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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 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 moving in a potential . In this situation, the total energy of the particle is conserved, and can be separated at every point into kinetic and potential energy as .

If , the particle will always have an energy higher than the potential at any point . Hence, there will always be some non-zero kinetic energy at every point . 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 , a classical particle will reach a maximum position such that 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 .

## 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 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')))