Quantum Mechanics

Finite Square Well, potential step and barrier
Simple Harmonic Oscillator (SHM)
Five postulates of QM
1. post.2 --> Properties of Hermitian operators
2. post.1 --> wavefunctions of dynamical variables
3. post.3 --> x and p operators, others for dynamical variables (as classically)
4. post.4 --> prob. densities by sum of expansion coefficients (squared)
5. post. 5 --> time dependence as by TDSE
  1. Wavefunction collapses by interferences with the system
Commutators (Herm.) vs. anticommutators (anti-Herm.)
1. Compatibility of observables --> common set of eigenstates (e.g. H and p for free particle)
  1. If Q & R commute and have unique eigenvalues, then are compatible
Dirac notation: see summary table in lect.10 notes
Expectation values and uncertaity
1. <Q>=sum((a_n)^2 *q_n) or <Q>= int(psi* Q psi)
2. Ehrenfest Theorem: eq. of motion for expectation values of observables follow classical counterparts
3. RMS spread: Delta(q)= int(mod(Q psi)^2) or <Q^2>-<Q>^2
4. HUP: apply Schwartz inequality to difference operators (e.g. Q') with commutators and anticommutators;
  1. Find that <[Q', R']> = <[Q,R]>
    1. Consider only imaginary part, with [x,p] included, for the HUP statement
Continuous eigenvalues: swap Kronecker with Dirac delta + integration rather than summation (see notes lect.14)
1. Continuous eigenstates cannot be normalised - mostly multiply by a Gaussian factor to get wavepackets

Resource summary

	Created by franz.sciortino over 10 years ago