Hamiltonian operator classical

Up until now, little has been said about time. In classical mechanics, complete knowledge about the system at any time t suffices to predict with absolute certainty the properties of the system at any other time t. The situation is quite different in quantum mechanics, however, as it is not possible to know everything about the system at any time t. Nevertheless, the temporal behavior of a quantum-mechanical system evolves in a well defined way drat depends on the Hamiltonian operator and the wavefiinction T" according to the last postulate... 

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... 

In order to apply quantum-mechanical theory to the hydrogen atom, we first need to find the appropriate Hamiltonian operator and Schrodinger equation. As preparation for establishing the Hamiltonian operator, we consider a classical system of two interacting point particles with masses mi and m2 and instantaneous positions ri and V2 as shown in Figure 6.1. In terms of their cartesian components, these position vectors are... 

If we replace the z-component of the classical angular momentum in equation (6.87) by its quantum-mechanical operator, then the Hamiltonian operator Hb for the hydrogen-like atom in a magnetic field B becomes... 

In the quantum mechanical applications of the two-body problem, the classical energy of the system becomes the Hamiltonian operator The conversion... 

With these results for the angular-momentum operators it is possible to obtain die Hamiltonian for the rotation of a symmetric top by direct substitution in Eq. (13). The leader is warned that care must be taken in this substitution, as die order of the derivatives is to be rigorously respected. However, given sufficient patience one can show that the classical energy becomes the Hamiltonian operator in the form (problem 12)... 

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 electronic Hamiltonian operator He may be derived from the classical energy expression by replacing all momenta p, by the derivative operator, pt => —ihV(i) = —ih(8/8ri), where the first i is the square root of —1. Thus,... 

Note that the Hamiltonian operator in Eq. (4.3) is composed of kinetic energy and potential energy parts. The potential energy terms (the last three) appear exactly as they do in classical mechanics. The kinetic energy for a QM particle, however, is not expressed as p 2/2 j, but rather as the eigenvalue of the kinetic energy operator... 

We have bracketed the potential energy U with vib because both quantities depend on the coordinates of the nuclei relative to one another, whereas Trot averaged over the vibrations does not. The quantum-mechanical Hamiltonian operator is obtained by replacing the classical quantities with operators ... 

Here H is the Hamiltonian operator of the system without external field, M denotes the operator corresponding to the magnetization, and the Trace is the quantum mechanical equivalent of the classical integral over phase space. 

The total energy of a mechanical system is also given by the sum of its kinetic and potential energies, E = p2/2m + V. The operator equivalent of this classical expression, known as the Hamiltonian operator, is obtained by substituting the momentum operator into this equation, i.e. 

These two functions L and H allow us to solve classical problems, by focusing on the energies of the problem, rather than on forces, and thus present certain conceptual advantages the mathematical labor is the same From F = mdx2/dt2 it takes two integrations to obtain x(t) from F to either L and H involves one integration, but then one must do one more integration of L or H with respect to f, to obtain x t).L and FI also become important when the Hamiltonian operator is developed in quantum mechanics (Section 3.1). 

After using the classical-to-quantum correspondence px= — i ti l(d/dx), and so on, and fully expanding the quadratic forms, the quantum-mechanical Hamiltonian operator H becomes... 

In this H is thus now a symbol, not for a quantity but for an operation, to be applied to the wave function cp (see p. 123) H is called the HAMiLTONian operator. This operator H, therefore, has the same form as the function H in classical mechanics when the above-mentioned formal rules are borne in mind. 

We are interested in what happens when a magnetic moment fJt interacts with an applied magnetic field B0—an interaction commonly called the Zeeman interaction. Classically, the energy of this system varies, as illustrated in Fig. 2.1a, with the cosine of the angle between l and B0, with the lowest energy when they are aligned. In quantum theory, the Zeeman appears in the Hamiltonian operator... 

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

The Schrodinger equation is the starting point for molecular problems. The symbol H is a differential operator called the Hamiltonian operator, which is analogous to the classical Hamiltonian, in as much as it is a sum of kinetic and potential energy terms. E is the total energy for the system. The wavefunction P depends on the position of all the particles comprising the system. proposed that I Fp, and not P,... 

In accordance with the classical definition of the Hamiltonian, eqn (8.52), and recalling that the derivative of the Lagrangian with respect to a velocity (eqn (8.51)) is the momentum conjugate to the corresponding coordinate, the Hamiltonian operator is defined as... 

The quantity in square brackets is an operator —the Hamiltonian operator. All eigenvalue equations have this form and solutions occur only for certain eigenvalues of the factor E on the right hand side. This form immediately suggests the final generalisation from 1 to w electrons. Since the classical energy expression is... 

As an application of this formalism, we consider a two-level quantum system coupled to a classical bath as a simple model for a transfer reaction in a condensed phase environment. The Hamiltonian operator of this system, expressed in the diabatic basis L), P), has the matrix form [43]... 

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