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9 Wave Equations
157
Fig. 9.4. Plane and transversal Waves

Formanymathematicalpurposesitisconvenienttoscale theSchrödinger
equation by introducing macroscopic space-time coordinates. Denoting by ε the
microscopic/macroscopic aspect ratio, we obtain:
ε 2
n
iεu t = − Δu + V(x)u , x ∈ R , t ∈ R . (9.15)
2
In many applications, e.g. in quantum effect semiconductor devices, Bose–
6
Einstein condensates etc., ε is a small parameter. In order to understand the
dispersionpropertyandthetransportofoscillationspropertyoftheSchr¨ odinger
equation consider the free quantistic transport case V = 0 and prescribe an ε-
oscillatory plane wave initial wave function:

x

u(x, t = 0) = exp ik · . (9.16)
ε
It is an easy exercise to show that the solution of the initial value problem
(9.15), (9.16) in this case is given by the function:
x i t

u(x, t) = exp ik · − |k| 2 . (9.17)
ε 2 ε
Two issues are apparent:
1
a) The initial spatial oscillations with frequency of order of are propagated in
ε
spaceandoscillationsintimewithfrequencyofthesameorderaregenerated.
This is a typical property of linear wave equations.
k
b) The velocity of the wave with initial wave vector k is equal to .Thuswaves
ε ε
with different wave vectors move with different velocity vectors and possible
wave speeds are not restricted. This is a dispersion property, much stronger
than in the case of the linear second order wave equation discussed above.
Of particular interest is the so called classical limit of the Schrödinger equa-
tion ε → 0, which corresponds to observing particle motion on larger and
larger scales. Intuitively, quantistic effects should diminish in this process and
the motion should become dominated by classical mechanics, i.e. by Newton’s
second law. This was verbalized by the German Nobel Prize winning physicist
7
Max Planck , who stated in the year 1900:
Classical mechanics is the limit of quantum mechanics as tends to zero.

6
http://www.colorado.edu/physics/2000/bec/
7
http://www.dhm.de/lemo/html/biografien/PlanckMax/