Physics:Position operator

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Short description: Operator in quantum mechanics

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 [math]\displaystyle{ | x \rangle }[/math] we denote the unitary eigenvector of the position operator corresponding to the eigenvalue [math]\displaystyle{ x }[/math], then, [math]\displaystyle{ |x \rangle }[/math] represents the state of the particle in which we know with certainty to find the particle itself at position [math]\displaystyle{ x }[/math].

Therefore, denoting the position operator by the symbol [math]\displaystyle{ X }[/math] – in the literature we find also other symbols for the position operator, for instance [math]\displaystyle{ Q }[/math] (from Lagrangian mechanics), [math]\displaystyle{ \hat \mathrm x }[/math] and so on – we can write [math]\displaystyle{ X| x\rangle = x |x\rangle, }[/math] for every real position [math]\displaystyle{ x }[/math].

One possible realization of the unitary state with position [math]\displaystyle{ x }[/math] is the Dirac delta (function) distribution centered at the position [math]\displaystyle{ x }[/math], often denoted by [math]\displaystyle{ \delta_x }[/math].

In quantum mechanics, the ordered (continuous) family of all Dirac distributions, i.e. the family [math]\displaystyle{ \delta = (\delta_x)_{x \in \R}, }[/math] is called the (unitary) position basis (in one dimension), just because it is a (unitary) eigenbasis of the position operator [math]\displaystyle{ X }[/math]. Note that even though this family is ordered by the continuous coordinate [math]\displaystyle{ x }[/math], 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 [math]\displaystyle{ X }[/math] on the space of tempered distributions such that [math]\displaystyle{ X(\delta_x) = x \delta_x, }[/math] for every real point [math]\displaystyle{ x }[/math]. It's possible to prove that the unique above endomorphism is necessarily defined by [math]\displaystyle{ X(\psi) = \mathrm x \psi, }[/math] for every tempered distribution [math]\displaystyle{ \psi }[/math], where [math]\displaystyle{ \mathrm x }[/math] denotes the coordinate function of the position line – defined from the real line into the complex plane by [math]\displaystyle{ \mathrm x : \Reals \to \Complex : x \mapsto x . }[/math]

Introduction

In one dimension – for a particle confined into a straight line – the square modulus [math]\displaystyle{ |\psi|^2 = \psi^* \psi , }[/math] of a normalized square integrable wave-function [math]\displaystyle{ \psi : \Reals \to \Complex , }[/math] represents the probability density of finding the particle at some position [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ \psi }[/math] and assuming the wave function [math]\displaystyle{ \psi }[/math] be of [math]\displaystyle{ L^2 }[/math]-norm equal 1, [math]\displaystyle{ \|\psi\|^2 = \int_{-\infty}^{+\infty} |\psi|^2 d \mathrm x = 1, }[/math] then the probability to find the particle in the position range [math]\displaystyle{ [a,b] }[/math] is [math]\displaystyle{ \pi_X (\psi)([a,b]) =\int_a^b |\psi|^2 d \mathrm x . }[/math]

Hence the expected value of a measurement of the position [math]\displaystyle{ X }[/math] for the particle is the value [math]\displaystyle{ \langle X \rangle_{\psi} = \int_\R \mathrm x |\psi|^2 d \mathrm x = \int_\R \psi^* \mathrm x \psi \, d \mathrm x, }[/math] where:

  1. the particle is assumed to be in the state [math]\displaystyle{ \psi }[/math];
  2. the function [math]\displaystyle{ \mathrm x |\psi|^2 }[/math] is supposed integrable, i.e. of class [math]\displaystyle{ L^1 }[/math];
  3. we indicate by [math]\displaystyle{ \mathrm x }[/math] the coordinate function of the position axis.

Additionally, the quantum mechanical operator corresponding to the observable position [math]\displaystyle{ X }[/math] is denoted also by [math]\displaystyle{ X = \hat{\mathrm x} , }[/math] and defined [math]\displaystyle{ \left(\hat{\mathrm x} \psi\right) (x) = x\psi(x) , }[/math] for every wave function [math]\displaystyle{ \psi }[/math] and for every point [math]\displaystyle{ x }[/math] of the real line.

The circumflex over the function [math]\displaystyle{ \mathrm x }[/math] on the left side indicates the presence of an operator, so that this equation may be read:

The result of the position operator [math]\displaystyle{ X }[/math] acting on any wave function [math]\displaystyle{ \psi }[/math] equals the coordinate function [math]\displaystyle{ \mathrm x }[/math] multiplied by the wave-function [math]\displaystyle{ \psi }[/math].

Or more simply:

The operator [math]\displaystyle{ X }[/math] multiplies any wave-function [math]\displaystyle{ \psi }[/math] by the coordinate function [math]\displaystyle{ \mathrm x }[/math].

Note 1. To be more explicit, we have introduced the coordinate function [math]\displaystyle{ \mathrm x : \Reals \to \Complex : x \mapsto x , }[/math] 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) [math]\displaystyle{ \psi }[/math] can be reinterpreted as a scalar product: [math]\displaystyle{ \langle X \rangle_{\psi} = \int_\R \mathrm x |\psi|^2 = \int_\R \psi^* (\mathrm x \psi) = \langle \psi | X(\psi) \rangle , }[/math] assuming the particle in the state [math]\displaystyle{ \psi \in L^2 }[/math] and assuming the function [math]\displaystyle{ \mathrm x \psi }[/math] be of class [math]\displaystyle{ L^2 }[/math] – which immediately implies that the function [math]\displaystyle{ \mathrm x |\psi|^2 }[/math] Is integrable, i.e. of class [math]\displaystyle{ L^1 }[/math].

Note 3. Strictly speaking, the observable position [math]\displaystyle{ X }[/math] can be point-wisely defined as [math]\displaystyle{ \left(\hat{\mathrm x} \psi\right) (x) = x\psi(x) , }[/math] for every wave function [math]\displaystyle{ \psi }[/math] and for every point [math]\displaystyle{ x }[/math] of the real line, upon the wave-functions which are precisely point-wise defined functions. In the case of equivalence classes [math]\displaystyle{ \psi \in L^2 }[/math] the definition reads directly as follows [math]\displaystyle{ \hat{\mathrm x} \psi = \mathrm x \psi , }[/math] for every wave-function [math]\displaystyle{ \psi \in L^2 }[/math].

Basic properties

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 [math]\displaystyle{ D_X }[/math] of [math]\displaystyle{ L^2 }[/math] formed by those equivalence classes [math]\displaystyle{ \psi }[/math] whose product by the imbedding [math]\displaystyle{ \mathrm x }[/math] lives in the space [math]\displaystyle{ L^2 }[/math] as well. In this case the position operator [math]\displaystyle{ X : D_X \to L^2 : \psi \mapsto \mathrm x \psi }[/math] reveals not continuous (unbounded with respect to the topology induced by the canonical scalar product of [math]\displaystyle{ L^2 }[/math]), with no eigenvectors, no eigenvalues, consequently with empty eigenspectrum (collection of its eigenvalues).
  2. The position operator is defined on the space [math]\displaystyle{ \mathcal S_1 }[/math] 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 [math]\displaystyle{ \mathrm x }[/math] lives always in the space [math]\displaystyle{ \mathcal S_1 }[/math], which is a subset of [math]\displaystyle{ L^2 }[/math]. In this case the position operator [math]\displaystyle{ X : \mathcal S_1 \to \mathcal S_1 : \psi \mapsto \mathrm x \psi }[/math] reveals continuous (with respect to the canonical topology of [math]\displaystyle{ \mathcal S_1 }[/math]), 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 [math]\displaystyle{ L^2 }[/math] in the sense that [math]\displaystyle{ \langle X (\psi)|\phi\rangle = \langle \psi|X(\phi)\rangle, }[/math] for every [math]\displaystyle{ \psi }[/math] and [math]\displaystyle{ \phi }[/math] belonging to its domain [math]\displaystyle{ \mathcal S_1 }[/math].
  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 [math]\displaystyle{ \mathcal S'_1 }[/math] of complex valued tempered distributions (topological dual of the Schwartz function space [math]\displaystyle{ \mathcal S_1 }[/math]). The product of a temperate distribution by the imbedding [math]\displaystyle{ \mathrm x }[/math] lives always in the space [math]\displaystyle{ \mathcal S'_1 }[/math], which contains [math]\displaystyle{ L^2 }[/math]. In this case the position operator [math]\displaystyle{ X : \mathcal S'_1 \to \mathcal S'_1 : \psi \mapsto \mathrm x \psi }[/math] reveals continuous (with respect to the canonical topology of [math]\displaystyle{ \mathcal S'_1 }[/math]), 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 [math]\displaystyle{ L^2 }[/math] in the sense that its transpose operator [math]\displaystyle{ {}^tX : \mathcal S_1 \to \mathcal S_1 : \phi \mapsto \mathrm x \phi , }[/math] which is the position operator on the Schwartz function space, is self-adjoint: [math]\displaystyle{ \left\langle\left. \,{}^tX (\phi)\right|\psi \right\rangle = \left\langle \phi| \,{}^tX(\psi)\right\rangle , }[/math] for every (test) function [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ \psi }[/math] belonging to the space [math]\displaystyle{ \mathcal S_1 }[/math].

Eigenstates

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 [math]\displaystyle{ \psi }[/math] is an eigenstate of the position operator with eigenvalue [math]\displaystyle{ x_0 }[/math]. We write the eigenvalue equation in position coordinates, [math]\displaystyle{ \hat{\mathrm x}\psi(x) = \mathrm x \psi(x) = x_0 \psi(x) }[/math]

recalling that [math]\displaystyle{ \hat{\mathrm x} }[/math] simply multiplies the wave-functions by the function [math]\displaystyle{ \mathrm x }[/math], in the position representation. Since the function [math]\displaystyle{ \mathrm x }[/math] is variable while [math]\displaystyle{ x_0 }[/math] is a constant, [math]\displaystyle{ \psi }[/math] must be zero everywhere except at the point [math]\displaystyle{ x_0 }[/math]. 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 [math]\displaystyle{ L^2 }[/math]-norm would be 0 and not 1. This suggest the need of a "functional object" concentrated at the point [math]\displaystyle{ x_0 }[/math] and with integral different from 0: any multiple of the Dirac delta centered at [math]\displaystyle{ x_0 }[/math]. Q.E.D.

The normalized solution to the equation [math]\displaystyle{ \mathrm x \psi = x_0 \psi }[/math] is [math]\displaystyle{ \psi(x) = \delta(x - x_0), }[/math] or better [math]\displaystyle{ \psi = \delta _{x_0}. }[/math]

Proof. Here we prove rigorously that [math]\displaystyle{ \mathrm x \delta_{x_0} = x_0 \delta_{x_0} . }[/math]

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 [math]\displaystyle{ \mathrm x \delta_{x_0} = \mathrm x (x_0) \delta_{x_0} =x_0 \delta_{x_0} . }[/math] Q.E.D.

Meaning of the Dirac delta wave. Although such Dirac states are physically unrealizable and, strictly speaking, they are not functions, Dirac distribution centered at [math]\displaystyle{ x_0 }[/math] can be thought of as an "ideal state" whose position is known exactly (any measurement of the position always returns the eigenvalue [math]\displaystyle{ x_0 }[/math]). Hence, by the uncertainty principle, nothing is known about the momentum of such a state.

Three dimensions

The generalisation to three dimensions is straightforward.

The space-time wavefunction is now [math]\displaystyle{ \psi(\mathbf{r}, t) }[/math] and the expectation value of the position operator [math]\displaystyle{ \hat \mathbf{r} }[/math] at the state [math]\displaystyle{ \psi }[/math] is [math]\displaystyle{ \left\langle \hat \mathbf{r} \right\rangle _\psi = \int \mathbf{r} |\psi|^2 d^3 \mathbf{r} }[/math] where the integral is taken over all space. The position operator is [math]\displaystyle{ \mathbf{\hat{r}}\psi = \mathbf{r}\psi. }[/math]

Momentum space

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 [math]\displaystyle{ \eta = \left(\left[ (2\pi\hbar)^{-\frac{1}{2}} e^{(\iota/\hbar) (\mathrm x|p)}\right]\right) _{p \in \R}. }[/math]

In momentum space, the position operator in one dimension is represented by the following differential operator [math]\displaystyle{ \left(\hat{\mathrm x}\right)_P = i\hbar\frac{d}{d \mathrm p} = i\frac{d}{d \mathrm k}, }[/math]

where:

  • the representation of the position operator in the momentum basis is naturally defined by [math]\displaystyle{ \left(\hat{\mathrm x}\right)_P (\psi)_P = \left(\hat{\mathrm x}\psi\right)_P }[/math], for every wave function (tempered distribution) [math]\displaystyle{ \psi }[/math];
  • [math]\displaystyle{ \mathrm p }[/math] represents the coordinate function on the momentum line and the wave-vector function [math]\displaystyle{ \mathrm k }[/math] is defined by [math]\displaystyle{ \mathrm k = \mathrm p / \hbar }[/math].

Formalism in L2(R, C)

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) [math]\displaystyle{ L^2(\Reals,\Complex) }[/math] of complex-valued and square-integrable (with respect to the Lebesgue measure) functions on the real line.

The position operator in [math]\displaystyle{ L^2(\Reals, \Complex) }[/math], [math]\displaystyle{ Q : D_Q \to L^2(\Reals, \Complex) : \psi \mapsto \mathrm q \psi, }[/math] is pointwise defined by:[2][3]

[math]\displaystyle{ Q (\psi)(x) = x \psi (x) = \mathrm q(x) \psi (x), }[/math] for each pointwisely defined square integrable class [math]\displaystyle{ \psi \in D_Q }[/math] and for each real number x, with domain [math]\displaystyle{ D_Q = \left\{ \psi \in L^2(\R) \mid \mathrm q \psi \in L^2(\R) \right\}, }[/math] where [math]\displaystyle{ \mathrm q : \Reals \to \Complex }[/math] is the coordinate function sending each point [math]\displaystyle{ x \in \R }[/math] to itself.

Since all continuous functions with compact support lie in D(Q), 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)

As with any quantum mechanical observable, in order to discuss position measurement, we need to calculate the spectral resolution of the position operator [math]\displaystyle{ X : D_X \to L^2(\Reals, \Complex) : \psi \mapsto \mathrm x \psi }[/math] which is [math]\displaystyle{ X = \int_\R \lambda \, d \mu_X(\lambda) = \int_\R \mathrm x \, \mu_X = \mu_X (\mathrm x), }[/math] where [math]\displaystyle{ \mu_X }[/math] is the so-called spectral measure of the position operator.

Since the operator of [math]\displaystyle{ X }[/math] is just the multiplication operator by the embedding function [math]\displaystyle{ \mathrm x }[/math], its spectral resolution is simple.

For a Borel subset [math]\displaystyle{ B }[/math] of the real line, let [math]\displaystyle{ \chi _B }[/math] denote the indicator function of [math]\displaystyle{ B }[/math]. We see that the projection-valued measure [math]\displaystyle{ \mu_X : \mathcal{B}(\R) \to \mathrm{Pr}^\perp \left(L^2(\Reals, \Complex)\right) }[/math] is given by [math]\displaystyle{ \mu_X(B)(\psi) = \chi_B \psi , }[/math] i.e., the orthogonal projection [math]\displaystyle{ \mu_X(B) }[/math] is the multiplication operator by the indicator function of [math]\displaystyle{ B }[/math].

Therefore, if the system is prepared in a state [math]\displaystyle{ \psi }[/math], then the probability of the measured position of the particle belonging to a Borel set [math]\displaystyle{ B }[/math] is [math]\displaystyle{ \|\mu_X(B)(\psi)\|^2 = \|\chi_B \psi\|^2 = \int_B |\psi|^2\ \mu =\pi_X(\psi)(B), }[/math] where [math]\displaystyle{ \mu }[/math] 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 [math]\displaystyle{ \frac{\mu_X(B) \psi}{\|\mu_X(B) \psi \|} = \frac{\chi_B \psi}{\| \chi_B \psi \|} }[/math] or [math]\displaystyle{ \frac{(1 - \chi_B)\psi}{\|(1 - \chi_B)\psi\|}, }[/math] where [math]\displaystyle{ \| \cdot \| }[/math] is the Hilbert space norm on [math]\displaystyle{ L^2(\Reals, \Complex) }[/math].

See also

References

  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. https://archive.org/details/isbn_9780071471411. 
  3. Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics (2nd ed.). McGraw Hill. ISBN 978-0071623582.