Classical Hamiltonian function

The classical function A is an observable, meaning that it is a physically measurable property of the system. For example, for a one-particle system the Hamiltonian operator H corresponding to the classical Hamiltonian function... 

If we restrict our interest to systems for which the potential energy 7 is a function only of the relative position vector r, then the classical Hamiltonian function H is given by... 

The interest here is in the energy levels of molecular systems. It is well known that an understanding of these energy levels requires quantum mechanics. The use of quantum mechanics requires knowledge of the Hamiltonian operator Hop which, in Cartesian coordinates, is easily derived from the classical Hamiltonian. Throughout this chapter quantum mechanical operators will be denoted by subscript op . If the classical Hamiltonian function H is written in terms of Cartesian momenta and of interparticle distances appropriate for the system, then the rule for transforming H to Hop is quite straightforward. Just replace each Cartesian momentum component... 

The quantum Hamiltonian of the classical kicked rotor, defined by the classical Hamiltonian function (5.1.2), is easily obtained by canonical quantization. On replacing the classical angular momentum L by the quantum angular momentum operator L according to... 

In this section we interpret the Hamiltonian (10.2.4) as a classical Hamiltonian function of the conjugate variables xi,pi) and x2,P2)- Transforming to action and angle variables with the help of the canonical transformation (6.1.18), the Hamiltonian (10.2.4) becomes ... 

Vi = d/dri. In deriving Eq. (B.106) we employed the space representation of the Hamiltonian operator [see Eq. (2.95)], and the fact that the classic Hamiltonian function can be split into kinetic- and potential-energy contributions iiccording to Eq. (2.100). Terms proportional to in Eq. (B.106) arise from the kinetic part of // applied to the product of terms on the right side of Eq. (B.104) (using, of course, the product rule of conventional calculus). 

In the usual procedure a classical Hamiltonian function for the model is formulated. Attention should be given to the choice of the molecular coordinate system. As the frame and top are rigid, a convenient choice is the principal axis system of the whole molecule.8 As a consequence of the symmetry of the top, the orientation of the coordinate system within the molecule is independent of the torsion angle. Another choice, which is called the internal axis system, is defined in such a way that the angular momenta produced by internal rotation of the top and frame compensate each other.10 The Hamiltonian functions in both coordinate systems are related by a contact transformation, which guarantees the invariance of Poisson brackets11 and, subsequently, of the commutation relations. 

By a routine mathematical technique, the equation can be split into a space dependent part and a time dependent part. These two parts of the Schroedinger equation are set equal to the same constant (the energy E) and solved separately. For our purposes, we are interested in the space dependent, time independent part, which describes a system that is not in a state of change. In Eq. (3.4), H is an operator called the Hamiltonian operator by analogy to the classical Hamiltonian function, which is the sum of potential and kinetic energies, and is equal to the total energy for a conservative system... 

The classical Hamiltonian function for the system is given in terms of these variables by... 

Hamiltonian An operator which when operating on the wave function of a quantum chemical system returns the energy of that system. The classical Hamiltonian function H = T + Fis the sum of the kinetic energy function Land the potential energy function V representing the total energy E of a system. 

Following the standard procedure of the Lagrange-Hamilton formalism, the classical Hamiltonian function is obtained ... 

The entirely classical Hamiltonian function, H, can be converted into the Hamiltonian operator, H, by applying some simple rules, which can be stated as follows ... 

Poisson bracket 0,H for the same observable O = 0 p, cj) (and the classical Hamiltonian function H) that does not explicitly depend on time. In addition, we note that if 6 ) commutes with the Hamiltonian operator tl, the time derivative of will be zero and, hence, a constant of the motion — in close analogy to the classical case given in Eq. (2.88). 

The problem for us is therefore to derive the classical Hamiltonian function for an electron in the presence of electromagnetic fields, which is normally done from the classical Lagrangian. Hamilton s and Lagrange s generalizations of classical mechanics are essentially the same theory as Newton s formulation but are more elegant and often computationally easier to use. In our context, their importance lies in the fact that they serve as a springboard to quantum mechanics. 

From this Lagrangian one can then define a classical Hamiltonian function, which translated to operator form yields the desired Hamiltonian operator. The classical Hamiltonian W is a function of time, of the generalized position coordinates r and of their conjugated generalized momenta p. It is defined as °... 

Assuming a stationary nucleus, the classical Hamiltonian function of the electrons in a helium-like atom with Z protons in the nucleus is... 

Classical mechanical formulas must agree with those obtained by taking the limit of quantum mechanical formulas as masses and energies become large (the correspondence limit). This limit does not affect the formula representing the equilibrium canonical probability density, so it must therefore be the same function of the energy as that of quantum statistical mechanics. For a one-component monatomic gas or liquid of N molecules without electronic excitation but with intermolecular forces, the classical energy (classical Hamiltonian function Jf) is expressed in terms of momentum components and coordinates ... 

The classical Hamiltonian function, in this coordinate system, will then be... 

The procedure for constructing the Hamiltonian operator is identical with that followed before in the classical Hamiltonian function the... 

Let us consider that the molecule is subjected, not to the infiuence of an electromagnetic field similar to that associated with light, but to a constant electric field in the z direction, of strength Ez, For this case, the vector potential A may be taken equal to zero the scalar potential is —zFa. The classical Hamiltonian function for a system of charged particles will then be... 

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