uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators. The position operator and momentum operator do not commute, but rather satisfy the canonical commutation relation: Given a quantum state, the Born rule lets us compute expectation values for both and , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation, we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators and . The commutator of these two operators is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space. This is why in quantum equations in position space, the momentum is replaced by , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times . Composite systems and entanglement When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let and be two quantum systems, with Hilbert spaces and , respectively. The Hilbert space of the composite system is then If the state for the first system is the vector and the state for the second system is , then the state of the composite system is Not all states in the joint Hilbert space can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if and are both possible states for system