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# 4.5. Transport¶

Expected prerequisites

Before the start of this lecture, you should be able to:

- Describe the behaviour of tunnelling through a rectangular barrier
- Write down the WKB wave function
- Apply the connection formulas to connect the WKB wave functions across a turning point
- Rewrite integrals

Learning goals

After this lecture you will be able to:

- Write down the transmission probability of a tunnelling problem.
- Describe the effect of the potential on the probability current for a travelling wave.

## 4.5.1. Setting up a transport problem¶

Having looked into bound states, we will now focus on quite the opposite situation:

- Far away from the potential, where the energy is
*larger*than the potential,

In such an *open* case, we will not have bound states but there will be eigenstates for a continuous range of energies. Far away from some central potential, the eigenstates can be written as a superposition of plane waves. One particular choice for the eigenstate is to choose to have only an incoming wave on one side (this is always possible) and to consider its reflection and transmission as shown in the following sketch:

Assuming that the potential becomes constant for some point away from the central potential, the wave function can be written as where are the wave vectors on the left and right side. These can be different if the potential left and right are not equal.

In such a setting, the interesting question is:

*How much of the incoming wave is transmitted to the right, and how much reflected to the left?*

In fact, you solved exactly such a system already in your first quantum mechanics class when you computed the tunnel probability through a square barrier! With WKB, we can now find approximate solutions for any kind of potential.

The central quantity of interest is the tunneling probability . As in the example of the square tunnel barrier, it is generally defined as
where and are the velocities to the left and the right of the central potential. The reasoning behind the velocity entering the equation is that transmission probability corresponds to the probability *current* and the question of how much of the incident current goes through and how much is reflected.

It is possible to get rid of the awkward velocity term by a small redefinition: We can instead write: The point is that with this convention, the waves to the left and right have a constant current if e.g. the coefficients and are the same. In fact, we can immediately see that for the WKB wave functions we have already chosen such a normalization where :

The tunnel probability is thus simply given by

## 4.5.2. : Reflectionless transport¶

Let us start with the simplest case, where for all . For that case we already previously wrote down the WKB wave function for all . In particular, if we only consider an incoming wave from the left, we obtain We see here that there is no reflected wave, so the transmission probability is . Indeed, we argued initially that for a smooth potential there is no backreflection, so no surprises here!

It can still be interesting to think about how the wave function interacts with the potential. When deriving the WKB wave function intuitively, we used current conservation to derive the amplitude of the wave function. In particular,

where was the amplitude of the wave function. We thus see that for regions:

- where the momentum decreases, i.e. for larger , the amplitude needs to increase.
- Conversely, when increases, the amplitude needs to decrease.

This phenomenon can be observed in the following animation:

import common from IPython.display import HTML, display common.configure_plotting() from wkb import wkb_static_animation import numpy as np x = np.linspace(500, 2500, 1000) anim = wkb_static_animation(x, E=1.5) display(HTML(anim.to_jshtml(default_mode='loop')))