commutation relations but not the Weyl relations.) For the usual position and momentum operators and on the real line, no such counterexamples can occur. As long as and are defined in the state , the Heisenberg uncertainty principle holds, even if fails to be in the domain of or of . Examples (Refs ) Quantum harmonic oscillator stationary states Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators in terms of the creation and annihilation operators: Using the standard rules for creation and annihilation operators on the energy eigenstates, the variances may be computed directly, The product of these standard deviations is then In particular, the above Kennard bound is saturated for the ground state , for which the probability density is just the normal distribution. Quantum harmonic oscillators with Gaussian initial condition In a quantum harmonic oscillator of characteristic angular frequency ω, place a state that is offset from the bottom of the potential by some displacement x0 as where Ω describes the width of the initial state but need not be the same as ω. Through integration over the propagator, we can solve for the -dependent solution. After many cancelations, the probability densities reduce to where we have used the notation to denote a normal distribution of mean μ and variance σ2. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as From the relations we can conclude the following: (the right most equality holds only when Ω = ω) . Coherent states A coherent state is a right eigenstate of the annihilation operator, which may be represented in terms of Fock states as In the picture where the coherent state is a massive particle in a quantum harmonic oscillator, the position and momentum operators may be expressed in terms of the annihilation operators in the same formulas above and used to calculate the variances, Therefore, every coherent state saturates the Kennard bound with position and momentum each contributing an amount in a "balanced" way. Moreover, every squeezed coherent state also saturates the Kennard bound although the individual contributions of position and momentum need not be balanced in general. Particle in a box Consider a particle in a one-dimensional box of length . The eigenfunctions in position and momentum space are and where and we have used the de Broglie relation . The