A Short Course on Quantum Mechanics and Methods of Quantization 71.2 Overview of Quantum Mechanics
1.2.1 Fundamental definitions and examples
Whenever teaching introductory courses in either classical or quantum physics, one
is lead to first answer the following question: what are the minimal conceptual and
mathematical structures required for the description of a physical system?
According to modern theoretical setup, we essentially need a few main ingre-
dients that can be identified with:
- a space of statesS, a state being an object which is able to encode and
describe all degrees of freedom of the system; - a space of observablesO, which can be applied to a state to change it in
some specified manner; - a law of evolution, which fixes (possibly in a unique way) how states
and/or observables change as we let the system evolve, eventually under
the influence of external and internal forces; - apairingμ:S×O→R, which produces a real number out of a state and
an observable and corresponds to a measurement process.
In CM,Sis the collections of independent coordinates and momenta{qi,pi}de-
scribing the system whereas mathematicallyS is a symplectic manifold; Ois
given by (usually differentiable) real functions onS; the evolution laws are given
in terms of differential equations, which can be derived as a special observable,
the Hamiltonian, is specified; whileμis a probability measure and the process of
measure corresponds to evaluate the average value of an observable function on
the space of states.
Similarly, the so-called “Postulates” of QM give the mathematical framework
and explain the conceptual physical interpretation for a system governed by quan-
tum rules. We will try to illustrate them by means of some simple but paradigmatic
examples. For a more detailed exposition, we refer the interested reader to QM
textbooks[8; 15]or books on Quantum Information and Computation[ 32 ].
The interested reader may find in the second part of this volume a discussion
of the postulates of QM with emphasis on the algebraic approach, which includes
a detailed explanation of the non-boolean logic and of the von Neumann algebra
of observables.
1.2.1.1 The space of states
Postulate 1. A physical (pure) state of a QM system is represented by a ray
in a complex and separable Hilbert spaceH, i.e. by an equivalence class of unit
vectors under the relation:|ψ 1 〉 |ψ 2 〉⇔|ψ 2 〉=eiα|ψ 1 〉,α∈R.