A new type of basis set for quantum mechanical problems is introduced. These basis states are adapted to describing the dynamics of a Hamiltonian Ĥ which is dependent upon a parameter X . A function f ( E ) is defined which is an analytic function of E , and which is negligibly small when | E |>>δ E , where δ E is large compared to the typical level separation. The energy-shell basis set consists of states |ξ n ( X )⟩ which are derived by applying the operator f (Ĥ( X )-Ē n ( X )) to elements of a fixed basis set, where Ē n ( X ) is an analytic approximation to an eigenvalue E n ( X ). The energy-shell basis states are combinations of states close to energy E n , but vary more slowly as a function of X than the eigenfunctions |ϕ n ( X )⟩ of Ĥ( X ). This feature gives the energy-shell basis states some advantages in analysing solutions of the time-dependent Schrödinger equation.
The Hermitian operators behind Wigner's phase space function (1932) are recognised to be simple ordered exponentials of the dynamical variables. This operator basis is highly symmetric; it supplies a perfectly unbiased formulation of operator equations in terms of phase space functions, which is as close to classical physics as it possibly can be. The author demonstrates how the symmetry properties of the basis can be exploited to simplify the computation of Wigner functions enormously. The ordered-operator methods are also applicable to the phase space functions of Kirkwood (1933) and Glauber (1963) type. Both kinds are briefly discussed with emphasis on how they differ from Wigner's description.