From Last Time. Course essay reference & outline. The wavefunction and quantum jumps. More unusual aspects of quantum mechanics

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1 Course essay reference & outline Wednesday, Nov. 16, due in class Reference to main article Outline of essay. Essay is 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) etc From Last Time Heisenberg Uncertainty principle Arises from wave nature of particles. Particle cannot have precise position and momentum. Highly accurate momentum (wavelength) means position is uncertain Can localize particle by superimposing many wavelengths, so momentum is uncertain. Quantum mechanical tunneling. 1 2 More unusual aspects of quantum mechanics Quantum jumps: wavefunction of particle changes throughout all space when it changes quantum state. Superposition: quantum mechanics says wavefunction can be in two very different configurations, both at the same time. Entanglement: two quantum-mechanical objects can be intertwined so that their behaviors are instantly correlated over enormous distances. The wavefunction and quantum jumps A quantum system has only certain discrete quantum 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 at a particular spatial location. When particle changes it s quantum state, wavefunction throughout all space changes. 3 4 Hydrogen atom quantum jump n=4 n=3 n=2 Photon emitted hf=e 3 -E 1 n=1 Wavefunction changes from 3p to 1s throughout all space. 5 The electron jumps from one quantum state to another, changing its wavefunction everywhere. During the transition, we say that the electron is in a superposition between the two states. 6 1

2 Classical particle in a box Box is stationary, so average speed is zero. But remember the classical version L Particle bounces back and forth. On average, velocity is zero. But not instantaneously Sometimes velocity is to left, sometimes to right Quantum version Quantum state is both velocities at the same time " = 2L One halfwavelength L Ground state is a standing wave, made equally of Wave traveling right ( positive momentum +h/λ ) Wave traveling left ( negative momentum - h/λ ) Quantum ground state is equal superposition of two very different motions. momentum p = h " = h 2L 7 8 Making a measurement Suppose you measure the speed (hence, momentum) of the quantum particle in a tube. How likely are you to measure the particle moving to the left? A. 0% (never) B. 33% (1/3 of the time) C. 50% (1/2 of the time) The wavefunction Wavefunction = Ψ = moving to right> + moving to left> The wavefunction for the particle is an equal superposition of the two states of precise momentum. When we measure the momentum (speed), we find one of these two possibilities. Because they are equally weighted, we measure them with equal probability A Measurement We interpret this as saying that before the measurement, particle exists equally in states momentum to right momentum to left When we measure the momentum, we get a particular value (right or left). The probability is determined by the weighting of the quantum state in the wavefunction. The measurement has altered the wavefunction. The wavefunction has collapsed into a definite momentum state. 11 Double-slit particle interference Reduce intensity until only single photon at a time goes through slits. Which slit does the photon go through? 12 2

3 Interference of a single photon Which slit? In the two-slit experiment with one photon, which slit does the photon go through? A. Left slit B. Right slit C. Both slits 1/30 sec exposure 1 sec exposure 100 sec exposure Photon on both paths Path 1: photon goes through left slit Path 2: photon goes through right slit Wavefunction for the photon is a superposition of these two states. Quantum mechanics says photon is simultaneously on two widely separated paths. Superposition of quantum states We made a localized state made by superimposing ( adding together ) states of different wavelength (momenta). Quantum mechanics says this wavefunction physically represents the particle. The amplitude squared of each contribution is the probability that a measurement will determine a particular momentum. Copenhagen interpretation says that before a measurement, all momenta exist. Measurement collapses the wavefunction into a particular momentum state (this is the measured momentum) Measuring which slit A superposition state Measure induced current from moving charged particle Suppose we measure which slit the particle goes through? Interference pattern is destroyed! Wavefunction changes instantaneously over entire screen when measurement is made. 17 Margarita or Beer? This QM state has equal superposition of two. Each outcome (drinking margarita, drinking beer) is equally likely. Actual outcome not determined until measurement is made (drink is tasted). 18 3

4 What is object before the measurement? Quantum dice game What is this new drink? Is it really a physical object? Exactly how does the transformation from this object to a beer or a margarita take place? This is the collapse of the wavefunction. Suppose we have a six-sided quantum die. It is not fair, but comes up 2 most often. It s wavefunction could be A B C Not universally accepted 5th Solvay Conf, Brussels 1927 Electrons and Photons Historically, not everyone agreed with this interpretation. Einstein was a notable opponent God does not play dice These ideas hotly debated in the early part of the 20th century. Particulary famous venue was 1927 Solvay conference in Belgium 21 Seventeen of the 29 attendees were or became Nobel Prize winners. A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schroedingr, E Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin, P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L.V. de Broglie, M. Born, N. Bohr I. Langmeir, M. Planck, Mme. Curie, H.A. Lorntz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. 22 Wed. Richardson Mar. 29, 2006 Phys107 Lecture 26 Absent: W.H. Bragg, MM. H. Deslandres, E. van Aubel Schroedinger s cat A home movie shot by Irving Langmuir, (the 1932 Nobel Prize winner in chemistry). It captures 2 minutes of an intermission in the proceedings. Twenty-one of the 29 attendees are on the film. Voice-over is by Nancy Thorndike Greenspan. Solvay movie Some founders of quantum mechanics were not happy with this interpretation. Schroedinger developed scenario involving a living object to illustrate his skepticism. His scenario involved a cat, a radioactive atom, and a canister of poison gas

5 A cat, along with a decaying radioactive nucleus, a radiation detector, and a flask of poisonous gas are enclosed in a box. The radiation detector will break the flask if it detects emission from decay of nucleus (which decays with 50% probability). This kills the cat. But maybe it doesn t decay (also with 50% probability). This doesn t kill the cat. This wavefunction is a superposition of A nucleus that has not decayed A radiation detector that has not detected radiation An unbroken flask A live cat and A decayed nucleus A radiation detector that has detected radiation A broken flask A dead cat Wavefunction collapse Quantum computing This wavefunction collapses into one or the other state with 50% probability when you open the box and observe the situation. How can the cat be both dead and alive? Wouldn t it know? Who qualifies as making the measurement, the cat or the box opener? In the last several years, it has been discovered that these superposition ideas could be used in a novel way to do complex calculations. A normal computer uses bits Each bit can take a value of 0 or 1. A qubit (quantum bit) is a physical device that is in a linear superposition of two quantum states. Can think of these two quantum states as the 0 and 1 of a classical binary computer Qubits Qubits label two levels of an atom as 0> and 1> Quantum computing Create superposition state of N atoms Parallel processing of quantum data. 0> 1> Superposition state ψ>= a 0> + b 1> Create a superposition state of 2 atoms, for example ψ>= a 00> + b 11> This is an entangled state

6 Quantum computing Quantum computing algorithms Suppose we have a 3-bit quantum computer. What is the number of simultaneous computations it could perform? A. 3 B. 6 C. 8 Factoring large numbers is a problem in which a quantum computer works much better than a classical computer. If we could do this with 10,000 qubits we could factor a number larger than any classical computer can. So far people can do experiments with < 10 qubits Building a Quantum Computer One Successful Quantum Computer Atoms Choose two convenient energy levels for qubit 1-qubit operations with laser light 2-qubit operations with vibrational mode coupling (ion traps) or interchange of photons (cavity QED) Superconductors Qubit is presence or absence of flux 1-qubit operations with applied field 2-qubit operations with exchange of flux Electrons on Liquid Helium Electron spin as qubit 1-qubit operations with applied fields 2-qubit operations with spatial overlap Nuclear Magnetic Resonance Nuclear spins as qubits 1-qubit operations with applied fields 2-qubit operations by means of naturally present exchange interactions 33 7 qubit implementation of Shor s algorithm. Molecule chemically synthesized to factor 15 in prime factors. Nuclear spin plays role of Qubit. Detected by nuclear magnetic resonance technique. Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance LIEVEN M. K. VANDERSYPEN, MATTHIAS STEFFEN, GREGORY BREYTA, COSTANTINO S. YANNONI, MARK H. SHERWOOD & ISAAC L. CHUANG Nature 414, (20 December 2001); 34 Scenario for coupled solid state Qubits (Courtesty Prof. M. Eriksson, UW-Madison Physics) Electrostatics Wave Function 2 qubits 4 qubits Coupling off Coupling on Uncoupled J 0 Swap J >