Abstract

CHAPTER 1. OVERVIEW 8
for predicting the actions of objects that are of moderate size (that is, objects that
are not as large as galaxies but are not as small as atoms).
Quantum mechanics, though, is a probabilistic description of nature. The fun-
damental idea of quantum mechanics is that matter exhibits a wave-particle duality.
This means matter acts in some respects as both a particle and a wave. For a particle
in classical mechanics, we can describe its instantaneous state by giving its position
and velocity. In quantum mechanics [26],[23], to describe a particle’s instantaneous
state, we need an object called a wavefunction. We will now list some postulates of
quantum mechanics. For these postulates, we are considering just a single particle
and not a system of particles, and these particles are confined to move in one space
dimension. These postulates can be easily extended to a more general case:
1. The quantum state of a particle is described by a wavefunction ψ(x) whichis
an element of the complex Hilbert space L 2 . A physical interpretation of the
wavefunction is as probability distribution, where the probability of finding a
particle in a region dx about the point x is given by |ψ(x)| 2 dx. Since the particle
must exist somewhere in (−∞, ∞), this imposes the normalization condition
that ∞
−∞ |ψ(x)|2 dx =1.
2. Every physical observable quantity has a linear, Hermitian operator associated
with it. Further, for a given operator Θ, we can calculate its expected value