Chapter 1
Mathematical Foundations
Before I begin to
introduce some basics of complex vector spaces and discuss the mathematical
foundations of quantum mechanics, I would like to present a simple
(seemingly classical) experiment from which we can derive quite a few quantum rules.
1.1 The quantum mechanical
state space
When we talk about physics, we attempt to
find a mathematical description of the world. Of course, such a description
cannot be justified from mathematical consistency alone, but has
to agree with experimental evidence. The mathematical concepts that are
introduced are usually motivated from our experience of nature.
Concepts such as position and momentum or the state of a system are
usually taken for granted in classical physics. However, many of these
have to be subjected to a careful re-examination when we try to carry
them over to quantum physics. One of the basic notions for the
description of a physical system is that of its ’state’. The ’state’ of a
physical system essentially can then be defined, roughly, as the description
of all the known (in fact one should say knowable) properties of that
system and it therefore represents your knowledge about this system. The set of
all states forms what we usually call the state space. In
classical mechanics for example this is the phase space (the variables are
then position and momentum), which
is a real
vector space.
For a classical point-particle moving in one dimension,
this space is two dimensional, one dimension for position, one dimension for
momentum. We expect, in fact you probably know this from your second
year lecture, that the quantum mechanical state space differs from
that of classical mechanics. One reason for this can be found in the
ability of quantum systems to exist in coherent superpositions of states
with complex amplitudes, other differences relate to the description of
multi-particle systems. This suggests, that a good choice for the
quantum mechanical state space may be a complex vector space.
Before I begin to
investigate the mathematical foundations of quantum mechanics, I would like to
present a simple example (including some live experiments)
which motivates the choice of complex vector spaces as state spaces a
bit more. Together with the hypothesis of the existence of photons
it will allow us also to ’derive’, or better, to make an educated guess for
the projection postulate and the rules for the computation of
measurement outcomes. It will also remind you of some of the features of
quantum mechanics which you have already encountered in your second year course.
Sumber : Plenio, Martin (2002).Quantum Mechanics.Imperial College: Blackett.
