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 All information is contained in the wavefunction  Probabilities are determined by the overlap of wavefunctions The time evolution of the wavefn given by …plus a few more

Expectation Values Probability Density at r Prob of finding the system in a region d 3 r about r Prob of finding the system anywhere

Average value of position r Average value of momentum p Expectation value of total energy

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

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.

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

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

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

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.

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)

Simple/Bare Coulomb Eigenfns: Bare Coulomb - stationary states  nlm (r  ) or R nl (r) Y lm (  )

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

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

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