HPS 0410 | Einstein for Everyone | Spring 2012 |

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For submission

1. (a) What experiment gives us good
reason to think that light consists of waves? How does it lead to that
result?

(b) What experiment gives us good reason to think that high frequency light has
its energy localized at points in space, like a particle? How does it lead to
that result?

2. (a) What model of the atom tells us
that electrons could be found anywhere in the vicinity of an atom's very small
nucleus? On what physical theory is that model based?

(b) How does the theory of atomic spectra suggest that the theory of (a) is
wrong.

(c) What theory of the atom results from taking the atomic spectra
seriously?

3. (a) How does de Broglie's theory of
matter waves connect the energy and momentum of particles with the frequency
and wavelength of waves?

(b) How does this theory make sense of the theory of the atom of 2.(c)?

For discussion in the recitation

A. Consider the sequence of theories that set us on the way to modern quantum theory. They mixed together components of classical physics with new quantum notions and, to use the "old quantum theory" one had to invoke both classical and quantum notions at the same time:

• Planck's analysis of heat radiation assumed that heat radiation was generated by emission and absorbtion of light from classically described electric resonators. His analysis seemed to require that electric resonators only be allowed to adopt discrete energy levels, although classical physics told us that they could adopt a continuous range of energies.

• Einstein's 1905 light quantum hypothesis held that high frequency light energy is localized at points in space. Yet at the same time Einstein still allowed that interference phenomena were possible for light and that requires that the light be spread out in space.

• Bohr's 1913 theory of the atom took the classical theory of electron orbits in which electrons may orbit at any distance from the nucleus, but cannot do so stably. To it he added the assumption that these electrons can orbit stably, but only at very few discrete distances from the nucleus.

In all these cases, the theorists seem to make essential use of logically incompatible assumptions. Electrons cannot both be stable and not be stable, for example. The presence of a logical inconsistency is usually taken to be fatal to a physical theory. Yet here were successful theories that seemed to depend essentially on contradictory assumptions.

(a) Should we require our physical theories to be consistent?

(b) Do you know any examples of theories that were discarded when they were found to be based on contradictory assumptions?

(c) Are there other examples of successful theories that are based on inconsistent assumptions?

B.To sharpen the problems above, consider this. If a theory is contradictory, then it allows both the truth of some proposition A and also the truth of its negation not-A. In classical logic, one can deduce anything at all from a contradition. Here's the proof. (If you have had a logic class, this will seem entirely trivial. If not, you may be a bit startled by how easy it is to infer anything from a contradiction.) The inference combines two standard argument forms:

Addition
C |
Disjunctive syllogism
C or D |

To prove any proposition B from a contradiction (A and not-A)

1. A (Assumption) 2. not-A (Assumption) 3. A or B (From 1,2 by Addition) 4. B (From 2, 3 by Disjunctive Syllogism) |
For example: 1. Electron orbits are stable. (Assumption) 2. Electron orbits are not stable. (Assumption) 3. Electron orbits are stable OR bananas are high in Potassium. (From 1, 2 by Addition) 4. Bananas are high in Potassium. (From 2, 3 by Disjunctive Syllogism) |

What this tells us is that, in an inconsistent theory, we can deduce anything. So should we be so surprised that Planck, Einstein and Bohr can deduce their results from inconsistent premises? From inconsistent premises, we could deduce that planets orbit in squares; or that everything is made of licorice!

Or is there something more subtle at work? Planck, Einstein and Bohr seem to have found some deep truths about the world. How can they be extracted from the snake pit of logical inconsistency?