the observable B: In other words, this is the time interval (Δt) after which the expectation value changes appreciably. An informal, heuristic meaning of the principle is the following: A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy. For example, in spectroscopy, excited states have a finite lifetime. By the time–energy uncertainty principle, they do not have a definite energy, and, each time they decay, the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, while slow-decaying states have a narrow linewidth. The same linewidth effect also makes it difficult to specify the rest mass of unstable, fast-decaying particles in particle physics. The faster the particle decays (the shorter its lifetime), the less certain is its mass (the larger the particle's width). For the number of electrons in a superconductor and the phase of its Ginzburg–Landau order parameter A counterexample Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable , which we may take to lie in the interval . Define "position" and "momentum" operators and by and where we impose periodic boundary conditions on . The definition of depends on our choice to have range from 0 to . These operators satisfy the usual commutation relations for position and momentum operators, . Now let be any of the eigenstates of , which are given by . These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator is bounded, since ranges over a bounded interval. Thus, in the state , the uncertainty of is zero and the uncertainty of is finite, so that Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that is not in the domain of the operator , since multiplication by disrupts the periodic boundary conditions imposed on . Thus, the derivation of the Robertson relation, which requires and to be defined, does not apply. (These also furnish an example of operators satisfying the canonical