Quantum Mechanics

franz.sciortino
Mind Map by , created over 5 years ago

Mind Map on Quantum Mechanics, created by franz.sciortino on 04/02/2014.

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franz.sciortino
Created by franz.sciortino over 5 years ago
Schroedinger Equation
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Quantum Mechanics
1 Finite Square Well, potential step and barrier
2 Simple Harmonic Oscillator (SHM)
3 Five postulates of QM
3.1 post.2 --> Properties of Hermitian operators
3.2 post.1 --> wavefunctions of dynamical variables
3.3 post.3 --> x and p operators, others for dynamical variables (as classically)
3.4 post.4 --> prob. densities by sum of expansion coefficients (squared)
3.5 post. 5 --> time dependence as by TDSE
3.5.1 Wavefunction collapses by interferences with the system
4 Commutators (Herm.) vs. anticommutators (anti-Herm.)
4.1 Compatibility of observables --> common set of eigenstates (e.g. H and p for free particle)
4.1.1 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
6.1 <Q>=sum((a_n)^2 *q_n) or <Q>= int(psi* Q psi)
6.2 Ehrenfest Theorem: eq. of motion for expectation values of observables follow classical counterparts
6.3 RMS spread: Delta(q)= int(mod(Q psi)^2) or <Q^2>-<Q>^2
6.4 HUP: apply Schwartz inequality to difference operators (e.g. Q') with commutators and anticommutators;
6.4.1 Find that <[Q', R']> = <[Q,R]>
6.4.1.1 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)
7.1 Continuous eigenstates cannot be normalised - mostly multiply by a Gaussian factor to get wavepackets

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