We consider the most general dilaton gravity theory l + l dimensions. By suitably parametrizing the metric and scalar field we find a simple expression that relates the energy of a generic solution to the magnitude of the corresponding Killing vector. In theories that admit black hole solutions, this relationship leads directly to an expression for the entropy S = ~~T o / G , where TO is the value of the scalar field (in this parametrization) at the event horizon. This result agrees with the one obtained using the more general method of Wald. Finally, we point out an intriguing connection between the black hole entropy and the imaginary part of the "phase" of the exact Dirac quantum wave functionals for the theory. PACS number(s): 04.70.D~

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The most general dilaton gravity theory in 2 spacetime dimensions is considered. A Hamiltonian analysis is performed and the reduced phase space, which is two dimensional, is explicitly constructed in a suitable parametrization of the fields. The theory is then quantized via the Dirac method in a functional Schrodinger representation. The quantum constraints are solved exactly to yield the (spatial) diffeomorphism invariant physical wave functionals for all theories considered. These wave functionals depend explicitly on the single configuration space coordinate as well as on the imbedding of space into spacetime (i.e. on the choice of time).