Configuration Space
In classical mechanics, the state of a system with s degrees of freedom is represented by 2s time-depen- dent generalized coordinates {
Here, the dot on ), denotes the time derivative -y-. The space formed from the q(t) values is called the configuration space. During the time evolution from time moment ta to time moment tb> the system will follow a certain path q(t) in configuration space. The starting point of classical mechanics is the concept of the state of motion, that is, the set of information specifying the history of a point particle as the function of the time. The differential equation defining the path of the system in the configuration space is called the equation of motion of the system. Newton’s equation of motion is the second order in the time derivative. So, we need generalized coordinates {
it is therefore very possible to use the coordinate or the momentum or both to define the state of motion with the main object of the restriction being the trajectory, q(t). In relation 3.2, h is the universal Planck constant with dimension equal that of an action and the number in parentheses indicates the experimental uncertainty of the last two digits before it. Knowledge of the trajectory of a particle by the continuous monitoring of its location will permit knowledge of the coordinate and momentum simultaneously. The aim of classical as well as quantum mechanics is to predict the time evolution of the system, starting from knowledge of the state of the system at a given point in time. We consider the simplest quantum mechanical system with one degree of freedom of a non-relativistic particle described by the Hamiltonian . operator H p,q :
Here = | and p = jpfj are respectively the operators of the generalized coordinate and momentum with s being the number of degrees of freedom and m , the mass of the particle; U^q j, the operator of the potential energy with q = {q,| being the generalized coordinate.
