4. Quantum mechanics Fall 2012
Physics 451
Quantum mechanics
Fall 2012
Karine Chesnel

5. Today: Review - Monday: Practice test
Phys 451
Announcements
Test 1 next week
Mo Sep 24 – Th Sep 27
Today: Review - Monday: Practice test
Be prepared to present the solution of
your chosen problem during class (~ 5 to 10 min)

6. EXAM I Phys 451 Time limited: 3 hours Closed book Closed notes
Useful formulae provided
Review lectures,
Homework
and sample test

7. EXAM I Phys 451 Wave function, probabilities and expectation values
2. Time-independent Schrödinger equation
3. Infinite square well
4. Harmonic oscillator
5. Free particle

8. Phys 451
Review I
What to remember?

9. 1. Wave function and expectation values
Review I
Quantum mechanics
1. Wave function and expectation values
Density of probability
Normalization:

10. 1. Wave function and expectation values
Review I
Quantum mechanics
1. Wave function and expectation values
“Operator” p
“Operator” x

11. What is the correct expression for the operator
Quiz 9a
What is the correct expression for the operator
T= Kinetic energy? A. B. C. D. E.

12. Uncertainty principle
Review I
Quantum mechanics
1. Wave function and expectation values
Variance:
Heisenberg’s
Uncertainty principle

13. 2. Time-independent Schrödinger equation
Review I
Quantum mechanics
2. Time-independent Schrödinger equation
Here
The potential is independent of time
General solution:
“Stationary state”

14. 2. Time-independent Schrödinger equation
Review I
Quantum mechanics
2. Time-independent Schrödinger equation
Function of
time only
Function of
space only
Solution:
Stationary state

15. 2. Time-independent Schrödinger equation
Quantum mechanics
Review I
2. Time-independent Schrödinger equation
is independent of time
for each
Stationary state
where
A general solution is

16. The particle can only exist in this region
Review I
Quantum mechanics
3. Infinite square well
with a x
The particle can only exist in this region

17. 3. Infinite square well Review I Quantum mechanics
Excited states
Quantization of the energy
Ground state a x

18. Review I
Quantum mechanics
3. Infinite square well

19. Quiz 9b The particle is in this sinusoidal state.
What is the probability of measuring the energy E0 in this state?
A. 0
B. 1
C. 0.5
D. 0.3 E. a x

20. 4. Harmonic oscillator Review I Quantum mechanics Operator positionx
V(x)
Operator position
Operator momentum or

21. 4. Harmonic oscillator Review I Quantum mechanics Ladder operators:
Raising operator:
Lowering operator:

22. Quantum mechanics
Review I
5. Free particle
with
Wave
packet

23. Free particle Quantum mechanics Review I Method:
1. Identify the initial wave function
2. Calculate the Fourier transform
3. Estimate the wave function at later times

24. 5. Free particle Quantum mechanics Review I Dispersion relation
here
here
Physical interpretation:
velocity of the each wave at given k:
velocity of the wave packet: