Quantum Theory 2 . Variational functional method { [4] = < 4, H 4 > 1 Find 114 112 114 112 2 Hamiltonian, HY sometimes calculate derivatives separately 3 < 4, HY > 4 then divide by 114/12 for E[4] = < 4, HY> . Find the best upper bound for the ground state energy - Emin 1 Derive e.g & (E [ 4]) = 0 1x 114 11 2 2 Equate to variable e.g do 3 Substitute into [4] as Emin= [4×0] · Integrating spherical co-ords J . J . J . 4 r2 dr.sin @ dodd = 4T J : 4 ,2 dr = 4TT · Hamiltonian operators · harmonic oscillator, H =- 2 22+ /x2 (HD) · box H= h2 (2 + 22) H =- h?2+Ex 2m ? Spherical coords, H =- hz (/ or(~2 d)+ i'm at2 - h2 V(r) if inclep . of 0, 0 then L' = 0 Eigenvalue eq: HY=EY , HYn°=Enº 4n° > energy eigenvalues: En' = < 4n° H, Un> = )= In° H, Un dx · Raising and Lowering operators La En = En°+1En' Hi could be perturbation. dy D+= d = y, y=x, x= [mw1 -> Ho =- hw(D+D-1) -> if Ho =x = yx === (DI-D+) D+4nº=\2n+2 Unt, D- 4n°= Van Un°, D- 4% = 0 => x4m=(van 4h-,-/20+2 4n+i) · Hydrogen-like atom - 4: (r)= x" NTT 3/2 e-ar , x = Z Mez · Second order perturbation: En2 = { \<4m, H, 4,0>|2 · Spin operators min [S], Sk]=it €jk, S. jKLE {1,2,3} Enº - Emº S = (S1, S2, S3 ) , S+ = S, +iS2 > expected values: < ,>= t Re(2(+)}(+) · Reduced radial eq "(r ) + ( k 2 - (+) U (r)= 0 · Total cross-section& amplitude { S2 > = h 1m (x (+)}(+) ) <S3>=?(1x(t)12-1?(+)(2) 1 = 0 O (K ) = 4TT [ ( 21 + 1) sin' di