2. Wavepackets etc. Plane wave eigenstates: scattering states non-normalizable continuous spectra x Free particle

3. Wavepackets withNormalization: Linear superpositions of plane wave solutions

4. Example 1 Unfortunately, the integrand is badly behaved and difficult to evaluate even numerically.

5. Propagator is a free particle propagator “imaginary time”

6. Calculate evolution of the wavefunction using the propagator

7. Example 2: Gaussian wavepacket

8. Example 3: moving Gaussian wavepacket How do we know this is a moving wavepacket? Calculate the momentum expectation value:

9. Example 4: wavepacket reflected from a hard wall Boundary condition: Eigenstates: Propagator: 0 x0

11. Wavepackets reflected from steps & barriers For very nice and instructive animations visit http://www.quantum-physics.polytechnique.fr/en/pages/p0103.html