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 of this basis set is Aleph nought, instead of Aleph one. 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
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 is
Hence the expected value of a measurement of the position for the particle is the value
the particle is assumed to be in the state ;
the function is supposed integrable, i.e. of class ;
we indicate by the coordinate function of the position axis.
Additionally, the quantum mechanical operator corresponding to the observable position is denoted also by
for every wave function and for every point of the real line.
The circumflex over the function on the left side indicates the presence of an operator, so that this equation may be read:
The result of the position operator acting on any wave function equals the coordinate function multiplied by the wave-function .
Or more simply:
The operator multiplies any wave-function by the coordinate function .
Note 1. To be more explicit, we have introduced the coordinate function
which simply imbeds the position-line into the complex plane. It is nothing more than the canonical embedding of the real line into the complex plane.
Note 2. The expected value of the position operator, upon a wave function (state) can be reinterpreted as a scalar product:
assuming the particle in the state and assuming the function be of class – which immediately implies that the function Is integrable, i.e. of class .
Note 3. Strictly speaking, the observable position can be point-wisely defined as
for every wave function and for every point of the real line, upon the wave-functions which are precisely point-wise defined functions. In the case of equivalence classes the definition reads directly as follows
for every wave-function .
In the above definition, as the careful reader can immediately remark, does not exist any clear specification of domain and co-domain for the position operator (in the case of a particle confined upon a line). In literature, more or less explicitly, we find essentially three main directions for this fundamental issue.
The position operator is defined on the subspace of formed by those equivalence classes whose product by the imbedding lives in the space as well. In this case the position operator
reveals not continuous (unbounded with respect to the topology induced by the canonical scalar product of ), with no eigenvectors, no eigenvalues, consequently with empty eigenspectrum (collection of its eigenvalues).
The position operator is defined on the space of complex valued Schwartz functions (smooth complex functions defined upon the real-line and rapidly decreasing at infinity with all their derivatives). The product of a Schwartz function by the imbedding lives always in the space , which is a subset of . In this case the position operator
reveals continuous (with respect to the canonical topology of ), injective, with no eigenvectors, no eigenvalues, consequently with void eigenspectrum (collection of its eigenvalues). It is (fully) self-adjoint with respect to the scalar product of in the sense that
for every and belonging to its domain .
This is, in practice, the most widely adopted choice in Quantum Mechanics literature, although never explicitly underlined. The position operator is defined on the space of complex valued tempered distributions (topological dual of the Schwartz function space ). The product of a temperate distribution by the imbedding lives always in the space , which contains . In this case the position operator
reveals continuous (with respect to the canonical topology of ), surjective, endowed with complete families of eigenvectors, real eigenvalues, and with eigenspectrum (collection of its eigenvalues) equal to the real line. It is self-adjoint with respect to the scalar product of in the sense that its transpose operator
which is the position operator on the Schwartz function space, is self-adjoint:
for every (test) function and belonging to the space .
Informal proof. To show that possible eigenvectors of the position operator should necessarily be Dirac delta distributions, suppose that is an eigenstate of the position operator with eigenvalue . We write the eigenvalue equation in position coordinates,
recalling that simply multiplies the wave-functions by the function , in the position representation. Since the function is variable while is a constant, must be zero everywhere except at the point . Clearly, no continuous function satisfies such properties, and we cannot simply define the wave-function to be a complex number at that point because its -norm would be 0 and not 1. This suggest the need of a "functional object" concentrated at the point and with integral different from 0: any multiple of the Dirac delta centered at . Q.E.D.
The normalized solution to the equation
Proof. Here we prove rigorously that
Indeed, recalling that the product of any function by the Dirac distribution centered at a point is the value of the function at that point times the Dirac distribution itself, we obtain immediately
Meaning of the Dirac delta wave. Although such Dirac states are physically unrealizable and, strictly speaking, they are not functions, Dirac distribution centered at can be thought of as an "ideal state" whose position is known exactly (any measurement of the position always returns the eigenvalue ). Hence, by the uncertainty principle, nothing is known about the momentum of such a state.
The generalisation to three dimensions is straightforward.
The space-time wavefunction is now and the expectation value of the position operator at the state is
where the integral is taken over all space. The position operator is
Usually, in quantum mechanics, by representation in the momentum space we intend the representation of states and observables with respect to the canonical unitary momentum basis
In momentum space, the position operator in one dimension is represented by the following differential operator
the representation of the position operator in the momentum basis is naturally defined by , for every wave function (tempered distribution) ;
represents the coordinate function on the momentum line and the wave-vector function is defined by .