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Date Created: 10/20/15
Lecture 26 Outline Guts of Hermitian o Generalised Statistical Interpretation Section 34 o Determinate States Section 322 o Eigenfunctions of Hermitian Operators Section 33 Generalised Statistical Interpretation 0 If you measure an observable Qa p on a particle of state 1115 25 you are certain to get one of the eigenvalues of the hermitian operator eg Qa ih Qfgt qfgt whereas QI9gt Q 9gt o If Q has a discrete spectrum the probability of observing the particular eigenvalue qn associated With orthonormalised eigenvector fngt is ICnI2 ltfn 1 gt2 0 Upon measurement the wavefunction collapses to the corresponding eigenstate 111newgt fngt 0 That s the generalised statistical interpretation along With Schrodinger eqn time evolution QM foundation Collapse of Wavefunction Collapse of Wavefunction Eigenfunctions Two theorems o For normalisable eigenfunctions of hermitian operators Qlfgt qfgt ltQfl E ltQlfl Qlfgtl qlfgtl qltf a Eigenvalues are real Proof is straightforward ltfQfgt ltQlflfgt ltQflfgt a qltflfgt qltflfgt Thus since fgt 75 0gt we must have q q b Eigenfunctions belonging to distinct eigenvalues are orthonormal ltfjgt 617 Proof is given Qlfgt qlfgt and nggt q lggt then ltfIQ9gt ltQ9lfgt gt 61 ltf9gt qltf9gt We know that q q are real distinct thus lt f g 0 Completeness Axiom 7 Eigenfunctions of an observable operator1 are complete ie any function2 can be expressed as a linear combination of them 1 Hermitian 2 function that is in the Hilbert space Observable operator has complete set of eigenfunctions I I C W 0gt Zn Cnlfngt on mm 2 fzltxgtwltmgtdx So on is how much of fngt is in before For a normalised llgt total probability Zn cn2 l Completeness of Wavefunction Determinate States a Observables are represented by hermitian ops QT o Determinate States are Eigenfunctions of 0 These are states Where every Q measurement q 0393 ltQ lt12gt lt 1 Q 12110 ltQ Q 1 Q 1110 0 0 Where q Q Note Q is Herrnitian so is q o The only vector that has zero inner product is 0 lt00gt IltQ M 0gt so on gm 0 Time independent Schrodinger eqn HWY Enwngt 0 eg energy of stationary states En Physics 410 Quantum Mechanics Administrivia info for me and you syllabus homework O exams timeline o questionaire Lecture 1 Outline History of Quantum theory summary of modern physics old quantum theory in a page 1900 Planck s quantum theory gt E n60 hV 1905 Einstein s photoelectric effect theory 1 2 gt mvmax hV W 1911 Rutherford Geiger Marsden gt nucleus 10 14 m atom 10 10 m 1913 Bohr s model of the hydrogen atom gt hVEb Ea andLnh 1914 Franck and Hertz gt energy quantisation of atoms 1921 1922 Stern and Gerlach gt L2 mh and more 1923 1924 de Broglie matter waves gt v Eh A hp modern quantum mechanics Many formulations we concentrate on the rst two 1925 Schrodinger wave mechanics non relativistic E2 IJ 3 V2IJ VIJ 2m h 2 825 19251926 Heisenberg Born Jordan gt matrix mechanics non relativistic 1926 gt1928 Dirac s quantum eld theory relativistic gt 1950s Quantum electrodynamics 1960s till now Quantum chromodynamics and the standard model from httphscicasoueduSolvayCongressr19277Brusselsjpg Lecture 2 Outline Probably a The statistical interpretation and indeterminancy 12 0 Probability theory Discrete variables 131 0 Probability theory continuous variables 132 Statistical interpretation measurement 0 Suppose that we do measure the particles near point c i before measurement l 139 l 2m I l 139 l 2Xt ii measurement a this is known as wavefunction collapse ie a repeating the measurement also nds particles near c a Statistical interpretation implies a indeterminancy Quantum indeterminancy a but where was the particle s before the measurement 1 Realist position the particle was at c hidden variable 2 Orthodox position the particle really wasn t anywhere 3 Agnostic position refuse to answer just don t matter 0 1964 Bell showed that there is an observable difference between options 1 and 2 which eliminates 3 o Read section 124 for homework the cat a Optionally read chapter 12 at the end of the course Copenhagen Interpretation 0 Orthodox position developed by Bohr Heisenberg etc o The particle wasn t located anywhere before the measurement forced it to take a stand 1 Each Observer knows a system wavefunction which is probabilistic but completely describes a system Probability of a measurement event is Pw oc lltw gt2 Measurement is a classical process using classical devices 039 wa Matter can behave as either wave or particle depending on the experiment a see also Bohm s 77nonlocal hidden variable theory 0 Alternate views are a breeding ground for fruit loops Probability theory Discrete variables 0 Given set of NU eg N20 1 N21 2 N22 4 N23 2 M24 1 N30 1 M34 1 M35 1 N38 1 o What useful things can we say about the data set 1 Total number in sample N 230 NU l4 2 Probability of Choosing a j at random PU NanN eg Plt23gt17 3 Sum of probabilities 230 Pj 1 4 Most probable maXPj eg j 22 Discrete variables expectation values 0 Same set of Nj data N20 1 N21 2 N22 4 N23 2 M24 1 N30 1 M34 1 M35 1 N38 1 5 Median J 225 for which Egg PU 22 PU 6 In general expectation values ZfjPj 7 which is best seen in average mean j 202 gtlt 214gtlt 22353814 35714 255 ltjgt 21131 8 Average of squares 3392 20214 2 x 21214 38214 684 Note ltjgt2 2552 65025 7A 02gt Discrete variables spread 9 The wrong way to describe a spread Aj j The problem is sometimes Aj gt 0 sometimes Aj lt 0 ltAjgt Z 339 ltjgtPltj ZJPU 2130 ltjgt ltjgt 0 10 The best way is variance 02 E ltAj2gt 2 0 02 E j ltjgt2Pj 02gt ltjgt2 continuous vs discrete variables 0 Probability of measurement between a and a dx ie 135 pada Where prob density 0 thus probability of measurement between a and b b Pab dx Discrete Continuous 1 230 PO 1 fig pltasgtdx ltjgt 230ij ltasgt Hf aspltasgtdas ltfjgt 220 fjPj ltf93gt filo f93p93d93 0392 ltAj2gt 02gt ltjgt2 0392 ltA932gt lt932gt lt93gt2 Rock falling off cliff cliff height h Rock accelerates g position 5132 9252 Velocity iii f gt ie dt 3 5 Total ight time T 23 ie 513T h Probability camera ashes in interval dt is 1 da T gt 2h 9 2 2h 2 Probability density pa W 0 g a g h Rock still falling off cliff Grif ths Example 11 contd 0 Probability density pa 03933 b hltgtd 1 h ld 12V1 Ll LUZ a 2 x 62 0p NEo NE 0 h 1 h 1 1 2 3 h as a asdas asidas a ltgt 0 p NEo 2E3 0 Lecture 9 Outline The 00 well 0 stuck in an in nite well 22 in nite square well recall WOO WOO n3 a was g sin 77 mg 7127 n 2 E 2m 277m2 n 1 The normalisation A is independent of n SideNote A2 2a but we have Choice of phase of A Solutions are alternatively even and odd Each solution has n l nodes Ort hogonality o importantly solutions nx are mutually orthogonal was was do 0 m 7A n 71 d1 3061 sin sin d9 otsam mm i sin n7r sin m n mn 7m dx 7T Orthonormality Mutually orthogonal f mum wncr dx O Vm 7 n and What about normalisation Z WW dx 3061 81112 dx 1 De ning Kronecker delta f mum wncr dx 6mm Function is orthonorrnal when 6mm O V m 7 n 1Vmn Completeness These wncr functions are also complete in that any1 function can be expressed in terms of them fx ch 11 VEch sin This is also known as the Fourier series of f and the equation itself is called Dirichlet s Theorem 1 even if the function has a nite number of nite discontinuities 0 Using Fourier s trick multiply by mum and integrate WW f93 d9 f Cn WW 7M1quot d9 icn mn Cm 0 Therefore cn fx dx Stationary States 0 Which are Ilnx t nx gpnt nx 6 iEnth 2 n27r2 Ilnxt sin e 1 2ma2ht a 0 General soln to time dependent Schrodinger equation 00 2 TNT n27r2h x t Z on sin e 1 2ma2 t n21 a a 0 Expand initial wavefunction xnf O Z on on nxllx 0 dx g sin Q4930 dx 0 on tells you the amount of w in x t Coef cients and energy 0 With xnf O Zen o lCn2 tells you probability of measuring En 00 Z cn2 1 n1 0 Proof 1 Mac ogt2dx 2 am wequot Zen we aim m1 n1 Z Z mm was da m1 n21 Z Zcfncn mn Z lCn2 m1 n21 n1 Energy and Coef cients o What about the total energy of the system 0 At t O remembering that 13 W 11gt IJUO H11a0dx gem wear wan d9 g cgann was was dx 2n Enjcgann mn n cn2En Physics 410 Quantum Mechanics AdministriVia info for me and you syllabus homework 0 exams timeline o questionaire Lecture 1 Outline History of Quantum theory summary of modern physics old quantum theory in a page 1900 Planck quanta theory gt E n50 nhu nhCA 1905 Einstein s photoelectric effect theory 1 2 gt 577mm by W 1911 Rutherford Geiger Marsden gt nucleus 10 L4 m atom 10 LO m 1913 Bohr s stationary circular states of the hydrogen atom gt by Eb EL and L mm nh 1914 Franck and Hertz gt energy quantisation of atoms 1921 1922 Stern and Gerlach gt L2 mh and 6 1923 1924 de Broglie matter waves gt v Eh A hp modern quantum mechanics Many formulations we concentrate on the rst two 1925 Schrodinger wave mechanics non relativistic 8111 E2 39h v2x11 W Z 82 2m 19251926 Heisenberg Born Jordan gt matrix mechanics non relativistic 1926 gt1930 Dirac s quantum eld theory relativistic gt 1950s Quantum electrodynamics QED 1960s till now Quantum chromodynamics QCD and the standard model And what is dark energy matter anyway 1927 So vay Caneremce m Brusse s from httphscicasoueduSolvayCongre5571927rBrusselsjpg 1D Classical Mechanics a Force Fat m aat m FXt gt I N Xt Veg A p d2a 8V Newton s Law 771 dt2 81 1D Schr dinger Equation 0 Given a wavefunction 1115 25 a complex number a Bern s statistical interpretation prob of measuring the particles between points a and b Ila t2da PZOjM a b Z7181145132 71 282111xt 8t 2m 8132 Vltx7tgtqlltx7tgt Statistical interpretation measurement 0 Suppose that we do measure the particles near point c i before measurement l 139 l 2m I l 139 l 2Xt ii measurement a this is known as wavefunction collapse ie a repeating the measurement also nds particles near c a Statistical interpretation implies a indeterminancy