Quantum Mechanics

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.