Motion of a Free Particle

MOTION OF A FREE PARTICLE

Let us consider electrons propagating freely in space in the positive x-direction and not acted upon by any force.

As the electrons are not acted upon by any force, their potential energy V is zero. Schrodinger equation

The general solution of the above equation is

where A and B are constants. As it is assumed that the waves propagate only in the positive x-direction, we can write

There are no boundary conditions to be considered and hence there are no restrictions on k. All values of the energy are allowed. The allowed energy values form a continuum and are given by

A freely moving electron therefore possess a continuous energy spectrum as shown in fig. 6.7.

It is noted from equation (2) that

The k known as wave vector describes the wave properties of the electrons. Further, it is seen from the relation (2) that

The plot of E as a function of k gives a parabola, as explained in fig. 6.8.

The momentum is well defined in this case. Therefore, according to uncertainty principle it is difficult to assign a position to the electron. The uncertainty in position will be infinity which means that the electron position is indeterminate.