Quantum Mechanics 103 Quantum Implications for Computing SchrГ¶dinger and Uncertainty пЃ¶ пЃ¶ пЃ¶ пЃ¶ пЃ¶ Going back to TaylorвЂ™s experiment, we see that the wavefunction of the photon extends through both slits Therefore the photon has вЂњtraveledвЂќ through both openings simultaneously The wavefunction of a вЂњparticleвЂќ will contain every possible path the particle could take until the particle is вЂњdetectedвЂќ by scattering or being absorbed These paths can interfere with each other to produce diffraction-like probability patterns BUT, SchrГ¶dinger took this explanation to an extreme SchrГ¶dingerвЂ™s Famous Cat Suppose a radioactive substance is put in a box with a c for a period of time вЂў During that time, there is a 50% chance that one of the nuclei will decay and trigger a Geiger Counter вЂў If the Geiger Counter triggers, a gun is discharged and the cat is killed пЃ¶ SchrГ¶dingerвЂ™s Famous Cat пЃ¶ Until an observer opens the box to make a вЂњmeasurementвЂќ of the system, вЂў The nucleus remains both decayed and undecayed вЂў The Geiger counter remains both triggered and untriggered вЂў The gun has both fired and not fired вЂў The cat is both dead and alive Disclaimer: To be truly indeterministic, this experiment must be performed in a sound-proof room with no window Paradox? пЃ¶ пЃ¶ пЃ¶ Paradoxical as it may seem, the concept of вЂњsuperposition of statesвЂќ is borne out well in experiment пЃ¶ Like superposition of waves producing interference effects Quantum Mechanics is one of the most-tested and bestverified theories of all time But it seems counter-intuitive since we live in a macroscopic world where uncertainty on the order of пЃЁ is not noticeable Quantum paradox #2 пЃ¶ Einstein-Podolsky-Rosen (EPR) paradox пЃ¶ пЃ¶ Consider two electrons emitted from a system at rest; measurements must yield opposite spins if spin of the system does not change We say that the electrons exist in an вЂњentangled stateвЂќ More EPR пЃ¶ пЃ¶ пЃ¶ пЃ¶ If measurement is not done, can have interference effect since each electron is superposition of both spin possibilities But, measuring spin of one electron destroys interference effects for both it and the other electron; It also determines the spin of the other electron How does second electron вЂњknowвЂќ what its spin is and even that the spin has been determined Interpreting EPR Measuring one electron affects the other electron! пЃ¶ For the other electron to вЂњknowвЂќ about the measurement, a signal must be sent faster than the speed of light! пЃ¶ Such an effect has been experimentally verified, but it is still a topic of much debate пЃ¶ Interference effects пЃ¶ пЃ¶ пЃ¶ Remember this Mach-Zender Interferometer? Can adjust paths so that light is split evenly between top U detector and lower D detector, all reaches U, or all reaches D вЂ“ due to interference effects Placing a detector (either bomb or non-destructive) on one of the paths means 50% goes to each detector ALL THE TIME Interpretation пЃ¶ пЃ¶ пЃ¶ пЃ¶ Wave theory does not explain why bomb detonates half the time Particle probability theory does not explain why changing position of mirrors affects detection Neither explains why presence of bomb destroys interference Quantum theory explains both! пЃ¶ пЃ¶ пЃ¶ Amplitudes, not probabilities add - interference Measurement yields probability, not amplitude - bomb detonates half the time Once path determined, wavefunction reflects only that possibility - presence of bomb destroys interference Quantum Theory meets Bomb пЃ¶ пЃ¶ пЃ¶ Four possible paths: RR and TT hit upper detector, TR and RT hit lower detector (R=reflected, T=transmitted) Classically, 4 equally-likely paths, so prob of each is 1/4, so prob at each detector is 1/4 + 1/4 = ВЅ, independent of path length difference Quantum mechanically, square of amplitudes must each be 1/4 (prob for particular path), but amplitudes can be imaginary or complex! пЃ¶ This allows interference effects What wave function would give 50% at each detector? пЃ™ пЂЅ a TR пЂ« b RT пЂ« c RR пЂ« d TT пЃ¶ Must have |a|2 = |b|2 = |c|2 = |d|2 = 1/4 пЃ¶ Need |a + b|2 = |c+d|2 = 1/2 пЃ™пЂЅ 1пЂ« i 2 2 TR пЂ« 1пЂ i RT пЂ« 2 2 aпЂ«b 2 пЂЅ 1пЂ« i 2 2 2 2 пЂЅ 2 пЂЅ 2 2 2 4 пЂЅ 8 2 2 cпЂ«d RR пЂ« 2 пЂЅ 4 8 1 2 пЂЅ 1 2 1пЂ i 2 2 TT If Path Lengths Differ, Might Have пЃ™пЂЅ 1 TR пЂ« пЂ1 2 RT пЂ« 2 пЃ¶ Lower detector: RR пЂ« 2 2 2 пЃ¶ 1пЂ« i пЃ™ пЂЅ 1 пЂ« Upper detector: пЃ™ пЂЅ 2 1пЂ« i 2 2 TT 2 2 пЂ1 2 1пЂ« i 2 пЂЅ0 2 пЂ« 1пЂ« i 2 2 Voila, Interference! 2 пЂЅ 2 пЂ« 2i 2 2 2 пЂЅ1 When Measure Which Path, пЃ™пЂЅ пЃ™пЂЅ 1 2 1 1 TR пЂ« 2 1 RR пЂ« 2 пЃ¶ TT RT 2 Lower detector: пЃ™ 2 пЂЅ 1 2 пЂЅ 2 2 пЃ¶ Upper detector: пЃ™ 2 пЂЅ 1 2 1 2 пЂЅ 1 2 Voila, No Interference! Quantum Storage пЃ¶ пЃ¶ пЃ¶ Consider a quantum dot capacitor, with sides 1 nm in length and 0.010 microns between вЂњplatesвЂќ How much energy required to place a single electron on those plates? Can make confinement of dot dependent upon voltage пЃ¶ пЃ¶ Lower the voltage, let an electron on вЂ“> 1 Lower voltage on other side, let the electron off -> 0 What must a computer do? Deterministic Turing Machine still good model пЃ¶ Two pieces: Read/write head in some internal state пЃ¶ вЂњInfiniteвЂќ tape with series of 1s, 0s, or blanks пЃ¶ пЃ¶ Follows algorithms by performing 3 steps: Read value of tape at headвЂ™s location пЃ¶ Write some value based on internal state and value read пЃ¶ Move to next value on tape пЃ¶ Can we improve this model? пЃ¶ пЃ¶ пЃ¶ пЃ¶ пЃ¶ Probabilistic Turing Machine sometimes better Multiple choices for internal state change Not 100% accurate, but accuracy increases with number of steps Can solve some types of problems to sufficient accuracy much more quickly than deterministic TM can Similar concept to Monte Carlo integration Limits on Turing Machines пЃ¶ Some problems are solvable in theory but take too long in practice пЃ¶ пЃ¶ e.g., factoring large numbers Can label problems by how the number of steps to compute grows as the size of the numbers used grows addition grows linearly пЃ¶ multiplication grows as the square of digits пЃ¶ Fourier transform grows faster than square пЃ¶ factoring grows almost exponentially пЃ¶ Examples of factoring time пЃ¶ пЃ¶ пЃ¶ пЃ¶ пЃ¶ MIP-year = 1 year of 1 million processes per second Factoring 20-digit decimal number done in 1964, requiring only 0.000009 MIP-years 45-digit decimal number (1974) needs 0.001 MIPyears 71-digit decimal number (1984) needs 0.1 MIPyears 129-digit decimal number (1994) needs 5000 MIP-years Quantum Cryptography Current best encryption uses public key for encoding пЃ¶ Need private key (factors of large integer in public key) to decode пЃ¶ Really safe unless пЃ¶ Someone can access your private key пЃ¶ Quantum computers become prevalent пЃ¶ Quantum Cryptography II Quantum Computers can factor large numbers near-instantly, making public key encryption passe пЃ¶ But, can send quantum information and know whether it has been intercepted пЃ¶ What problems face QC? пЃ¶ Decoherence: if measurement made, superposition collapses пЃ¶ пЃ¶ пЃ¶ Quantum error correction пЃ¶ пЃ¶ пЃ¶ Even if measurement not intentional! i.e., if box moves, cat becomes alive or dead, not both No trail of path taken (or else no superposition) Proven to be possible; that doesnвЂ™t mean itвЂ™s easy! HUGE Technical challenges пЃ¶ пЃ¶ пЃ¶ electronic states in ion traps (slow, leakage) photons in cavity (spontaneous emission) nuclear spins in molecule (small signal in large noise)

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