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Expectation values and uncertainties II (including free particles)

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. Stationary state:. Free (non-interacting) particle:. kinetic energy operator. So it can not be normalized to unity:

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Expectation values and uncertainties II (including free particles)

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  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

  2. Stationary state:

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

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

  5. Momentum eigenfunctions

  6. Completeness of the momentum eigenfunctions Fourier transform

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

  8. Top hat wave packet:

  9. Normalizable

  10. Dirac delta function:

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

  12. Gaussian wave packet

  13. 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 →

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

  15. when specified for all completely specify wave function

  16. 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

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