In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the position operator is the
operator that corresponds to the position
observable of a
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
.
When the position operator is considered with a wide enough domain (e.g. the space of
tempered distributions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
), its
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s are the possible
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
s of the particle.
In one dimension, if by the symbol
we denote the unitary eigenvector of the position operator corresponding to the eigenvalue
, then,
represents the state of the particle in which we know with certainty to find the particle itself at position
.
Therefore, denoting the position operator by the symbol
in the literature we find also other symbols for the position operator, for instance
(from Lagrangian mechanics),
and so on we can write
for every real position
.
One possible realization of the unitary state with position
is the Dirac delta (function) distribution centered at the position
, often denoted by
.
In quantum mechanics, the ordered (continuous) family of all Dirac distributions, i.e. the family
is called the (unitary) position basis (in one dimension), just because it is a (unitary) eigenbasis of the position operator
. Note that even though this family is ordered by the continuous coordinate
, the
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of this basis set is
Aleph nought, instead of
Aleph one
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...
. This is because the Dirac distributions in this family are required to be
square-integrable (see the relevant section below), which means that the Hilbert space spanned by this basis has countably infinite many basis states. One way to understand this is to treat the Dirac delta functions as the limit of very tiny lattice segments of the continuous position space, and therefore as the lattice spatial period goes to zero, the number of these lattice sites goes to countable infinity.
It is fundamental to observe that there exists only one linear continuous endomorphism
on the space of tempered distributions such that
for every real point
. It's possible to prove that the unique above endomorphism is necessarily defined by
for every tempered distribution
, where
denotes the coordinate function of the position line defined from the real line into the complex plane by
Introduction
In one dimension for a particle confined into a straight line the
square modulus
of a normalized square integrable wave-function
represents the
probability density of finding the particle at some position
of the real-line, at a certain time.
In other terms, if at a certain instant of time the particle is in the state represented by a square integrable wave function
and assuming the wave function
be of
-norm equal 1,
then the probability to find the particle in the position range