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QM Review

This comprehensive guide delves into the fundamental concepts of quantum mechanics, covering postulates, expectation values, eigenfunctions, eigenvalues, and techniques for solving the Schrödinger Equation both analytically and numerically. Learn about wavefunctions, probability density, eigenvalue problems, creation-annihilation operators, and various solving techniques with real-world examples and applications. Discover how physics tools like Newton's equations and conservation laws tie into quantum mechanics for a deeper understanding of the subject.

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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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