1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Phys 452 Homework • 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. Phys 452 Solution Schrödinger equation Variational principle Hamiltonian H system • Energies • States

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

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

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

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

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

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

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

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

12. 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

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

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

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

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