Quantum Mechanics - Axiomatic Formulation
Overview
State of a Quantum System
The state of a quantum system is a unit vector in a
complex
Hilbert space.
Following Dirac, this is usually
denoted as a Dirac ket.
{% | \psi > %}
Observable
Observable properties of the system are represented by operators on the Hilbert space. When the state of the system is
one of the
eigenvectors
of the observable, then the system is guaranteed to be observed with the observable equal to the
eigenvalue
corresponding to the given eigenvector. The eigenvectors are assumed to form a complete orthogonal basis for the
Hilbert space.
- {% A |a {>} = a | a {>} %}
- eigenvectors are states with a definite value for the observable {% A %} given by the
eigenvalue {% a %}
- {% {<} a | a' {>} = \delta(a- a') %}
-
eigenvectors are orthogonal
- {% \int da |a {>} {<} a | = 1 %}
-
eigenvectors form a complete basis
When the system is not in an eigenvector of the observable operator, then it can be expressed as a linear combination of the
eigenvectors.
{% | \psi {>} \; = \sum_i a_i | \phi_i > %}
When an observation corresponding the given observable is made, the system will collapse into one of the eigenvectors states and
be observable will be seen to be the eigenvalue corresponding to that state. The probability that the state collapses to any
given state {% | \phi > %}
is given by the inner product associated with the Hilbert space squared.
{% | {<} \phi | \psi {>} | ^2 %}
The probability amplitude is defined to be
{% {<} \phi | \psi {>} %}
which is a complex number.
Commuting Observables
Position and Momentum
When a particle is observed at a particular point {% q %},
the state then collapses into a vector that represents the given point, using Diracs notation,
{% | q {>} %} .
Likewise, if the point is measured to have a given momentum {% p %}, it will then be in the
state {% | p {>} %}
Topics
