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This 9 page Class Notes was uploaded by Jamaal McGlynn on Tuesday October 20, 2015. The Class Notes belongs to PHYS410 at San Diego State University taught by Staff in Fall. Since its upload, it has received 39 views. For similar materials see /class/225321/phys410-san-diego-state-university in Physics 2 at San Diego State University.
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Date Created: 10/20/15
Lecture 24 Outline Continuous Spectra 0 Will cover the following twice in terms of 3 and 13 Determinate States Section 322 Eigenfunctions of Hermitian Operators Section 332 Generalised Statistical Interpretation Section 34 o Generalised Uncertainty Principle Section 35 GenStatInterp Discrete o If you measure an observable Qx p on a particle of state x If you are certain to get one of the eigenvalues of the hermitian operator Qx ih o If Q has a discrete spectrum the probability of observing the particular eigenvalue qnassociateol With orthonormaliseol eigenvector is ICn2 ltfn 1 gt2 o wavefunction collapses to corresponding eigenstate GenStatInterp Continuous o If C has a continuous spectrum with real eigenvalues qz and Dirac orthonormalised eigenvectors f2x the probability of getting a result in the range dz is CZ2dz ltfzq1gtl2dz o wavefunction collapses to a range i many identical systems before measurement l Ly l 2m MM ii many independent measurements 0 range depends on measuring device precision Eigenfunctions Discrete Spectra o For normalisable eigenfunctions of hermitian operators Qfx qfx and O lt f fadx lt oo o Proved tvvo theorems last time a Eigenvalues are real q q and b Eigenfunctions belonging to distinct eigenvalues are orthonorrnal llj Eigenfunctions Continuous Spectra o Hermitian operators can have continuous spectra Example 33 Find eigenvalues eigenfunctions of Li xg x Ag x meansthat gAx A6x A ie the eigenfunctions must be zero except at a A o remember defn is a Fa6x a De ne Dirac orthonormality of eigenfunctions gt0 0 gmwmwwxMV x XMW wa A26 X 60x X 0 Not square integrable functions but are complete fKdeWMA fOWW AMA GenStatInterp measurement Every real number A is an eigenfunction of 3 Dirac orthonormalised eigenfunctions are gAx 6x A co we OO 00 gAx x tdx A t the probability of getting a result in the range d is P CAl2dgt lt9A 1 Mgt20lA wavefunction collapses to a range which depends on measuring device precision Eigenfunctions 15 Spectra Example 32 Find eigenvalues eigenfunctions of 13 ihfpx pfpx solutions fpx AeWh ie the eigenfunctions are again not square integrable and not in Hilbert space for any complex p but can again de ne a kind of Dirac orthonormality 0 00 9px9pxdx A2 iP p xhdx A227Th 61 p 51 pl haVil lg Chosen A 1 27th the eigenfunctions satisfy ltfp lfpgt 61 p and are complete GenStatInterp p measurement Complete write any square integrable function as x t ff cpfprdp 1 ff cpeipxhdp Where the expansion coefficients are a function 1 27Th Thus the probability of getting a result in the range dp is P Cpl2dp lt1gtptl2dp cltpgt ltpr 1 gt 00 was was cw t Again range depends on measuring device precision Eigenfunctions of 13 have wavelength A 27Thp proof of de Broglie formula but no particle has exact p momentum space 131 15 position space x t Generalised Uncertainty Principle Last lecture de ned determinate state as 03 Q ltQgt2gt Q Q PIQ Q I gt 0 0 But for indeterminate states and two observables 031 4 ltAgt 1 l1 ltAgt 1 gt His 3 ltBgt Pl3 ltBgt 1 gt 0 can show using Schwartz inequality and more that 1 A A 2 2 gt ltlA Blgt UAUB 7 o remembering 33 iii thus 0926013 2 732 0 ie incompatible observables cannot have a complete set of common eigenfunctions
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