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

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)

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

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

Phys 451 Review I What to remember?

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

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

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.

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

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”

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

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

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

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

Review I Quantum mechanics 3. Infinite square well

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

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

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

Quantum mechanics Review I 5. Free particle with Wave packet

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

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: