Physics 452

1. Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

2. Phys 452 Homework • Thursday Feb 9 • Assignment # 8: • 7.1, 7.3, 7.4, 7.5

3. Phys 452 ??? Schrödinger Equation… … very hard to solve! Variational principle Hamiltonian Many particles

4. Phys 452 Ground state Expectation value on any normalized function y Variational principleThe trick:

5. Phys 452 Quiz 12 What do we need to know, essentially, in order to use the variational principle? • The Hamiltonian of the system • The Hamiltonian, its eigenstates and eigenvalues • The Hamiltonian and the ground state • The Zero-order energy in the perturbation theory • The First-order energy in the perturbation theory

6. Phys 452 • Calculate • Minimize • You get an estimate • of ground state energy Variational principleThe method: • Define your system, and the Hamiltonian H • Pick a normalized wave function y

7. Phys 452 In general, we need to calculate: For the kinetic part: Need to calculate: For the potential part: or Variational principleapplication:

8. Phys 452 Hamiltonian: Test function: Result: Exact result Variational principleExamples Delta-function Potential

9. Phys 452 Hamiltonian: for Test function: elsewhere Result: Exact result 0 a x Variational principleExamples Infinite square well

10. Phys 452 You need then (a) (b) Variational principleAbout homework: Pb 7.1

11. Phys 452 Delta-function potential Trial function: Triangular Build Build Build Then, calculate: and Variational principleAbout homework: Deal with discontinuities Pb 7.3

12. Phys 452 First excited state Expectation value on a normalized function y that is normal to ground state Derivation: To choose : symmetry considerations Variational principleFor the first excited state: Pb 7.4

13. Phys 452 For ground state Variational principleCombining with perturbation theory: Pb 7.5