Question

1. Explain the variational principle and illustrate it with some example (different from the one in the following point) 2. A trial function for ls electron in hydrogen atom has a form of (r) = e-ara. Derive the nor- malization constant. Explain the difference between this trial function and the true l-electron hydrogen-like orbital. 3. The expression for energy as a function of a for the H-atom using above trial function is given by: E(a) = 3h2a 2me e2a1/2 21/26073/2 Derive a that minimizes the energy and confirm that the variational principle holds. 4. Explain the perturbation theory and illustrate it with some example. 5. Using first-order perturbation theory, calculate the energy of H-atom in an electric field of of strength ε. The Hamiltonian operator for this system is given by: W = - h2 2me + eer cos e 47€or Two useful relations: 20 = 47 €0h2 mee2 mee4 ΕΗ 1672 ezh2

Answer 1

ed 1/2 3) The expression of the energy is given by 3thoc E(&) 22 Co T312 for, minimize the energy then we have me de 372 2-1/2 е da =0 ame 2312 € 312 2-12 ed 2 312 67 3/2 352 2me

This gives - x 112 = x-1/2 = 3 (20) 3/2 tot 2 2 2 meet 2 a = 2 La => a= amel 3 (217) 3/2 to t² 4 met 9/21)3 624 This is the value of a to a to minimize the energy. 4 If we disturb the system by some amount (very little), then the Hamiltonian will cha change with disturbance. That disturbance acts as perturbed Hamil- -tonian and added to the renperturbed one. If Ho as the unperfurbed Hamiltonian and it is the small disturbance to the system then the perturbed Hamel- tonian is geven by respect to the H = 4-6 + H. This change in Hamiltonian will head to the change in Wavefunction corresponding to the system and the energy of the system will also change a accordingly. This is perturbation theory in short, ar acc