In one dimension, the square modulus of the wave function, , represents the probability density of finding the particle at position . Hence the expected value of a measurement of the position of the particle is
Accordingly, the quantum mechanical operator corresponding to position is , where
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.
The eigenfunctions of the position operator, represented in position space, are Dirac delta functions. To show this, suppose that is an eigenstate of the position operator with eigenvalue . We write the eigenvalue equation in position coordinates,
recalling that simply multiplies the function by in the position representation. Since is a variable while is a constant, must be zero everywhere except at . The normalized solution to this is
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 ). Hence, by the uncertainty principle, nothing is known about the momentum of such a state.
The generalisation to three dimensions is straightforward. The wavefunction is now and the expectation value of the position is
where the integral is taken over all space. The position operator is
In momentum space, the position operator in one dimension is
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:
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.
Since Q is just multiplication by x, its spectral resolution is simple. For a Borel subset B of the real line, let denote the indicator function of B. We see that the projection-valued measure ΩQ is given by
where μ is the Lebesgue measure. After the measurement, the wave function collapses to either
, where is the Hilbert space norm on L2(R).
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- McMahon, D. (2006). Quantum Mechanics Demystified (2nd ed.). Mc Graw Hill. ISBN 0 07 145546 9.
- Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics (2nd ed.). McGraw Hill. ISBN 978-0071623582.