Quantum Mechanics

Quantum Mechanics

1. Finite Square Well, potential step and barrier

2. Simple Harmonic Oscillator (SHM)

3. Five postulates of QM

   • post.2 --> Properties of Hermitian operators
   • post.1 --> wavefunctions of dynamical variables
   • post.3 --> x and p operators, others for dynamical variables (as classically)
   • post.4 --> prob. densities by sum of expansion coefficients (squared)
   • post. 5 --> time dependence as by TDSE
     • Wavefunction collapses by interferences with the system

4. Commutators (Herm.) vs. anticommutators (anti-Herm.)

   • Compatibility of observables --> common set of eigenstates (e.g. H and p for free particle)
     • If Q & R commute and have unique eigenvalues, then are compatible

5. Dirac notation: see summary table in lect.10 notes

6. Expectation values and uncertaity

   • <Q>=sum((a_n)^2 *q_n) or <Q>= int(psi* Q psi)
   • Ehrenfest Theorem: eq. of motion for expectation values of observables follow classical counterparts
   • RMS spread: Delta(q)= int(mod(Q psi)^2) or <Q^2>-<Q>^2
   • HUP: apply Schwartz inequality to difference operators (e.g. Q') with commutators and anticommutators;
     • Find that <[Q', R']> = <[Q,R]>
       • Consider only imaginary part, with [x,p] included, for the HUP statement

7. Continuous eigenvalues: swap Kronecker with Dirac delta + integration rather than summation (see notes lect.14)

   • Continuous eigenstates cannot be normalised - mostly multiply by a Gaussian factor to get wavepackets