# Scattering States, Transmission and Reflection

### From FSUPhysicsWiki

We will now discuss scattering states in a one-dimensional potential. Scattering states are states that are not bound. Such states have energies larger than the potential at at least one of and their energy spectrum forms a continuous band, rather than a discrete set as the bound states do. Unlike the bound case, the wave function does not have to vanish at infinity, though a particle cannot reflect from infinity, often giving a useful boundary condition. At any discontinuous changes in the potentials, the wave function must still be continuous and differentiable as for the bound states.

We are interested in obtaining the wave functions for these scattering states in order to discuss transmission and reflection of waves from one-dimensional potentials, and to find the transmission and reflection coefficients and which give the probability that an incident wave will be transmitted and reflected, respectively.

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## The Step Potential

As a first example, let us consider a one-dimensional potential step. The potential in this case is given by

The corresponding Schrödinger equation is

Let us first consider states with energy We will divide the one-dimensional space into two regions - region I, for which and region II, for which The Schrödinger equations for the two regions are

and

and thus the corresponding wave functions are

and

where and We may consider the first term in to be an incident wave from the left, in which case the second term, is a reflected wave from the potential barrier and the first term in , is a transmitted wave. Similarly, we can think of the second term in to be an incident wave from the right, in which case is now the reflected wave and is the transmitted wave. These interpretations of the various terms will become more obvious shortly.

The boundary conditions at require

and

which give us

and

If we assume that the wave is incident from the left, then we can set . In this case, reflection occurs at the potential step, and there is transmission to the right. We then have

and

From the above equations, we obtain

and

To determine the probability of reflection and transmission, we must now find the current density,

on each side of the barrier. Doing so, we obtain

We thus see more clearly that the terms that we earlier identified as incident, reflected, and transmitted waves are as we labeled them. We define the ratio of the reflected current density to the incident current density as the *reflection coefficient*, and the ratio of the transmitted current density to the incident current density as the *transmission coefficient*,

The continuity equation for the current density in one dimension implies conservation of current density, and thus that

If we determine the reflection and transmission coefficients for the problem at hand, we obtain

and

We see that these expressions satisfy as expected.

Now let us consider the case in which If we again assume that the wave is incident from the left, the wave functions become

and

where Applying the same boundary conditions as before, we obtain

and

From the second wave function, we see that the transmitted wave decreases exponentially over a length scale given by .

If we determine the current density in this case, we find that

We see that the wave is completely reflected; i.e., and

The reflected wave acquires a phase difference relative to the incident wave. To see this, we simply rewrite the wave function as

where the phase difference of the reflected wave with respect to the incident wave is given by

or

## The Square Potential Barrier

For a square potential barrier, given by

we can write the general solution of the Schrödinger equation for

where and

The boundary conditions are the same as before; the boundary conditions at give us

Similarly, the boundary conditions at give us

For the convenience, let us express the coefficients of these linear homogeneous relations in terms of matrices:

If we combine these two equations, we have

where and

Note that

Let us now assume that there is only an incident wave coming from the left; i.e., By similar arguments as before, we may identify the transmission coefficient as

and the reflection coefficient as

## Finite Asymmetric Square Well

We now consider an asymmetric square well potential, given by

In this case, the wave functions are given by

where and

Applying the boundary conditions at we obtain

and

while those at give us

and

We may express these in matrix form as

and

If we combine these, we obtain

Let us once again assume that there is only an incident wave from the left, so that Using the same arguments as before, we may identify the transmission coefficient as

and the reflection coefficient as

## The Dirac Delta Function Potential

We now consider scattering from a Dirac delta function potential, For the Schrödinger equation is just that for a free particle,

where

The general solution for is

while that for is

As in the other cases, the wave function must be continuous at so

The derivative of the wave function, however, is not continuous, as noted when we studied the bound states. The discontinuity of the derivative is given by

which yields the relation,

or

where

We once again assume that the incoming particles are coming from the left, so that Consideration of the current densities on each side of the potential tells us that the reflection coefficient and that the transmission coefficient is We thus wish to solve for and in terms of doing so, we obtain

and

The reflection coefficient is thus

and the transmission coefficient is

In terms of the energy, these become

and