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4.1. A particle moving in a potential

Expected prerequisites

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

• Describe the motion of a classical particle in a potential.

Learning goals

After this lecture you will be able to:

• Describe the motion of a quantum mechanical wave packet in a smooth or abrupt potential.

4.1.1. Classical intuition

The WKB approximation is used to describe a particle of mass $m$ in a one-dimensional (1D) potential with some initial velocity. To call on some intuition, let us recall the classical case: consider a particle of energy $E$ moving in a potential $V(x)$. In this situation, the total energy of the particle is conserved, and can be separated at every point $x$ into kinetic and potential energy as $E = E_\text{kin}(x) + V(x)$.

If $E > V(x)$, the particle will always have an energy higher than the potential at any point $x$. Hence, there will always be some non-zero kinetic energy at every point $x$. As a consequence, the particle will continuously move in one direction (determined by the initial velocity). This happens irrespective of the potential shape: for example, it does not matter whether the potential is smooth or changes abruptly.

If $E < V(x)$, a classical particle will reach a maximum position $x_0$ such that $E = V(x_0)$ and the kinetic energy becomes zero. Then, the particle will be reflected. Again, this will happen for any potential shape.

In summary, the global motion of a particle in classical mechanics in a 1D potential is determined only by the value of the total energy compared to the potential $V(x)$.

4.1.2 Quantum case

Let us now consider the quantum case: instead of a point particle, we will use its quantum analogue, a wave packet, and let it time-evolve in different potentials.

Smooth potential

Firstly, we consider the case where $V(x)$ is a smooth potential describing a dip. One can confirm with the animation below that the intuition from classical mechanics still holds because the particle moves across the potential with a sightly increased velocity while in the dip.

import common
from IPython.display import HTML, display
common.configure_plotting()
from wkb import make_wave_packet_animation
import math

def pot_gauss(x):
    return -0.02*math.exp(-(x-1500.0)**2/200**2)

anim = make_wave_packet_animation(L=3500, pot_func=pot_gauss, 
                                  zero_pos=750, width=100,
                                  energy=0.01)

display(HTML(anim.to_jshtml(default_mode='loop')))

Once Loop Reflect

Abrupt potential

One the other hand, if $V(x)$ describes an abrupt potential, one can observe in the following animation that the wave function reflects back whenever it crosses an abrupt step.

import common
from IPython.display import HTML, display
common.configure_plotting()
from wkb import make_wave_packet_animation
import math

def pot_step(x):
    if 1500 < x < 2200:
        return -0.02
    else:
        return 0

anim = make_wave_packet_animation(L=3500, pot_func=pot_step, 
                                  zero_pos=750, width=100,
                                  energy=0.01)

display(HTML(anim.to_jshtml(default_mode='loop')))

Once Loop Reflect

This effect is not present in the classical case.

4.1.3. Summary

Take-home message

For a quantum particle with $E > V(x)$:

• When a particle moves along a smooth potential, it resembles the classical case.
• When a particle moves along an abrupt potential, there's always some reflection.