Weyl Transform in Nonrelativistic Quantum Dynamics

The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase‐space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time‐dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t₀, Q₀, K₀), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t₀, Q₀, K₀) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q₀, K₀ | t₀, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator A_H(t) with Weyl transform A_H(t, Q, K), the function $\mathcal{A}\mathrm{(t,}\mathbf{Q}\mathrm{)=\int d}\mathbf{K}{A}_H\mathrm{(t,}\mathbf{Q,K})$ is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum‐mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of $\mathcal{A}\mathrm{(t,}\mathbf{Q})$ can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t₀, Q₀) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non‐Cartesian coordinate systems is given.