Q. Write down the Schrodinger's time independent wave equation for a free particle confined in one dimensional box of size a. Obtain eigen values and normalized wave functions for this particle.

Ans.

Particle in one Dimensional Box ( 1 - D Box )

In quantum mechanics a one dimensional box means a system in which a particle is constraint to move in one direction in a box of potential V = 0. The limits of the potential function for the particle is defined as follows

Suppose a particle of mass m is in motion along the x axis. Let no other force act on this particle, so that the potential energy of the particle is constant. For convenience, the constant potential energy is taken to be zero. 

(A)  Eigen Value of the Wave Function : The wave function corresponding to E0 are called eigen functions of the particle. The integer n corresponding to the energy E0 is called the quantum number of the energy level E0. Fig. shows the schematic representation of energy level for a particle in one dimensional box.

 Calculation of Constant A (Normalization of the eigen function) : Since the solutions of the wave equation are independent of the magnitude of A this permits us to choose a value for A in order to satisfy any condition we want to impose. The total probability that the particle is somewhere in the box must be unity.