Quantum mechanics II Winter 2011 Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

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

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

Variational principle The trick: Phys 452 Variational principle The trick: Ground state Expectation value on any normalized function y

Quiz 12 Phys 452 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

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

Variational principle application: Phys 452 Variational principle application: In general, we need to calculate: For the kinetic part: Need to calculate: For the potential part: or

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

Variational principle Examples Phys 452 Variational principle Examples x a Hamiltonian: Test function: for elsewhere Infinite square well Result: Exact result

Variational principle About homework: Phys 452 Variational principle About homework: then You need Pb 7.1 (a) (b)

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

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

Variational principle Combining with perturbation theory: Phys 452 Variational principle Combining with perturbation theory: Pb 7.5 For ground state