A system with n degrees of freedom, and n constants of motion, such that the Poisson bracket of any pair of constants of motion vanishes, is known as a completely integrable system. Such a collection of constants of motion are said to be in involution with each other. For a closed system (Lagrangian not explicitly dependent on time), the energy of the system is a constant of motion (a conserved quantity). In quantum mechanics An observable quantity Q will be a constant of motion if it commutes with the hamiltonian, H, and it does not itself depend explicitly on time. This is because where is the commutator relation. Derivation Say there is some observable quantity Q which depends on position, momentum and time, And also, that there is a wave function which obeys Schrödinger's equation Taking the time derivative of the expectation value of Q requires use of the product rule, and results in {| | | |- | | |- | | |- | | |- | | |} So finally, {|cellpadding="2" style="border:2px solid #ccccff" | |} Comment For an arbitrary state of a Quantum Mechanical system, if H and Q commute, i.e. if and Q is not explicitly dependent on time, then But if is an eigenfunction of Hamiltonian, then even if it is still the case that provided Q is independent on time. Derivation {| | | |- | | |} Since {| | |} then {| | | |- | | |} This is the reason why Eigenstates of the Hamiltonian are also called stationary states. Relevance for quantum chaos In general, an integrable system has constants of motion other than the energy. By contrast, energy is the only constant of motion in a non-integrable system; such systems are termed chaotic. In general, a classical mechanical system can be quantized only if it is integrable; as of 2006, there is no known consistent method for quantizing chaotic dynamical systems. Integral of motion A constant of motion may be defined in a given force field as any function of phase-space coordinates (position and velocity, or position and momentum) and time that is constant throughout a trajectory. A subset of the constants of motion are the integrals of motion, or first integrals, defined as any functions of only the phase-space coordinates that are constant along an orbit. Every integral of motion is a constant of motion, but the