Quantum Mechanics

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Mindmap am Quantum Mechanics, erstellt von franz.sciortino am 02/04/2014.
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Quantum Mechanics
  1. Finite Square Well, potential step and barrier
    1. Simple Harmonic Oscillator (SHM)
      1. Five postulates of QM
        1. post.2 --> Properties of Hermitian operators
          1. post.1 --> wavefunctions of dynamical variables
            1. post.3 --> x and p operators, others for dynamical variables (as classically)
              1. post.4 --> prob. densities by sum of expansion coefficients (squared)
                1. post. 5 --> time dependence as by TDSE
                  1. Wavefunction collapses by interferences with the system
                2. 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
                  2. Dirac notation: see summary table in lect.10 notes
                    1. Expectation values and uncertaity
                      1. <Q>=sum((a_n)^2 *q_n) or <Q>= int(psi* Q psi)
                        1. Ehrenfest Theorem: eq. of motion for expectation values of observables follow classical counterparts
                          1. RMS spread: Delta(q)= int(mod(Q psi)^2) or <Q^2>-<Q>^2
                            1. 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
                            2. 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
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