Page 105 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
α
This inequality is to hold for arbitrary real . Thus,
2
—
( 〈 ∆x ˆ) 〉 (〈 ∆p ˆ ) 〉 ≥ ----- (3.13)
2
2
4
must hold. (3.13) is called the Heisenberg uncertainty principle. Heisen-
berg derived it in 1925, and it gave rise to the creation of a new physical
discipline called quantum mechanics. The simple significance is that the
product of the variances of position and momentum has a minimum,
2
which is — ⁄ 4 . Since it clearly derives from the commutator relations its
physical interpretation is that position and momentum may not be deter-
mined by measurement with arbitrary precision. Note that (3.13) only
gives a lower limit. Now, turning back to the example of a gaussian wave-
function we see that it is a function, we see that it is a function fulfilling
(3.13) exactly with the “=”-sign. Thus we call it a state of “minimal
uncertainty”.

3.1.2 The Schrödinger Equation

The quantitative description of wavefunctions was given by Erwin
Schrödinger in 1924. An example of a classical wave equation is a partial
differential equation of second order in space and time. The corpuscular
E
nature of the light field as discovered by Einstein relates the energy of
the photon to the angular frequency of the electromagnetic wave by
E = —ω , while its momentum is given by p = —k . The assumption of
De Broglie for those relations to hold also for particles together with
2 2
E = — k ⁄ ( 2m) led Schrödinger to take an equation of first order in
time. This equation of motion must read

∂ — 2
,
(
(
,
i— Ψ x t) = – -------∇ 2 Ψ x t) (3.14)
t ∂ 2m
which is a partial differential equation of first order in time and second
order in space. Note the fact that an imaginary coefficient of the time
derivative turns (3.14) into a wave equation. The spatial derivative of sec-

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