Quantum Theory 2 Chapter 1 - Variational Methods Brief Summary of QT ! TOSE: - h2 74(x,+)+ V(x)4(x, t)= it 24 (x, t) > Normalisation: 14(x +) \2 dx=1 2m dt Time variable: 4(2, t) = 4(x ) e = Et => TISE: - h2 4(x)+ Ve=E4 Y eigenfunction. 2m · Physical observables represented by Hermitian operator : A = A+ Examples 1 Particle in a box 2 H= - h2 d2 -> boundary conditions 4(0) = == 4(L) 2m doc2 0 L 2 Harmonic Oscillator H =- T2 d' + mw2x2 => HYn = En In where En = tw (n +1/2) 2m doc' 3 Hydrogen atom V (x) = - e' => H= - t2 V- e= = > En= - Ry 121 2m 121 * Variational Function E [4] = < 4, H 4 > 1 E [ 4] > Eo , ground state 1411 we study when 114 11 2 Examples 1 Harmonic Oscillator a) 4x(x)= N > normalise Y a > N = 2 x 5 > { [ 4 x ] = < 4x, H 4 x > trial JC2+x2 TT function > Minimise { [4 ] (find derivative) b) 4x (x)= N exp(-x2x2) > normalise 4x > N= Vx > {[4x] = < 42, HY=> T 14 * Minimise E [4x] 2 Hydrogen-like atoms - spherical coords: x= (rsinecose, rsinesind, rcose ) V(r) =- ze r L2 Yum = t2 ((1+1) Yim , LEYum= tom Yum trial functions : Rx (r) = Ne-ar -> N = 2 x 3/2 3 Helium - like system H= - 2 A1 -t2 12 - Ze2 - Ze2 + e2 > same method 2m 2m 1211 1041 104-2021 trial function: Ya (n, r2) = Ne-xr-xrz N=x3 - apply H 1xx-OG21= \n2+12-2rincost · Legendre Polynomials : Pn(x)=1 2nn! dr" 2n+1 d' (x2-1)" > ) -. Pn Pm doc = 2 Onim > Variational functional in this case: {[Ya] = h2x2 - 2 Ze x + 5 e2x m 8
Chapter 2 - Stationary Perturbation Theory Assume we know Hoth = En In° > perturbations: H=H3+ AH, Yri, En unperturbed perturbation * Non-degenerate perturbation theory > multiplicity = 1 HYn= En Yn => (Ho +RHI) (4n°+AYn'+x24n2+.) = (En+) En + X2 En+) (4nÂș + +++++++++.) compare coeff : 1°: Hotn° = En UnÂș N': Ha Un' + H, Un = En In+ En' Un (Ho-EnÂș) 4r' = - (H1-E0) Un° AK: Ho4n"+ H.4K-n = En° Unk + En'Unk-/+ En24n=+.+ EnKY LHS => (Ho - En) Ink= { En Ynk-1 - H1 4n *** <Yn, (Ho-En) Yn">= 0 RHS : E En' <Yn, Unk-> _< 4nÂș, H, 4nk-> => Enk = < 4n°, H,Ynk-1>- [En]<4nÂș, Unk-> K -! JEI To find En' = < 4n°, H,Un>, Un=4n°+14n + Q(x2) <4n, 4n°>= 1 note (YnÂș, un'> = 0 => Un'= { {4m2, H. YnÂș > 4m => En2 = [ |<4m, H. Yn°>\2 Min EnÂș- EmÂș Min En -Em Examples 1 Anharmonic Oscillator H =- h2 d2 + mw2x2 +1x 4