 59.78 741079 sule - 59.787 x 1079 loweed a222 E = m² s JE; = 37.36687 ev