to write the real part of the ★-genvalue equation in the form of a modified Laguerre equation (not Hermite's equation!), the solution of which involves the Laguerre polynomials as introduced by Groenewold, with associated ★-genvalues For the harmonic oscillator, the time evolution of an arbitrary Wigner distribution is simple. An initial evolves by the above evolution equation driven by the oscillator Hamiltonian given, by simply rigidly rotating in phase space, Typically, a "bump" (or coherent state) of energy can represent a macroscopic quantity and appear like a classical object rotating uniformly in phase space, a plain mechanical oscillator (see the animated figures). Integrating over all phases (starting positions at t = 0) of such objects, a continuous "palisade", yields a time-independent configuration similar to the above static ★-genstates , an intuitive visualization of the classical limit for large-action systems. Free particle angular momentum Suppose a particle is initially in a minimally uncertain Gaussian state, with the expectation values of position and momentum both centered at the origin in phase space. The Wigner function for such a state propagating freely is where α is a parameter describing the initial width of the Gaussian, and . Initially, the position and momenta are uncorrelated. Thus, in 3 dimensions, we expect the position and momentum vectors to be twice as likely to be perpendicular to each other as parallel. However, the position and momentum become increasingly correlated as the state evolves, because portions of the distribution farther from the origin in position require a larger momentum to be reached: asymptotically, (This relative "squeezing" reflects the spreading of the free wave packet in coordinate space.) Indeed, it is possible to show that the kinetic energy of the particle becomes asymptotically radial only, in agreement with the standard quantum-mechanical notion of the ground-state nonzero angular momentum specifying orientation independence: Morse potential The Morse potential is used to approximate the vibrational structure of a diatomic molecule. Quantum tunneling Tunneling is a hallmark quantum effect where a quantum particle, not having sufficient energy to fly above, still goes through a barrier. This effect does not exist in classical mechanics. Quartic potential Schrödinger cat state References Hamiltonian mechanics Symplectic geometry Mathematical quantization Foundational quantum physics Articles containing video clips