Model QM Problems with Exact Solutions - Lecture Notes | CHEM 542, Study notes of Quantum Mechanics

Download Model QM Problems with Exact Solutions - Lecture Notes | CHEM 542 and more Study notes Quantum Mechanics in PDF only on Docsity! Lecture2A--Model QM Problems with Exact Solutions – (1-D) (Ch 2.2-Levine, 3-3 Atkins, Ch. 2-R&S) 1. Free Particle -- If there is no potential then Schroedinger Equation becomes T (x) = E (x) ==> -(h2/2m) d2/dx2 (x) = E (x) for this section: let underline "h" be "h-bar": h = h/2 easiest solution to this has exponential form: (d/dx)ex = ex but this has only one eigen value 1 and can’t well cope with eigen value could use = e-kx then k =(2mE/h2)1/2 but would get wrong sign so need e-ikx recall: e+ikx = cos kx + i sin kx These solutions are plane waves INSERT GRAPH starting point is due to phase (balance between sin & cos—arbitrary) but don’t measure Ψ just |Ψ∗Ψ| —thus phase not significant (normally) This describes motion of a free particle no potential —> no force (F = -dV/dx)) – continuous motion [Note these are eigen functions of momentum p ± = (-ih) d/dx[e ±ikx] = ±hk ±] What is it doing? e±ikx --> motion to the pos x --> <p> = +kh e-ikx --> motion to the pos x --> <p> = -kh 2 solution: Ψ± both solve the problem. These have different eigen values of momentum but different of energy depend on preparation—initial state cos kx also an eigen function T = Tcos(kx) = -k(-h2/2m)cos (kx) = E but p cos(kx) = -k(-ih)sin(kx) not eigen function so real component, cos(kx), of e±ikx has energy but ill defined momentum – real wave function—represents motion left (-) and right (+) --complex wave function—well defined linear momentum (-) or (+) Note if energy is not well defined – then k varies - means wavelength varies get wave packet -- super position of these with constructive interference -- if enough waves interfere see particle with some position, finite ∆x Time evolution: (x,t) = Ae-ikx e-Et/h = Aei[kx + (k2h/2m)t] evaluate at different times - point of constructive interference changes – phase shift *try it with graphing calculator or program 2. Particle in a box with infinite sides—restrict motion - contain with V(x) V = 0 à 0 < x < L V = all else (x < 0, x > L) INSERT GRAPH Region I, III H = E --> -(h2/2m)(d2/dx2) = (E - ) so particle of finite energy has no amplitude in this region Ψ (x)=0 (outside box) special result due to V(x) = outside (impenetrable) only need consider Region II hence locally V = 0 – just as above for free particle Ae-ikx but V(x) provides restrictions on motion (in box) so that leads to quantized behavior B.C. – boundary conditions, choose : = A cos kx + B sin kx At wall (0) = (L) = 0 – must be a continuous, finite function Ψ(0) = A sin kx (cos x 0 at 0 = x) Ψ(L) = 0 ==> k = (nπ/L) where n = 1,2,3. . .integer, (n = 0 not allowed) Energy is quantized by the B.C. (work it out) k = (2mE/h2)1/2 = n /L ===> En = n 2 2h2/L22m Note: still need Aeiφ, amplitude and phase-- Normalize: n* ndx = 1 ==> A = (2/L) 1/2 ===> n (x) = (2/L) 1/2 sin (n x/L) Important to see what happens as modify B.C. – as box enlarges Ψ toward free particle, i.e. the energy levels become continuous – smaller box more energy goes as ~1/L2 – note box constrains motion more, Ψ has more curvature, second derivative is curvature, |T| inc. with curvature – energy level separation expand with n2 -- property of steep sides What are they? H-atom--example more useful for smaller masses--eg. electron If spectral transition: h = E = En - En' - change between levels But—lower mass—electron~*2000 increase Energy over atom ---lower size--1Å (atomic) ~*100 inc Energy over 10Å (molecular)