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The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. . The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction ...
The value (,) of the Hamiltonian is the total energy of the system, in this case the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here p is the momentum mv and q is the space coordinate.
The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it.
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics , this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field .
Examples of integrals of motion are the angular momentum vector, =, or a Hamiltonian without time dependence, such as (,) = + (). An example of a function that is a constant of motion but not an integral of motion would be the function C ( x , v , t ) = x − v t {\displaystyle C(x,v,t)=x-vt} for an object moving at a constant speed in one ...
Moreover, the motion of three bodies is generally non-repeating, except in special cases. [ 8 ] However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a Puiseux series , specifically a power series in terms of powers of t 1/3 . [ 9 ]
Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the n = 0 state, called the ground state) is not equal to the minimum of the potential well, but ħω/2 above it; this is called zero-point energy.
The stationary nuclei enter the problem only as generators of an electric potential in which the electrons move in a quantum mechanical way. Within this framework the molecular Hamiltonian has been simplified to the so-called clamped nucleus Hamiltonian, also called electronic Hamiltonian, that acts only on functions of the electronic coordinates.