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Quantum Mechanics: Free Particles, Uncertainties & Expectations

Explore expectations, uncertainties, and values in quantum mechanics, focusing on free particles. Understand the non-normalizable kinetic energy operator and the continuum of energy eigenvalues for unbound solutions. Delve into momentum eigenfunctions, Fourier transforms, and wave packet evolution in time. Gain insights into the Gaussian wave packet amplitudes, completeness of eigenfunctions, and Heisenberg's uncertainty principle for both free and bound particles. Learn how energy eigenvalues and amplitudes specify the system's state, and grasp the implications of normalization in quantum systems.

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Quantum Mechanics: Free Particles, Uncertainties & Expectations

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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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