1Wed. Mar. 29, 2006 Phys107 Lecture 261Course essay reference & outline• Wednesday, Nov. 16, due in class– Reference to main article– Outline of essay.• Essay is 500-750 words(about 2 double-spaced typed pages)• So outline should be by paragraph• E.g.– I. Intro: (one-sentence summary)– II. Background/history (specific ‘high points’)– III. Specific example (identify the example)– etcWed. Mar. 29, 2006 Phys107 Lecture 262From Last Time…• Heisenberg Uncertainty principle• Arises from wave nature of particles.• Particle cannot have precise position andmomentum.• Highly accurate momentum (wavelength)means position is uncertain• Can localize particle by superimposing manywavelengths, so momentum is uncertain.• Quantum mechanical tunneling.Wed. Mar. 29, 2006 Phys107 Lecture 263More unusual aspects ofquantum mechanics• Quantum jumps: wavefunction of particle changesthroughout all space when it changes quantumstate.• Superposition: quantum mechanics sayswavefunction can be in two very differentconfigurations, both at the same time.• Entanglement: two quantum-mechanical objectscan be intertwined so that their behaviors areinstantly correlated over enormous distances.Wed. Mar. 29, 2006 Phys107 Lecture 264The wavefunctionand quantum ‘jumps’• A quantum system has only certain discretequantum states in which it can exist.• Each quantum state has distinct wavefunction,which extends throughout all space• It’s square gives probability of finding electron ata particular spatial location.• When particle changes it’s quantum state,wavefunction throughout all space changes.Wed. Mar. 29, 2006 Phys107 Lecture 265Hydrogen atom quantum jumpn=1n=2n=3n=4Wavefunction changes from 3pto 1s throughout all space.Photonemittedhf=E3-E1Wed. Mar. 29, 2006 Phys107 Lecture 266The electron jumps from one quantum state to another,changing its wavefunction everywhere.During the transition, we say that the electron is in asuperposition between the two states.2Wed. Mar. 29, 2006 Phys107 Lecture 267Classical particle in a box• Box is stationary, so average speed is zero.• But remember the classical version• Particle bounces back and forth.– On average, velocity is zero.– But not instantaneously– Sometimes velocity is to left, sometimes to rightLWed. Mar. 29, 2006 Phys107 Lecture 268Quantum version• Quantum state is both velocities at the same time• Ground state is a standing wave, made equally of– Wave traveling right ( positive momentum +h/λ )– Wave traveling left ( negative momentum - h/λ )! "= 2LOne half-wavelength! p =h"=h2LmomentumLQuantum ground state is equal superpositionof two very different motions.Wed. Mar. 29, 2006 Phys107 Lecture 269Making a measurementSuppose you measure the speed (hence,momentum) of the quantum particle in a tube.How likely are you to measure the particlemoving to the left?A. 0% (never)B. 33% (1/3 of the time)C. 50% (1/2 of the time)Wed. Mar. 29, 2006 Phys107 Lecture 2610The wavefunction• Wavefunction = Ψ= |moving to right> + |moving to left>• The wavefunction for the particle is an equal‘superposition’ of the two states of precisemomentum.• When we measure the momentum (speed), wefind one of these two possibilities.• Because they are equally weighted, we measurethem with equal probability.Wed. Mar. 29, 2006 Phys107 Lecture 2611A Measurement• We interpret this as saying that before themeasurement, particle exists equally in states– momentum to right– momentum to left• When we measure the momentum, we get aparticular value (right or left).• The probability is determined by the weighting ofthe quantum state in the wavefunction.• The measurement has altered the wavefunction.The wavefunction has ‘collapsed’ into a definitemomentum state.Wed. Mar. 29, 2006 Phys107 Lecture 2612Double-slit particle interference• Reduce intensity until only single photon at a time goesthrough slits.• Which slit does the photon go through?3Wed. Mar. 29, 2006 Phys107 Lecture 2613Interference of a single photon1/30 secexposure1 secexposure100 secexposureWed. Mar. 29, 2006 Phys107 Lecture 2614Which slit?In the two-slit experiment with one photon,which slit does the photon go through?A. Left slitB. Right slitC. Both slitsWed. Mar. 29, 2006 Phys107 Lecture 2615Photon on both pathsPath 1: photon goes through left slitPath 2: photon goes through right slitWavefunction for thephoton is a superpositionof these two states.Quantum mechanics saysphoton is simultaneously ontwo widely separated paths.Wed. Mar. 29, 2006 Phys107 Lecture 2616Superposition of quantum states• We made a localized state made by superimposing (‘addingtogether’) states of different wavelength (momenta).• Quantum mechanics says this wavefunction physicallyrepresents the particle.• The amplitude squared of each contributionis the probability that a measurementwill determine a particular momentum.• Copenhagen interpretation says thatbefore a measurement, all momenta exist.Measurement ‘collapses’ the wavefunction into a particularmomentum state (this is the measured momentum).Wed. Mar. 29, 2006 Phys107 Lecture 2617Measuring which slitMeasure induced current frommoving charged particle• Suppose we measure which slit the particle goes through?• Interference pattern is destroyed!• Wavefunction changes instantaneously over entire screenwhen measurement is made.Wed. Mar. 29, 2006 Phys107 Lecture 2618A superposition state• Margarita or Beer?• This QM state has equal superposition of two.• Each outcome(drinking margarita, drinking beer)is equally likely.• Actual outcome not determined untilmeasurement is made (drink is tasted).4Wed. Mar. 29, 2006 Phys107 Lecture 2619What is object before themeasurement?• What is this new drink?• Is it really a physical object?• Exactly how does the transformation from thisobject to a beer or a margarita take place?• This is the collapse of the wavefunction.Wed. Mar. 29, 2006 Phys107 Lecture 2620Quantum dice gameSuppose we have a six-sided quantum die. It isnot fair, but comes up ‘2’ most often. It’swavefunction could beA.B.C.! 1.2 1 + 1.5 2 + 1.3 3 + 1.7 4 + 0.8 5 +1.1 6! 1.11 + 1.4 2 + 1.3 3 + 1.3 4 + 0.8 5 + 1.1 6! 1.0 1 + 1.5 2 + 1.6 3 + 1.1 4 + 0.9 5 + 1.3 6Wed. Mar. 29, 2006 Phys107 Lecture 2621Not universally accepted• Historically, not everyone agreed with thisinterpretation.• Einstein was a notable opponent– ‘God does not play dice’• These ideas hotly
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