The problem, as Woolley addressed it, is that quantum mechanical calculations employ the fixed, or "clamped," nucleus approximation (the Born-Oppenheimer approximation) in which nuclei are treated as classical particles confined to "equilibrium" positions. Woolley claims that a quantum mechanical calculation carried out completely from first principles, without such an approximation, yields no recognizable molecular structure and that the maintenance of "molecular structure" must therefore be a product not of an isolated molecule but of the action of the molecule functioning over time in its environment.47 [Pg.297]

The solutions of classical diffusion equations are necessarily positive, and the nodes of quantum mechanical wave functions required by the exclusion principle render a direct solution of Eq. (2.8) impossible. Instead, a new function F is defined as the product of the unknown function and an approximation to it [Pg.22]

Macroscopic states involve variables that pertain to the entire system, such as the pressure P, the temperature T, and the volume V. For a fluid system of one substance and one phase, the equilibrium macrostate is specified by only three variables, such as P, T, and V. If we assume that classical mechanics is an adequate approximation, the microstate of such a system is specified by the position and velocity of every particle in the system. If quantum mechanics must be used for a dilute gas, there are several quantum numbers required to specify the state of each molecule in the system. This is a very large number of independent variables or a very large number of quantum numbers. Statistical mechanics is the theory that relates the small amount of information in the macrostates and the large amount of information in the microstates. [Pg.1040]

The derivation of these fundamental correspondence relations, (3), has been given previously,9 and one should see Ref. 9 for a more detailed discussion. To obtain the results it is necessary to assume only (2) (which is essentially a statement of the uncertainty principle), make use of classical mechanics itself, and invoke the stationary phase approximation14 to evaluate all integrals for which the phase of the integrand is proportional to h l. Since the stationary phase approximation14 is an asymptotic approximation which becomes exact as h -> 0, this is the nature of the classical-limit approximation in (3). In a very precise sense, therefore, classical-limit quantum mechanics is the stationary phase approximation to quantum mechanics. [Pg.81]

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