ASTRONOMY 9: HISTORY OF COSMOLOGY

Handout #21

J. E. Baker
UC Berkeley, Spring 2000

Life in the Quantum World

• The Nature of Light Revisited
  • Recall the ``ultraviolet catastrophe'': classical physics predicts blackbody spectrum keeps rising, infinite energies at high frequencies!
  • Max Planck (1900) makes a desperate hypothesis to explain the observed spectra:
  • Matter can only absorb and emit light in discrete (quantized) amounts:
    
    E = hf
    
    

  • Light of frequency f comes in indivisible little packets of energy hf
  • $h \approx 6.627\times 10^{-34}$ Joule sec is Planck's constant
    • Joule is an amount of energy (about enough to lift your textbook a few cm)
    • Small size of h means we don't usually see weird quantum effects in everyday life
  • Einstein (1905) studies photoelectric effect
    • Awarded 1921 Nobel prize for this work (not relativity!)
    • Shine light of frequency f on a metal
    • No electrons are ejected for f<f₀
    • For f>f₀, energy of ejected electrons is h(f-f₀)
    • Light is behaving like a particle (photon)
  • But sometimes light behaves like a wave (eg, it diffracts around corners) $\Rightarrow$ wave/particle duality
• The Nature of Matter: A Brief History of Atomic Theory
  • J. J. Thompson ($\sim$1900): ``plum pudding'' model
    • Electrons are plums
    • Diffuse positively charged pudding
  • Ernest Rutherford (1911)
    • Studies ``Rutherford scattering'' of charged particles hitting thin metal sheets
    • Almost all are just slightly deflected, but a few bounce straight back
    • Atomic nucleus is small and very dense, most of atom is empty space!
    • Electrons whiz around
  • Niels Bohr (1913)
    • Electron is accelerating as it moves around nucleus
    • Maxwell's equations say it should radiate all its energy and collapse to the nucleus!
    • Bohr's solution: electron can only exist in discrete atomic levels--atoms are quantized too!
    • Light is emitted or absorbed when electron jumps between levels (a ``quantum leap'': it doesn't go in between)
    • Each atom has a different set of levels, explains spectral lines
  • Louis de Broglie (1924)
    • Just as light can behave like particles, electrons can behave like waves!
    • Electron can exist only in orbits where integer number of waves ``fit'' around the orbit
    • Explains Bohr's quantization idea
• The Uncertainty Principle
  • Werner Heisenberg (1925)
  • If precisely measure momentum (p) of a particle, its position (x) becomes completely uncertain, and vice versa!
    
    $$\Delta x \, \Delta p > \frac{\hbar}{2}$$
    
    
    where $\hbar = h/2\pi$ and $\Delta$ means ``uncertainty in''
  • Can also be expressed as relationship between measurements of energy and time: $\Delta E \, \Delta t > \hbar/2$
  • Destroys old Laplacian philosophy of determinism!
  • Observer also plays a crucial role: what you choose to measure influences the physical state of the system!
• Max Born and Erwin Schrödinger (1927)
  • Quantum mechanics (QM) only describes probabilities
  • The wavefunction $\psi$ specifies all there is to know about a system
  • Schrodinger equation determines evolution of wavefunction in time and space
  • The wavefunction of an electron at point x represents the probability of finding it at x: $P\propto \vert\psi\vert^2$
  • Particles are ``fuzzy''!
• Quantum Statistics
  • Particles have an intrinsic ``spin'', also quantized: only comes in units of $\hbar/2$
  • Particles aren't really little spheres, spin is not really like our common-sense notion
  • Two types of particles, classified according to spin:
  • Fermions: Spins are a half-integer multiple (1/2, 3/2, 5/2, ...) of $\hbar$
    • Named after Enrico Fermi (nuclear physicist)
    • Examples: electron, proton, neutron (normal matter)
    • ``Anti-social'' particles: obey Pauli's Exclusion Principle
    • No two fermions may be in the same quantum state
  • Bosons: Spins are an integer multiple (0, 1, 2, ...) of $\hbar$
    • Named after Satyendranath Bose (Indian physicist)
    • Examples: photon (spin-0), graviton (spin-2, carries gravitational force just like photon carries electromagnetism)
    • Gregarious, ``social'' particles: all like to enter the same quantum state!
    • Note: spin-2 in a sense means you have to rotate by $2\times 360^\circ = 720^\circ$ before you look the same! Spin-1/2 means you only have to rotate by $180^\circ$
• Interpretation of Quantum Mechanics
  • Quantum mechanics works very well, in that it is a great tool for calculating things
  • Problems only arise when you try to interpret it in terms of our everyday experience
  • Very disturbing to many founders of quantum physics
  • Recent research is beginning to explore boundary between quantum and classical realms: how does quantum weirdness give way to macroscopic classical normality?

  1.

  Copenhagen (``standard'') interpretation (Bohr and others, 1920s)

  • Act of observation mysteriously ``collapses the wavefunction''
  • When you're not looking at it, the electron spreads out all over space, and different outcomes are superposed on top of each other
  • The act of observation somehow localizes things, or selects a particular outcome
  • Einstein objects: do you really think the Moon isn't there when you're not looking at it?
  • Example: Schrodinger's cat
    • Feline in a box with vial of poison, decay of a radioactive atom triggers release of poison
    • QM says the atom enters a ``superposition of states'', both decayed and non-decayed, until an observation is made
    • But then is the cat also simultaneously dead and not-dead? Implies macroscopic manifestation of quantum weirdness
    • What exactly is the act of ``observation''? Does human consciousness play a special physical role? Or do common environmental interactions collapse the wavefunction and prevent the cat from entering superposition?

  2.

  Many-worlds interpretation (Hugh Everett, 1957)

  • All possibilities embodied by the wavefunction actually occur!
  • Universe constantly forks into a multitude of different versions of itself (``parallel universes'')
  • In one universe, Schrodinger's cat is alive; in another it is dead
  • Universes are cut off from each other

  3.

  ``Hidden variables'' (historically favored by Einstein and others)

  • Idea: QM is a terribly incomplete theory, does not preserve locality
  • Example: Einstein-Podolsky-Rosen (EPR) ``paradox''
    • Send two particles off in a quantum superposition (eg, spins of each are pointing both up and down at the same time!)
    • Measurement of one particle causes it to choose one state or the other
    • Has instantaneous effect on the other particle, even if it has traveled half-way across the universe! Spooky
    • Note: doesn't violate special relativity, since still can't send a signal faster than light
  • Some unknown underlying ``variables'' affect the system, allow only local effects
  • Experimental evidence does not support this interpretation

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