Quantum Mechanics 103 - PowerPoint PPT Presentation

1 / 22
About This Presentation
Title:

Quantum Mechanics 103

Description:

If the Geiger Counter triggers, a gun is discharged and the cat is killed ... The Geiger counter remains both triggered. and undecayed. and untriggered. The gun ... – PowerPoint PPT presentation

Number of Views:37
Avg rating:3.0/5.0
Slides: 23
Provided by: Phys181
Learn more at: https://www.rpi.edu
Category:

less

Transcript and Presenter's Notes

Title: Quantum Mechanics 103


1
Quantum Mechanics 103
  • Quantum Implications for Computing

2
Schrödinger and Uncertainty
  • Going back to Taylors experiment, we see that
    the wavefunction of the photon extends through
    both slits
  • Therefore the photon has traveled through both
    openings simultaneously
  • The wavefunction of a particle will contain
    every possible path the particle could take until
    the particle is detected by scattering or being
    absorbed
  • These paths can interfere with each other to
    produce diffraction-like probability patterns
  • BUT, Schrödinger took this explanation to an
    extreme

3
Schrödingers Famous Cat
  • Suppose a radioactive substance is put in a box
    with a cat for a period of time
  • If the Geiger Counter triggers, a gun is
    discharged and the cat is killed

4
Schrödingers Famous Cat
  • Until an observer opens the box to make a
    measurement of the system,
  • The nucleus remains both decayed

and undecayed
  • The Geiger counter remains both triggered

and untriggered
  • The gun has both fired

and not fired
  • 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
5
Paradox?
  • Paradoxical as it may seem, the concept of
    superposition of states is borne out well in
    experiment
  • Like superposition of waves producing
    interference effects
  • Quantum Mechanics is one of the most-tested and
    best-verified theories of all time
  • But it seems counter-intuitive since we live in a
    macroscopic world where uncertainty on the order
    of ? is not noticeable

6
Quantum paradox 2
  • Einstein-Podolsky-Rosen (EPR) paradox
  • 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 entangled
    state

7
More EPR
  • 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 know what its spin is
    and even that the spin has been determined

8
Interpreting EPR
  • Measuring one electron affects the other
    electron!
  • For the other electron to know about the
    measurement, a signal must be sent faster than
    the speed of light!
  • Such an effect has been experimentally verified,
    but it is still a topic of much debate

9
Interference effects
  • 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 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

10
Interpretation
  • 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!
  • 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

11
Quantum Theory meets Bomb
  • Four possible paths RR and TT hit upper
    detector, TR and RT hit lower detector
    (Rreflected, Ttransmitted)
  • Classically, 4 equally-likely paths, so prob of
    each is 1/4, so prob at each detector is 1/4
    1/4 ½, 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!
  • This allows interference effects

12
What wave function would give 50 at each
detector?
  • Must have a2 b2 c2 d2 1/4
  • Need a b2 cd2 1/2

13
If Path Lengths Differ, Might Have
  • Lower detector
  • Upper detector

Voila, Interference!
14
When Measure Which Path,
  • Lower detector
  • Upper detector

Voila, No Interference!
15
Quantum Storage
  • Consider a quantum dot capacitor, with sides 1 nm
    in length and 0.010 microns between plates
  • How much energy required to place a single
    electron on those plates?
  • Can make confinement of dot dependent upon
    voltage
  • Lower the voltage, let an electron on gt 1
  • Lower voltage on other side, let the electron off
    -gt 0

16
What must a computer do?
  • Deterministic Turing Machine still good model
  • Two pieces
  • Read/write head in some internal state
  • Infinite tape with series of 1s, 0s, or blanks
  • Follows algorithms by performing 3 steps
  • Read value of tape at heads location
  • Write some value based on internal state and
    value read
  • Move to next value on tape

17
Can we improve this model?
  • 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

18
Limits on Turing Machines
  • Some problems are solvable in theory but take too
    long in practice
  • 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
  • multiplication grows as the square of digits
  • Fourier transform grows faster than square
  • factoring grows almost exponentially

19
Examples of factoring time
  • 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
    MIP-years
  • 71-digit decimal number (1984) needs 0.1
    MIP-years
  • 129-digit decimal number (1994) needs 5000
    MIP-years

20
Quantum Cryptography
  • Current best encryption uses public key for
    encoding
  • Need private key (factors of large integer in
    public key) to decode
  • Really safe unless
  • Someone can access your private key
  • Quantum computers become prevalent

21
Quantum Cryptography II
  • Quantum Computers can factor large numbers
    near-instantly, making public key encryption
    passe
  • But, can send quantum information and know
    whether it has been intercepted

22
What problems face QC?
  • Decoherence if measurement made, superposition
    collapses
  • Even if measurement not intentional!
  • i.e., if box moves, cat becomes alive or dead,
    not both
  • Quantum error correction
  • No trail of path taken (or else no superposition)
  • Proven to be possible that doesnt mean its
    easy!
  • HUGE Technical challenges
  • electronic states in ion traps (slow, leakage)
  • photons in cavity (spontaneous emission)
  • nuclear spins in molecule (small signal in large
    noise)
Write a Comment
User Comments (0)
About PowerShow.com