Expectation values and uncertainties II (including free particles)

1. Expectation values and uncertainties II(including free particles) SMXR355 2005 http://www.cmmp.ucl.ac.uk/~swz/courses/SMXR355.html For hard copy: give me your address

4. Stationary state:

7. Free (non-interacting) particle: kinetic energy operator

10. So it can not be normalized to unity: It is NOT a wave function that represents a realizable state of the particle

11. Momentum eigenfunctions

12. Completeness of the momentum eigenfunctions Fourier transform

14. For free particles: energy eigenvalues form a continuum of unbound solutions → instead of sums, integrals

15. Top hat wave packet:

16. Normalizable

17. Dirac delta function:

19. Time evolution Time dependence of amplitude function for the expansion of the initial wave packet in momentum and energy eigenfunctions Using energy eigenvalues

21. Gaussian wave packet

24. amplitudes: for a given set of eigenfunctions, the amplitudes completely specify the state of the system normalization → if system is in eigenstate, only one measurement →

25. completely specifies state at time t amplitude for position x Note:

26. when specified for all completely specify wave function

27. Heisenberg’s uncertainty principle for both free and bound particles It is impossible to prepare any state of a one particle quantum system for which the product (x)(px) is less than ħ/2