1. Quantum mechanics II Winter 2011
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. Variational principle
Phys 452
Variational principle
Hamiltonian
Schrödinger
Equation…
… very hard
to solve!
???
Many
particles

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

5. 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

6. 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

7. 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

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

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

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

11. 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

12. 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

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