Position operator

Summary

In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.

When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues are the possible position vectors of the particle.[1]

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 in the space of distributions dual to the space of wave-functions.

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

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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

 
where:
  1. the particle is assumed to be in the state  ;
  2. the function   is supposed integrable, i.e. of class  ;
  3. we indicate by   the coordinate function of the position axis.

Additionally, the quantum mechanical operator corresponding to the observable position   is denoted also by

 
and defined
 
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  .

Basic properties

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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.

  1. 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).
  2. 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  .
  3. 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  .

Eigenstates

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The eigenfunctions of the position operator (on the space of tempered distributions), represented in position space, are Dirac delta functions.

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  . The normalized solution to the equation
 
is
 
or better
 
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.

Three dimensions

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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
 

Momentum space

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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

 

where:

  • 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  .

Formalism in L2(R, C)

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Consider, for example, the case of a spinless particle moving in one spatial dimension (i.e. in a line). The state space for such a particle contains the L2-space (Hilbert space)   of complex-valued and square-integrable (with respect to the Lebesgue measure) functions on the real line.

The position operator in  ,

 
is pointwise defined by:[2][3]

 
for each pointwisely defined square integrable class   and for each real number x, with domain
 
where   is the coordinate function sending each point   to itself.

Since all continuous functions with compact support lie in  , Q is densely defined. Q, being simply multiplication by x, is a self-adjoint operator, thus satisfying the requirement of a quantum mechanical observable.

Immediately from the definition we can deduce that the spectrum consists of the entire real line and that Q has purely continuous spectrum, therefore no discrete eigenvalues.

The three-dimensional case is defined analogously. We shall keep the one-dimensional assumption in the following discussion.

Measurement theory in L2(R, C)

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As with any quantum mechanical observable, in order to discuss position measurement, we need to calculate the spectral resolution of the position operator

 
which is
 
where   is the so-called spectral measure of the position operator.

Since the operator of   is just the multiplication operator by the embedding function  , its spectral resolution is simple.

For a Borel subset   of the real line, let   denote the indicator function of  . We see that the projection-valued measure

 
is given by
 
i.e., the orthogonal projection   is the multiplication operator by the indicator function of  .

Therefore, if the system is prepared in a state  , then the probability of the measured position of the particle belonging to a Borel set   is

 
where   is the Lebesgue measure on the real line.

After any measurement aiming to detect the particle within the subset B, the wave function collapses to either

 
or
 
where   is the Hilbert space norm on  .

See also

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References

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  1. ^ Atkins, P.W. (1974). Quanta: A handbook of concepts. Oxford University Press. ISBN 0-19-855493-1.
  2. ^ McMahon, D. (2006). Quantum Mechanics Demystified (2nd ed.). Mc Graw Hill. ISBN 0-07-145546-9.
  3. ^ Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics (2nd ed.). McGraw Hill. ISBN 978-0071623582.