Problems with unbound particles.

Las time we examined the continuity equation. This equation helps us to examine the flow of probability for wavefunctions

è is the material density, the amount per unit volume.

è is the current density, the amount per unit volume flowing in the direction .

If the current is not diverging then the amount of stuff that flows into a small volume is equal to the amount of stuff leaving. This condition is mathematically stated

The validity or meaning of this differential equation is usually made clear in vector calculus, for example, when considering Gauss’s law.

If the divergence is not zero then the amount of material must accumulate or decline.

For a one dimensional problem this condition that guarantees the conservation of material becomes

If the flow changes as you examine it wrt x then there must be a change in the density.

• no change in flow anywhere on the x-axis the same amount of material is moving so there is no change in the density.
• flow increases with x anywhere on the x-axis the same amount of material that is moving is increasing so more stuff is moving the larger x becomes. The stuff is obtained by depleting the local density. Therefore

Since the “amount of particle” needs to be conserved in QM we will enforce a similar relationship. We need to consider the probability of finding a particle in some volume as the stuff that is flowing. If it becomes more likely to find a particle in one region as time proceeds we will want another region to see a decline in the probability to detect a particle. The probability density changes in response to a flow which maintains the overall probability of finding the particle somewhere to be one.

With

equal to the probability density of finding a particle and the wf satisfying the Shrodinger equation.

if is time independent and V depends on only x then

Thus the flow of probability in QM can be defined to be

Consider free particles hitting some type of scattering object represented by the figure below.

The incident, reflected and transmitted waves are depicted. The incident and reflected wave occupy the identical environment and so only the direction of propagation has changed. The magnitude of the momentum is the same for each of these waves. The transmitted wave could be in a different environment. For these problems we only consider regions of constant potential so the wave can have a different momentum but it still represents a free particle solution.

This is the general format for the problem. Several different problems which have a different scattering potential will be solved. In order to complete these problems we need to figure out how many of the incoming particles are reflected and transmitted. We use the current density defined above to quantify these results.

the wavenumber for the incident region

the wavenumber for the transmitted region

+ is a reflected wave, - is an incident or transmitted wave.

conservation of energy for transmitted and reflected waves.

Define the coefficient for reflection and transmission

The above solutions are obtained from the Shrodinger equation.

So in regions where the energy is greater that the potential, a propagating plane wave with momentum satisfies the Schrodinger equation and for regions where the potential is greater the energy, which represents a classically disallowed region, the wave function grows or decays exponentially.

A typical problem involves a barrier of height V_b between regions of no potential and an energy E, .

The problem is divided into three regions I, II, III and the solutions for each region found (as above).

I	II		III
A sets the incident beam term B sets the reflected			E is transmitted F incoming form right F=0
Boundary condition I èII		Boundary condition II èIII

The problem is a bit different if the Energy is tuned to match the barrier height such that

This is a special case where the solution for region II has a linear dependence on x rather than a sinusoidal or exponential dependence.