1.2 Basic principles of quantum mechanics
We’ll divide the conventional list of basic principles of quantum mechanics into
two parts, with the first covering the fundamental mathematics structures.
1.2.1 Fundamental axioms of quantum mechanics
In classical physics, the state of a system is given by a point in a “phase space”,
which can be thought of equivalently as the space of solutions of an equation
of motion, or as (parametrizing solutions by initial value data) the space of
coordinates and momenta. Observable quantities are just functions on this space
(e.g., functions of the coordinates and momenta). There is one distinguished
observable, the energy or Hamiltonian, and it determines how states evolve in
time through Hamilton’s equations.
The basic structure of quantum mechanics is quite different, with the for-
malism built on the following simple axioms:
Axiom(States).The state of a quantum mechanical system is given by a non-
zero vector in a complex vector spaceHwith Hermitian inner product〈·,·〉.
We’ll review in chapter 4 some linear algebra, including the properties of in-
ner products on complex vector spaces.Hmay be finite or infinite dimensional,
with further restrictions required in the infinite dimensional case (e.g., we may
want to requireHto be a Hilbert space). Note two very important differences
with classical mechanical states:
- The state space is always linear: a linear combination of states is also a
state. - The state space is acomplexvector space: these linear combinations can
and do crucially involve complex numbers, in an inescapable way. In the
classical case only real numbers appear, with complex numbers used only
as an inessential calculational tool.
We will sometimes use the notation introduced by Dirac for vectors in the state
spaceH: such a vector with a labelψis denoted
|ψ〉Axiom(Observables).The observables of a quantum mechanical system are
given by self-adjoint linear operators onH.
We’ll review the definition of self-adjointness forHfinite dimensional in
chapter 4. ForHinfinite dimensional, the definition becomes much more subtle,
and we will not enter into the analysis needed.
Axiom(Dynamics).There is a distinguished observable, the HamiltonianH.
Time evolution of states|ψ(t)〉∈His given by the Schr ̈odinger equation
i~d
dt|ψ(t)〉=H|ψ(t)〉 (1.1)