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).

%$E = T + V $%

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

Since %${T = \frac{ 1}{ 2} m { v}^{ 2} $%  where m and v are the mass and velecity of the particle, respectively, and %$v = \frac{ dx}{ dt} $%, we can obtain:

%MATHMODE{\frac{ dx}{ dt} = \sqrt { \frac{ 2 \left (E-V \right ) }{ m} } }%

Solution to this equation for a given total energy yields the position of the particle as a function of time, %$x(t)$%.

Newton's second law

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

%MATHMODE{ \frac{ dp}{ dt} = F = m \frac{ { d}^{ 2} x}{ d { t}^{ 2} } }%

Since the force acting on the particle moving in one dimention is given by the gradient of the potential energy V, namely %$ F = - \frac{ dV}{ dx} $%,

%MATHMODE{ \frac{ dp}{ dt} = - \frac{ dV}{ dx} }%

Solution to this equation yields the momentum as a function of time, %$p(t)$%.

Both %$x(t)$% and %$p(t)$% 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, %$x\left ( { t}_{ 0} \right )$% and %$p\left ( { t}_{ 0} \right )$%, 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