1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Homework Phys 452 Thursday Feb 9 Assignment # 8: 7.1, 7.3, 7.4, 7.5 Quiz “4”: points of test prep (quiz entries are shifted from quiz 4 to quiz 9 - having been entered as quiz 5 to quiz 10)

3. Variational principle Phys 452 system Hamiltonian H Schrödinger equation Solution Energies States

4. Variational principle Phys 452 Examples of problems solved One particle Harmonic Oscillator x V(x) Schrödinger equation Solution x V(x)

5. Variational principle Phys 452 Examples of problems solved One particle Infinite Square well x 0a Schrödinger equation Solution Ground state Excited states

6. Variational principle Phys 452 Free particle Examples of problems solved One particle Schrödinger equation Solution

7. Variational principle Phys 452 Examples of problems solved One electron Hydrogen atom Schrödinger equation Solution

8. Variational principle Phys 452 Fine structure Zeeman effect Hyperfine split Perturbation theory Examples of problems solved One electron Solution

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

10. Variational principle The trick: Phys 452 Ground state Expectation value on any normalized function  Derivation:

11. Quiz 11 Phys 452 What is the variational principle based on? A. the Schrödinger equation B. the first –order perturbation theory C. the uncertainty principle D. the quantization of the energy E. the fact that excited states are energetically higher than ground state

12. Variational principle The method: Phys 452 Define your system, and the Hamiltonian H Pick a normalized wave function  Calculate You get an estimate of ground state energy Minimize

13. Variational principle Examples Phys 452 x V(x) Harmonic oscillator Hamiltonian: Test function: Result:

14. Variational principle Examples Phys 452 Delta-function Potential Hamiltonian: Test function: Result: Exact result

15. Variational principle Examples Phys 452 Infinite square well Hamiltonian: Test function: Result: x 0 a Exact result

16. Variational principle About homework: Phys 452 In general, we need to calculate: For the kinetic part: Need to calculate: For the potential part: or