takes advantage of the fact that the quantum Fisher information is the convex roof of the variance times four. A simpler inequality follows without a convex roof which is stronger than the Heisenberg uncertainty relation, since for the quantum Fisher information we have while for pure states the equality holds. Phase space In the phase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given a Wigner function with star product ★ and a function f, the following is generally true: Choosing , we arrive at Since this positivity condition is true for all a, b, and c, it follows that all the eigenvalues of the matrix are non-negative. The non-negative eigenvalues then imply a corresponding non-negativity condition on the determinant, or, explicitly, after algebraic manipulation, Examples Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below. For position and linear momentum, the canonical commutation relation implies the Kennard inequality from above: For two orthogonal components of the total angular momentum operator of an object: where i, j, k are distinct, and Ji denotes angular momentum along the xi axis. This relation implies that unless all three components vanish together, only a single component of a system's angular momentum can be defined with arbitrary precision, normally the component parallel to an external (magnetic or electric) field. Moreover, for , a choice , , in angular momentum multiplets, ψ = |j, m⟩, bounds the Casimir invariant (angular momentum squared, ) from below and thus yields useful constraints such as , and hence j ≥ m, among others. In non-relativistic mechanics, time is privileged as an independent variable. Nevertheless, in 1945, L. I. Mandelshtam and I. E. Tamm derived a non-relativistic time–energy uncertainty relation, as follows. For a quantum system in a non-stationary state and an observable B represented by a self-adjoint operator , the following formula holds: where σE is the standard deviation of the energy operator (Hamiltonian) in the state , σB stands for the standard deviation of B. Although the second factor in the left-hand side has dimension of time, it is different from the time parameter that enters the Schrödinger equation. It is a lifetime of the state with respect to