Position operator

Introduction

In one dimension, the square modulus of the wave function, | ψ | 2 = ψ ∗ ψ , represents the probability density of finding the particle at position x . Hence the expected value of a measurement of the position of the particle is

⟨ x ⟩ = ∫ − ∞ + ∞ x | ψ | 2 d x = ∫ − ∞ + ∞ ψ ∗ x ψ d x

Accordingly, the quantum mechanical operator corresponding to position is x ^ , where

x ^ ψ ( x ) = x ψ ( x )

The circumflex over the x on the left side indicates an operator, so that this equation may be read The result of the operator x acting on any function ψ(x) equals x multiplied by ψ(x). Or more simply, the operator x multiplies any function ψ(x) by x.

Eigenstates

The eigenfunctions of the position operator, represented in position space, are Dirac delta functions. To show this, suppose that ψ is an eigenstate of the position operator with eigenvalue x 0 . We write the eigenvalue equation in position coordinates,

x ^ ψ ( x ) = x ψ ( x ) = x 0 ψ ( x )

recalling that x ^ simply multiplies the function by x in the position representation. Since x is a variable while x 0 is a constant, ψ must be zero everywhere except at x = x 0 . The normalized solution to this is

ψ ( x ) = δ ( x − x 0 )

Although such a state is physically unrealizable and, strictly speaking, not a function, it can be thought of as an "ideal state" whose position is known exactly (any measurement of the position always returns the eigenvalue x 0 ). 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 wavefunction is now ψ ( r , t ) and the expectation value of the position is

⟨ r ⟩ = ∫ r | ψ | 2 d 3 r

where the integral is taken over all space. The position operator is

r ^ ψ = r ψ

Momentum space

In momentum space, the position operator in one dimension is

x ^ = i ℏ d d p = i d d k x

Formalism

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 is L²(R), the Hilbert space of complex-valued and square-integrable (with respect to the Lebesgue measure) functions on the real line. The position operator, Q, is then defined by:

Q ( ψ ) ( x ) = x ψ ( x )

with domain

D ( Q ) = { ψ ∈ L 2 ( R ) | Q ψ ∈ L 2 ( R ) } .

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

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

Q = ∫ λ d Ω Q ( λ ) .

Since Q is just multiplication by x, its spectral resolution is simple. For a Borel subset B of the real line, let χ B denote the indicator function of B. We see that the projection-valued measure Ω_Q is given by

Ω Q ( B ) ψ = χ B ψ ,

i.e. Ω_Q is multiplication by the indicator function of B. Therefore, if the system is prepared in state ψ, then the probability of the measured position of the particle being in a Borel set B is

| Ω Q ( B ) ψ | 2 = | χ B ψ | 2 = ∫ B | ψ | 2 d μ ,

where μ is the Lebesgue measure. After the measurement, the wave function collapses to either

Ω Q ( B ) ψ ∥ Ω Q ( B ) ψ ∥

or

( 1 − χ B ) ψ ∥ ( 1 − χ B ) ψ ∥ , where ∥ ⋯ ∥ is the Hilbert space norm on L²(R).