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# 6.1. Adiabatic approximation¶

Expected prerequisites

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

- Describe the behaviour of a classical pendulum
- Describe the behaviour of a quantum particle in a box.

Learning goals

goals this lecture you will be able to:

- Explain the adiabatic theorem.
- Explain how the wave function evolves under an adiabatic or abrupt potential.

## 6.1.1. Adiabatic theorem¶

Warning

An adiabatic process, in the context of quantum mechanics, is defined by a
gradual change in external conditions with *time*. This is not to be confused with adiabatic
processes in thermodynamics, which is characterised by no energy exchange
between a system and its environment.

The **adiabatic theorem**, from Max Born and Vladimir Fock, states that:

A physical system remains in its

instantaneous eigenstateif a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.

Let's unpack what this means through a classical example to solidify our understanding, and then apply it to a case involving quantum mechanics.

## 6.1.2. Classical Example¶

### Pendulum¶

The classical period of a pendulum with arm length under a gravitational force is . If we now abruptly changed the length of the pendulum's arm, the motion of the mass at the end would be hard to predict. It certainly wouldn't resemble the gradual swinging we started with. In contrast, should we vary very slowly, we would preserve the oscillation of the mass, and indeed the period, as a function of time, would now be .

Now that we have this intuition, let's look at an example from quantum mechanics.

## 6.1.3. Quantum Example¶

### Particle in a Box¶

Assume we start with a particle in the ground state of an infinite one-dimensional box of width . The wave function then has the form

Now the question we want to answer is: if we change the width of the well by
moving the right wall outwards, how does the wave function change? The answer
depends on *how fast* we move one of the walls.

The adiabatic theorem tells us that if system Hamiltonian changes gradually enough from some initial form to a final form , then if the particle was in the eigenstate of the initial Hamiltonian to start with, it will be carried to the eigenstate of the final Hamiltonian. All the while, as the wall is moving from say to , the wave function will remain in that ground state associated with whatever width the well happens to have at any given moment.

We are also now in a position to appreciate the second half of the theorem, which says:

" [...] if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum."

In practice, this means that the energy of the eigenstate, which we start with, has to be non-degenerate and discrete, so that it makes sense to talk about the eigenstate of one system compared to the eigenstate of another.

### Abrupt change in potential¶

First, consider a system where the change in the potential is abrupt. Therefore, the wave function goes out of the ground state, and it mixes with high energy states in a non-trivial way. Such behaviour can be observed in the following animation.

import common from IPython.display import HTML, display import numpy as np common.configure_plotting() from adiabatic import make_adiabatic_potential_animation L = 200 def pot_abrupt(i, t): if t < 50: width = L/1.7 else: width = L return np.tanh((i - width + 30) / 5) + 1 - np.tanh((i - 30) / 5) + 1 anim = make_adiabatic_potential_animation(potential=pot_abrupt) display(HTML(anim.to_jshtml(default_mode='loop')))