Page 45 - Science at the nanoscale

RPS: PSP0007 - Science-at-Nanoscale
9:2
June 9, 2009
3.2. Basic Postulates of Quantum Mechanics
3.1.4
Wave Particle Duality
Since the introduction of light in the form of photons with partic-
ulate nature, scientists began to wonder if matter, considered to be
made of particles, might also have a wave nature. Louis de Broglie
was the first person to provide an insight into the wave nature of
matter. The wavelength λ of a particle, according to de Broglie, is
given by
λ = h/p
where p is the momentum of the particle. This wave nature of
particles was confirmed by the observation of electron diffraction.
Once the idea of the wave nature of a particle was established,
rapid developments followed that provided a theory to determine
the wave properties of a particle moving in the presence of a con-
servative field.
Heisenberg’s Uncertainty Principle
3.1.5
With the wave description, it is impossible to know simultane-
ously and with exactness both the position and the momentum of
a particle. Suppose we know the position, x, of a particle very pre-
cisely, then we cannot simultaneous determine the momentum, p,
of the particle very precisely. The uncertainty in the position, ∆x,
and the uncertainty in the momentum, ∆p, follows the Heisen-
berg’s Uncertainty Principle, ∆x × ∆p > h/2π. Any measure-
ment made has to satisfy the uncertainty relation and be of limited
precision. The classical concept of having an arbitrarily precise
knowledge of both x and p does not apply. (3.5) 35 ch03
3.2 BASIC POSTULATES OF QUANTUM MECHANICS
Consider a physical system consisting of a particle. Quantum
physics proposes a special function known as the wavefunction
that determines everything that can be known about the system.
The wavefunction is a function of position and time, ψ(r, t) and
is mathematically a complex function. The product of a wave-
∗ 2
function ψ(r, t) and its complex conjugate ψ (r, t) gives |ψ(r, t)|
that represents the probability density of finding the particle in a
particular state. Hence the probability of finding the particle in a