Quantum Mechanics: Week 3 overview W3.1

Transcription

1 Week 3 Monday Tuesday Wednesday Thursday Frday Recap Bell, CHSH, KS Interpretatons S.E. H atom Q. Logc gates Harm. Osc. Q. Game: CHSH (Relatvty No-clonng PEP, standard model Q. Teleportaton QFT Measurement Problem Contnuous Obs. Tunnelng GHZ GR, BH Decoherence 1 Generalzed S.E. Uncertanty prncple Jouza s alg. Strng theory. Densty matrx Schrödnger eqn. (SE (P vs NP Q. Compl. Q. Eraser Q. Computng Intro P=NP wth non-lnear. Interpretatons (start. Tme-Indep. SE, Q. Crypto. Afternoon topcs (tbd Solvng TISE, Q. Complexty (? (Relatvty Why QM? (? (secret stuff MZ nter. LIGO, Q.E. Scaran Ch 5,9 Outsde readngs Possble problems Day 9 problems/day 11 [none] / Day 11 problems (possbly more [none] possbly optonal TBD

2 Week 3 potental tems Relatvty (complete. Summary CHSH, KS, sgnfcance. Quantum game CHSH (afternoon problem. GHZ (problem, Mermn, Q. Game. Measurement problem. Schrödnger s cat. Decoherence 1 Densty matrx Decoherence Quantum Eraser, demo Interpretatons Uncertanty prncple Spn: Standard model PEP Why QM? (new. Q. Complexty (new Contnuous observables Gen. S.E. Maxwell s S.E. Solvng S.E. Spectra, H atom (demo. Tunnelng (STM Harmonc Oscllator Q. Computng, ntroducton Q. Cryptography. Q. Teleportaton. No clonng, monogamy. Jouza s algorthm. Grover s algorthm. Other quantum algorthms. Black hole nfo paradox. Advanced topcs: QFT, QG, etc.

3 Monday Tuesday Wednesday Thursday Frday Recap polarzaton Bell nequalty Interpretatons (cont. Contextualty Prncples of QM Operators Probablty, jpd. Interp. probablty Kochen-Specker Computatonal bass. Expectaton values Measurement problem Mxtures More experments Logc gates Multpartcle states (tensor product Mxtures Densty matrx GHZ Hermtan operators, Interpretatons Decoherence PR box Postulates of QM, CHSH EPR ntro. Resoluton of dentty Bascs of spn, PEP Afternoon topcs (tbd H atom Scaran Ch 1, Sussknd Ch 1, Outsde readngs Possble problems Projectons, polarzaton GHZ GHZ

4 All your polarzaton bases are belong to us Egenvalue: +1 1 Operator HV bass: H = 0 P M bass: P = 1 1 RL bass: R = 1 V = ( 0 1 M = 1 1 L = 1 ( 1 0 Ŝ HV = 0 1 ( 0 1 Ŝ P M = 1 0 ( 0 Ŝ RL = 0 Most general state: θ, φ = cos θ H + e φ sn θ V = ( cos θ e φ sn θ Operators n general: ˆΩ = ω ω ω P rob(to fnd ω when n state ψ = ω ψ.

5 All your polarzaton bases are belong to us Egenvalue: +1 1 Operator HV bass: H = 0 P M bass: P = 1 1 RL bass: R = 1 Fndng probabltes for specfc outcomes: V = ( 0 1 M = 1 1 L = 1 P rob(to fnd ω when n state ψ = ω ψ. Fndng the expectaton value of an observable Ω when n state ψ: [ ] ˆΩ = ψ ˆΩ ψ. Why? = ψ ω ω ω ψ = ω ψ ω ω ψ = ( 1 0 Ŝ HV = 0 1 ( 0 1 Ŝ P M = 1 0 ( 0 Ŝ RL = 0 ω p(ω. Resoluton of the dentty: ˆ1 = λ λ.

6 For example, to expand a state vector nto the RL bass we apply the dentty as expressed n that bass: ˆ1 ψ = R R ψ + L L ψ = ψ R R + ψ L L, where ψ R and ψ L are the components extendng along R and L respectvely.

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8 Physcal realzaton: photon polarzaton dmensonal Hlbert space H Physcal realzaton: electron spn ( ( 1 0 ψ = cos θ H + e φ sn θ V HV z ψ = cos θ eφ sn θ PM x RL 1 1 y

9 1: State The state, ncludng all you can know about t, s represented mathematcally by a normalzed ket ψ n Hlbert space. : Observables A physcal observable s represented mathematcally by a Hermtan operator A that acts on kets on the Hlbert space H. 3: Measurement The probablty of obtanng the egenvalue a n n a measurement of the observable A on the system n state ψ s P an = a n ψ where a n s the normalzed egenvector of A correspondng to the egenvalue a n. 4: Tme Evoluton The state ψ of the system at tme t 1 s related to the state ψ of the system at tme t by a untary operator U whch depends only on the tmes t 1 and t, ψ(t = Û(t, t 1 ψ(t 1.

10 Day 1 (3.3 Plan Man topcs: measurement problem, nterpretatons, decoherence, quantum eraser. (other? Interpretatons, ponts: von-neumann/wigner: Pro: The mathematcs of QM s clear. Atoms, and thngs made up of atoms are quantum. Thus, atom, detector, retna, etc. should all be n a superposton. The mnd (conscousness s presumably somethng dfferent. We may not know, but vn (and Roger Penrose for that matter mght argue that we wll eventually understand t. And there are arguments that t should be a non-algorthmc process as well (e.g. epphanes. Thus, t mght be that ths s the non-lnear source of wavefuncton collapse. Con: Mentoned before. Ths leads to a dualst vew of nature. Wgner s frend. Wgner steps back n and asks queston. To hm, the wavefuncton collapsed at that pont, or dd t? Perhaps he asks the student what he felt rght before he asked the queston. The student would no doubt say that she saw the lght flash. So he must conclude that the student collapsed t (or else he falls nto solpssm that everythng perceved s a fgment of your magnaton. Thus, Wgner concluded, the frst conscous beng collapsed the wavefuncton. Interestng suggeston. NPR had an author talkng about those wth? syndrome. Where feel you are dead. Would these people be able to collapse a wavefuncton? Somehow you would have them obtan

11 WPI, and later reveal the result. There are experments that ndcate that neural processng occurs well before conscous awareness. Mnds can react to thngs pror to the person beng conscously aware.