Tunneling, Transmission, and Reflection

First we will take a diversion and consider a particle that is free to travel along the x-axis (a box of infinite length). The potential is zero aU along the x-axis. This means that the particle possesses only kinetic energy all along the x-axis and there are no boundary conditions. The Schroedinger equation can be readily written for this system. 

Note that the general wavefunction for a free particle is the same as the general wavefunction for a Particle-in-a-Box. The coefficients A and B are determined by the boundary conditions, however, since there are no 

The momentum of the particle can be determined by applying the momentum operator (see Equation 2-6) to the two parts of the wavefunction in Equation 5-20 independently. 

The only difference between the solutions in Equations 5-21 and 5-22 is the sign, so it can be concluded that since momentum is a vector quantity, the two solutions represent the particle with a momentum of equal magnitude but opposite directions. If a particle is shot from a cannon in the positive x direction, the value of the coefficient B for the wavefunction associated with that particle is zero. Likewise, a dueUng particle shot in the negative x direction wiU have a wavefunction with a coefficient A equal to zero. 

It is interesting to note that the wavefunction for the particle in this system extends from negative to positive infinity along the x-axis. Since there are no boundary conditions for the particle, there is no quantization or regions where the particle has the greatest probability density. The wavefunction is evenly distributed throughout the x-axis just like a pure wave. It can be concluded that in part what Umits the wavelike nature of matter is the potentials that a particle experiences. In fact, part of the reason for our particulate view of matter is as a result of the potentials that aU matter in the universe experiences. 

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