System of Point Particles

The third equivalent formulation of classical mechanics to be briefly discussed here is the Hamiltonian formalism. Its main practical importance especially for molecular simulations lies in the solution of practical problems for processes that can be adequately described by classical mechanics despite their intrinsically quantum mechanical character (such as protein folding processes). However, more important for our purposes here is that it can serve as a useful starting point for the transition to quantum theory. The basic idea of the Hamiltonian formalism is to eliminate the / generalized velocities in favor of the canonical momenta defined by Eq. (2.54). This is achieved by a Legendre transformation of the Lagrangian with respect to the velocities. 

H is called the Hamiltonian of the system and depends only on the coordinates q, on the canonical momenta p conjugate to these coordinates, and on the time t. The 2/ variables q and p are also denoted as canonical variables of the system. They are treated as being independent of each other within the Hamiltonian framework and specify the state of the system completely. The 2/-dimensional space spanned by the canonical variables is called the phase space or T-space. Each point in phase space uniquely determines one state of the system, and the time evolution of the system is given as a trajectory in phase space. 

Since the 2/ canonical variables are independent of each other the Hamiltonian principle has to be modified. The variation of the action 

Executing this variation, integrating by parts, and reordering the terms yields for the variation of the action 

This variation vanishes exactly then for all independent variations Sq and (f = 1./) if and only if both expressions in parentheses vanish. This condition yields the Hamiltonian or canonical equations. 

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The first consistent attempt to unify quantum theory and relativity came after Schrddinger s and Heisenberg s work in 1925 and 1926 produced the rules for the quantum mechanical description of nonrelativistic systems of point particles. Mention should be made of the fact that in these developments de Broglie s hypothesis attributing wave-corpuscular properties to all matter played an important role. Central to this hypothesis are the relations between particle and wave properties E — hv and p = Ilk, which de Broglie advanced on the basis of relativistic dynamics. 

The equilibrium distribution of soft generalized coordinates for both stiff and rigid classical mechanical systems of point particles may be written in the generic form... 

In the first chapter no attempt will be made to give any parts of classical dynamics but those which are useful in the treatment of atomic and molecular problems. With this restriction, we have felt justified in omitting discussion of the dynamics of rigid bodies, non-conservative systems, non-holonomic systems, systems involving impact, etc. Moreover, no use is made of Hamilton s principle or of the Hamilton-Jacobi partial differential equation. By thus limiting the subjects to be discussed, it is possible to give in a short chapter a thorough treatment of Newtonian systems of point particles. 

THE WAVE EQUATION FOR A SYSTEM OF POINT PARTICLES IN THREE DIMENSIONS... 

This is Schrodinger s amplitude equation for a conservative system of point particles. 

In retrospect, one can see why any attempt to represent the wave-function of a many-electron system by a system of point particles should give a geometry which can t be visualized in three-space. For example, any set of representative (as opposed to instantaneous) interparticle distances, such as a set of expectation values, will generally define a structure which is higher than three-dimensional. In fact, the set of expectation values for an iV-electron atom can be expected to define an iV-dimensional figure, just like the subhamiltonian minimum. 

For a system of point particles at equilibrium, the function nn depends only on the separation distances between particles. Functionally, we must have... 

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