Schrodinger Equation H≡E Wavefunction Hamiltonian H=2(h2/2mav 2-(h2/2me)2Vi e ∑∑a Celtic Energy +∑∑e2/r Contents 1. Variation method 2. Hartree-Fock Self-Consistent field method
H y = E y SchrÖdinger Equation Hamiltonian H = (-h 2 /2m ) 2 - (h 2 /2me )ii 2 + ZZe 2 /r - i Ze 2 /ri + i j e 2 /rij Wavefunction Energy Contents 1. Variation Method 2. Hartree-Fock Self-Consistent Field Method
The variation method The variation theorem Consider a system whose hamiltonian operator H is time independent and whose lowest-energy eigenvalue is E. If o is any normalized, well behaved function that satisfies the boundary conditions of the problem, then φHφdτ>E
The Variation Method Consider a system whose Hamiltonian operator H is time independent and whose lowest-energy eigenvalue is E1 . If f is any normalized, wellbehaved function that satisfies the boundary conditions of the problem, then f * H f dt > E1 The variation theorem
00 Expand o in the basis set ( vr k kYk where fa, are coefficients k= Ek the jo o dt=2,E.osy - EkIaklek2exklal=er Since is normalized,∫φpdr=ΣαP=1
Proof: Expand f in the basis set { yk } f = k kyk where {k } are coefficients Hyk = Ekyk then f * H f dt = k j k * j Ej dkj = k | k | 2 Ek > E 1 k | k | 2 = E1 Since is normalized, f *f dt = k | k | 2 = 1
o: trial function is used to evaluate the upper limit of ground state energy El φ= ground state wave function, JφHψdτ=E1 iii. optimize paramemters in q by minimizing ∫φHφdτ/∫φφdτ
i. f : trial function is used to evaluate the upper limit of ground state energy E1 ii. f = ground state wave function, f * H f dt = E1 iii. optimize paramemters in f by minimizing f * H f dt / f * f dt
Application to a particle in a box of infinite depth Requirements for the trial wave function 1.zero at boundary ii. smoothness a maximum in the center Trial wave function: o=x(L-X
Requirements for the trial wave function: i. zero at boundary; ii. smoothness a maximum in the center. Trial wave function: f = x (l - x) Application to a particle in a box of infinite depth 0 l