Overiew of Classical Mechanics

To appreciate Quantum Mechanics, it is important to review two fundamental theories in classical mechanics that govern the motion of a particle, namely the Energy Conservation  and Newton's second laws.

Conservation of Energy

The total energy E of a system is constant.   (E is a continuous quantity and can take any real number).

where T is the kinetic energy arising from the motion of the particle and V is the potential energy arising from its position. 

Since   where m and v are the mass and velecity of the particle, respectively, and , we can obtain:

Solution to this equation for a given total energy yields the position of the particle as a function of time, .

Newton's second law

The rate of change of the momentum is equal to the force acting on the molecule.

Since the force acting on the particle moving in one dimention is given by the gradient of the potential energy V, namely ,

Solution to this equation yields the momentum as a function of time, .

Both and define the trajectory of the particle, which completely and exactly describes the state of the particle at any time t.

Significance:   For a given total energy, knowing the potential energy function V(x) and the initial potition and momentum of the particle, and , one can determine the potition and momentum of the particle at any time t (forward or backward).

This principle is applied in the anti-missile defense system to predict the location of the missle in a near future time and also used in determining the locations and speeds of the involved vehicles  prior to their collision.   

-- ThanhTruong - 12 Jul 2007