In this video Lecture you will found: The Eigen function and Eigen values of a free particle and relation with the momentum and de Broglie wavelength.What is free particle? a particle is free if that is not bound by any internal or external force (it means potential is zero). What is one dimensional box or potential well?
This is just pictureization of the concept to show any quantum mechanically system like an atom or nucleus, or related solid state or molecular systems.An electron revolve around the nucleus in an atom this can be treated as the particle in a box concept and also a neutron or proton in nucleus are simple example.

A free particle is confined in a one dimensional box, the dimension of the box is L along the X-axis. The potential is infinite at x=0 and x=L ; so there is zero probability to find the particle at these points. You can say that wave function associated with the particle is zero. we use these two extreme points as a boundary condition to find the value of some arbitrary constants.

Now, the potential is zero inside the well (or Box) but infinite at x=0 and x=L or beyond that. Also, potential energy is independent from the time, so using the time independent Schrodinger equation and putting the value of potential that is zero. This becomes second order differential equation. Assume the solution of this equation with two constant with A and B. Find the values of these constant using the boundary condition.

After finding the values of the arbitrary constants define the possible energy states of the particle in one dimensional potential well and normalize the wave function. Because without normalization of the wave function you can not find out the probability of the particle in any system, it is must.

This quantum particle is associated with the matter waves so there will be a relationship between the momentum of this particle and the wavelength associated with it as per the de Broglie wave concept. For this, as we know the total energy is the sum of kinetic energy and potential energy, here potential energy is zero so the particle’s total energy will be equal to the kinetic energy. This kinetic energy can be define in terms of the momentum as p^2/2m where p is the momentum of the particle and m is the mass. Compare it with the total energy and find out the momentum of the particle in any state (p=nh/2L), this momentum depends on the dimension of the box and the energy state of the particle.

Further, if you want to calculate the de Broglie wavelength of the wave associated with this particle then lambda=eh bar upon p, put the value of p in terms of lambda in the above result and you will find that lambda= 2L/n ; for any state (n=1,2,3….)you can calculate the wavelength of the wave. So by this way one can see the particle and wave nature of matter.