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709292
Quantum Mechanics
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Mind Map on Quantum Mechanics, created by franz.sciortino on 02/04/2014.
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franz.sciortino
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franz.sciortino
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Resource summary
Quantum Mechanics
Finite Square Well, potential step and barrier
Simple Harmonic Oscillator (SHM)
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
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
Dirac notation: see summary table in lect.10 notes
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
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
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