Statistical mechanics Schrodinger equation

Statistical mechanics is the mathematical means to calculate the thermodynamic properties of bulk materials from a molecular description of the materials. Much of statistical mechanics is still at the paper-and-pencil stage of theory. Since quantum mechanicians cannot exactly solve the Schrodinger equation yet, statistical mechanicians do not really have even a starting point for a truly rigorous treatment. In spite of this limitation, some very useful results for bulk materials can be obtained. 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

The approximate solution to the Schrodinger equation, defined by the effective Hamiltonian in Eq. (9-1), with either method described in the previous section, associates to every vector of molecule coordinates, R, together with the solvent-solvent interaction potential, an energy (R). From basic classical statistical mechanics an N-particle distribution function (PDF) n(R) is thus obtained ... 

The kind of statistics obeyed by the system depends on the symmetry properties of the quantum-mechanical wave functions describing the molecules composing the system [3-7], For example, in some cases the a values may be taken as either integers (0, 1,. . . ) or half-integers (, f,. . . ) the choice is based on the nature of the particular Schrodinger equation describing the molecule. 

The Statistical Rate Theory (SRT) is based on considering the quantum-mechanical transition probability in an isolated many particle system. Assuming that the transport of molecules between the phases at the thermal equilibrium results primarily from single molecular events, the expression for the rate of molecular transport between the two phases 1 and 2 , R 2, was developed by using the first-order perturbation analysis of the Schrodinger equation and the Boltzmann definition of entropy. 

Quantum effects can be recovered by quantum simulations. Currently there are two main types of quantum simulation methods used. One is based on the time-dependent Schrodinger equation. The other is based on Feynman s path integral (PI) quantum statistical mechanics. [7,8] The former is usually complicated in mathematical treatment and needs also large computational resources. Currently, it can only be used to simulate some very limited systems. [77] MD simulations based on the latter have been used more than a decade and are gaining more and more popularity. The main reason is that in PIMD simulations, the quantum systems are mapped onto corresponding classical systems. In other words the quantum effects can be recovered by making a series of classical simulations with different effective potentials. 

In the standard theory of quantum mechanics, two kinds of evolution processes are introduced, which are qualitatively different from each other. One is the spontaneous process, which is a reactive (unitary) dynamical process and is described by the Heisenberg or Schrodinger equation in an equivalent manner. The other is the measurement process, which is irreversible and described by the von Neumann projection postulate [26], which is the rigorous mathematical form of the reduction of the wave packet principle. The former process is deterministic and is uniquely described, while the latter process is essentially probabilistic and implies the statistical nature of quantum mechanics. 

Indeed, as the resolution of the nuclear Schrodinger equation can only be performed for very simple systems, one can often use a classical approach where nuclei are no longer considered as quantum particles but as classical ones moving on the potential energy hypersurface. Then, we may calculate reaction probabilities which are related to the reaction rate constant by an equation deduced in statistical mechanics. This can be formally written ... 

Calculation of the quantum dynamics of condensed-phase systems is a central goal of quantum statistical mechanics. For low-dimensional problems, one can solve the Schrodinger equation for the time-dependent wavefunction of the complete system directly, by expanding in a basis set or on a numerical grid [1,2]. However, because they retain the quantum correlations between all the system coordinates, wavefunction-based methods tend to scale exponentially with the number of degrees of freedom and hence rapidly become intractable even for medium-sized gas-phase molecules. Consequently, other approaches, most of which are in some sense approximate, must be developed. 

Next, let us review the history of theoretical chemistry before the development of the Schrodinger equation (Asimov 1979), because it is also significant to consider the historical orientation of quantum chemistry. This history is basically divided into three stages genesis, thermal physics-statistical-mechanics stage, and early quantum mechanics stage. Below are brief reviews of each stage. 

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