QM Review Outline Postulates of QM Expectation Values

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QM Review

QM Review

Outline • • Postulates of QM Expectation Values Eigenfunctions & Eigenvalues Where do we

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 Y •

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

Expectation Values • Probability Density at r • Prob of finding the system in

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 •

• 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

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

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

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

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

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 Diffy. Q to

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

Analytic Techniques • Simple Cases – Free particle (ER 6. 2) – Infinite square

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 (ER 6. 3 -7) – Finite Square Well (ER App H)

Eigenfns: Bare Coulomb - stationary states Ynlm(r, q, f) or Rnl(r) Ylm(q, f) Simple/Bare

Eigenfns: Bare Coulomb - stationary states Ynlm(r, q, f) or Rnl(r) Ylm(q, f) Simple/Bare Coulomb

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

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

Numerical Techniques ER 5. 7, App G • Using expectations of what the wavefn

Numerical Techniques ER 5. 7, App G • Using expectations of what the wavefn should look like… – – – – – Numerical integration of 2 nd order Diffy. Q Relaxation methods. . Joe Blow’s idea Willy Don’s idea Cletus’ lame idea. .

SHO Creation-Annihilation Op Techniques Define: If you know the gnd state wavefn Yo, then

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

Inadequacy of Techniques • Modern measurements require greater accuracy in model predictions. – Analytic

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