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reaction constant characteristic of the given reac- Heisenberg uncertainty principle In quan-
tion of Y. The equation is often encountered in tum mechanics, a principle set forth by Heisen-
a form with lg k or lg K written as a separate berg which asserts that the simultaneous exact
0
0
term on the right-hand side, e.g., measurements of the values of position and
momentum is impossible. If q is the range
lg k = ρσ + lg k
0 of values found for the coordinate q of a particle,
or and p is the range in the simultaneous meas-
lg K = ρσ + lg K 0 urement of the corresponding momentum, then
q · p ≥ h, where h is the Planck constant.
It then signifies the intercept corresponding to
X = H in a regression of lg k or lg K on σ.
helicity A quantitative measure of the
See also Yukawa-Tsuno equation.
amount of helix in a polymer molecule with heli-
harmonic form A k-form α on a manifold cal structure.
M is called harmonic if α = 0, where =
dδ + δd is the Laplace-deRham operator on M. helix In biochemistry, a molecular structure
having a given skew symmetry. For proteins,
harmonic frequency generation Produc- there is α-helix which was first proposed by
tion of coherent radiation of frequency kν(k = L. Pauling. For DNAs, there is a double helix.
2, 3,.. .) from coherent radiation of frequency
ν. In general, this effect is obtained through the helix-coil transition A mathematical model
interaction of laser light with a suitable optical originally develped for the probability distribu-
medium with nonlinear polarizability. The case tion of α-helix formation in a polypeptide. The
k = 2 is referred to as frequency doubling, k = 3 model is formulated based on the transfer matrix
is frequency tripling, and k = 4 is frequency method. Itisageneralizationofone-dimensional
quadrupling. Even higher integer values of k are Ising model (cf. D. Poland and H.A. Scher-
possible. aga, Theory of Helix-Coil Transitions, Academic
Press, New York, 1970).
harmonic function A function u satisfying
u = 0. See Laplace equation. n
Helmholtz equation Let U ⊂ R be open
n
harmonic oscillator The function H = and u : U ⊂ R → R. The Helmholtz equation
1 2 2 2 or eigenvalue equation for u is
2 (−d /dx + x ) is the Hamiltonian of the har-
monicoscillator(assumingm = 1). Theintegral
− u = λu.
curves (trajectories) are circles.
heat, q, Q Q Q Energy transferred from a hotter Hermitian operator See self-adjoint oper-
q, q,
to a cooler body due to a temperature gradient. ator.
n
heat equation Let U ⊂ R open and u : heterolysis (heterolytic) The cleavage of a
U × R → R. The heat equation or diffusion covalent bond so that both bonding electrons
equation for u is remain with one of the two fragments between
which the bond is broken.
u − u = 0.
t
n
n
heat kernel On R × R the function Higgs mechanism In quantum field theory,
the spontaneous breaking of gauge symmetry, in
whichamasslessgaugeboson(Goldstoneboson)
2 (
t
e (x, y) = (4πt) −n/2 exp − |x − y| and a massless scalar field combine to form a
4t massive gauge boson, called a Higgs boson.
The action of the heat kernel on a function f is
defined by Hilbert-Schmidt operator A bounded lin-
ear operator T on a Hilbert space H is called

∗
t t Hilbert-Schmidt if traceT T< ∞, that is,
(e f )(x) = e (x, y)f (y)dy.

∗
R n λ < ∞ for the eigenvalues of T T .
j
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