QM Review Outline Postulates of QM Expectation Values
- Slides: 20
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 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 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 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 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 Coulomb
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 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 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
- Western values vs eastern values
- Instumental values
- Terminal values and instrumental values
- A bit can have two possible values what value are those
- Define human value
- Sandwich paragraph example
- Expectation for webinar
- Operators in quantum mechanics
- Schrodinger wave equation
- Incident wave equation
- Expectation value of energy in quantum mechanics
- Law of iterated expectation
- Soil expectation value
- The process of making an expectation a reality is
- A particle limited to the x axis has the wave function
- We thank you
- Postulate of quantum mechanics
- Irony is the contrast between expectation and reality.
- Financial expectation
- Expectation value
- Service level expectation