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9 Wave Equations
155

where v denotes again a real positive parameter. Clearly, the velocity of propa-
k
gation of this plane wave is ±v . Therefore, in more than one dimension the
|k|
propagation velocities of plane wave solutions of the wave equation lie on the
sphere with radius v, but their directions depend on the wave vector. This is
a weak dispersion effect.
Quantum mechanics [19] is governed by a very particular wave equation,
named after the Nobel price winning Austrian theoretical physicist Erwin
4
Schrödinger . The Schrödinger equation, in its most basic form modeling the
quantistic transport of an elementary particle (say, an electron) with positive
mass m, is a linear partial differential equation for a complex valued function u,
thesocalledwavefunctionofthe particle.The equation reads:

2 n
i u t = − Δu + V(x)u , x ∈ R , t ∈ R (9.11)
2m
5
where is the so called normalized Planck constant and V(x) the real valued
electric potential field driving the motion of the electron. The wave function u
is an auxiliary quantity, the important physical observables are computed from
u by ‘post-processing’. They are quadratic in the wave function, e.g.

2
ρ(x, t) = |u(x, t)| (9.12)
is the (probabilistic) position density of the particle,

j(x, t) = Im u(x, t)grad u (x, t) (9.13)
∗
∗
its current density ( denotes complex conjugation) and
2
e(x, t) = |grad u(x, t)| + V(x)|u(x, t)| 2 (9.14)
2
2m
its energy density. Note that the total mass

M := ρdx
n

and the total energy

E := edx
n

are time-conserved by the Schrödinger equation.

4
http://nobelprize.org/physics/laureates/1933/schrodinger-bio.html
5
http://scienceworld.wolfram.com/physics/PlancksConstant.html