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QM Review

QM Review. Outline. Postulates of QM Expectation Values Eigenfunctions & Eigenvalues Where do we get wavefunctions from? Non-Relativistic Relativistic Techniques for solving the Schro Eqn Analytically Numerically Creation-Annihilation Ops. Postulates of Quantum Mechanics.

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QM Review

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  1. QM Review

  2. Outline • Postulates of QM • Expectation Values • Eigenfunctions & Eigenvalues • Where do we get wavefunctions from? • Non-Relativistic • Relativistic • Techniques for solving the Schro Eqn • Analytically • Numerically • Creation-Annihilation Ops

  3. Postulates of Quantum Mechanics • All information is contained in the wavefunction Y • Probabilities are determined by the overlap of wavefunctions • The time evolution of the wavefn given by …plus a few more

  4. Expectation Values • Probability Density at r • Prob of finding the system in a region d3r about r • Prob of finding the system anywhere

  5. Average value of position r • Average value of momentum p • Expectation value of total energy

  6. Eigenvalue Problems Sometimes a function fn has a special property eigenfn eigenvalue

  7. Where do we get the wavefunctions from? • Physics tools • Newton’s equation of motion • Conservation of Energy • Cons of Momentum & Ang Momentum The most powerful and easy to use technique is Cons NRG.

  8. Where do we get the wavefunctions from? Non-relativistic: 1-D cartesian KE + PE = Total E

  9. Where do we get the wavefunctions from? Non-relativistic: 3-D spherical KE + PE = Total E

  10. Non-relativistic: 3-D spherical Most of the time set u(r) = R(r) / r But often only one term!

  11. Techniques for solving the Schro Eqn. • Analytically • Solve the DiffyQ to obtain solns • Numerically • Do the DiffyQ integrations with code • Creation-Annihilation Operators • Pattern matching techniques derived from 1D SHO.

  12. Analytic Techniques • Simple Cases • Free particle (ER 6.2) • Infinite square well (ER 6.8) • Continuous Potentials • 1-D Simple Harmonic Oscillator (ER 6.9, Table 6.1, and App I) • 3-D Attractive Coulomb (ER 7.2-6, Table 7.2) • 3-D Simple Harmonic Oscillator • Discontinuous Potentials • Step Functions (ER 6.3-7) • Barriers (ER6.3-7) • Finite Square Well (ER App H)

  13. Eigenfns: Bare Coulomb - stationary statesYnlm(r,q,f) or Rnl(r) Ylm(q,f) Simple/Bare Coulomb

  14. http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgihttp://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi

  15. Numerical Techniques ER 5.7, App G • Using expectations of what the wavefn should look like… • Numerical integration of 2nd order DiffyQ • Relaxation methods • .. • .. • Joe Blow’s idea • Willy Don’s idea • Cletus’ lame idea • .. • ..

  16. SHO Creation-Annihilation Op Techniques Define: If you know the gnd state wavefn Yo, then the nth excited state is:

  17. Inadequacy of Techniques • Modern measurements require greater accuracy in model predictions. • Analytic • Numerical • Creation-Annihilation (SHO, Coul) • More Refined Potential Energy Fn: V() • Time-Independent Perturbation Theory • Changes in the System with Time • Time-Dependent Perturbation Theory

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