4.6. Track 2[[Student version, December 8, 2002]] 133
idea was right, then both Planck’s thermal radiation and the photoelectric experiments should in-
dependently determine a number, which we now call the Planck constant. Einstein showed that
both experiments gave thesame numerical valueof this constant.
- Einstein did not invent the equations for electrodynamics; Maxwell did. Nor was Einstein the first
to point out their curious invariances; H. Lorentz did. Einstein did draw attention to a consequence
of this invariance: the existence of a fundamental limiting velocity, the speed of lightc.Once again,
the idea seemed crazy. But Einstein showed that doggedly following it to its logical endpoint led
to a new, quantitative, experimentally testable prediction in an apparently very distant field of
research. In his very first relativity paper, also published in 1905, he observed that if the massmof
abodycould change, the transformation would necessarily liberate a definite amount energy equal
to ∆E=(∆m)c^2. - Einstein said some deep things about the geometry of space and time, but D. Hilbert was saying
many similar things at about the same time. Only Einstein, however, realized that measuring an
apple’s fall yields the numerical value of a physical parameter (Newton’s constant), which also con-
trols the fall of aphoton.His theory thus made quantitative predictions about both the bending of
light by the Sun and the gravitational blue-shift of a falling photon. The quantitative experimental
confirmation of the light-bending prediction catapulted Einstein to international fame.
4.3.1′
- We saw that typically the scaling exponent for a polymer in solvent is not exactly 1/2. One
special condition, called “theta solvent” actually does give a scaling exponent of 1/2, the same
as the result of our na ̈ıve analysis. Theta conditions roughly correspond to the case where the
monomers attract each other just as much as they attract solvent molecules. In some cases theta
conditions can be reached simply by adjusting the temperature. - The precise definition of the radius of gyrationRGis the root-mean-square distance of the
individual monomers from the polymer’s center of mass. For long polymer chains it is related to
the end-to-end distance by the relation (RG)^2 =^16 〈(rN)^2 〉. - Another test for polymer coil size is via light scattering; see (Tanford, 1961).
4.4.2′What if we don’t have everything uniform in theyandzdirections? The net flux of particles is
really avector,like velocity; ourjwasjust thexcomponent of this vector. Likewise, the derivative
dc/dxis just thexcomponent of a vector, thegradient,denoted∇c(and pronounced “gradc”).
In this language, the general form of Fick’s law is thenj=−D∇c,and the diffusion equation reads
∂c
∂t
=D∇^2 c.4.4.3′One can hardly overstate the conceptual importance of the idea that a probability distribution
may have deterministic evolution, even if the events it describes are themselves random. The
same idea (with different details) underlies quantum mechanics. There is a popular conception
that quantum theory says “everything is uncertain; nothing can be predicted.” But Schr ̈odinger’s
equation is deterministic. Its solution, the wavefunction, when squared yields theprobabilityof
certain observations being made in any given trial, just asc(x, t)reflects the probability of finding
aparticle nearxat timet.