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6.1. Adiabatic approximation

Expected prerequisites

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

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

Learning goals

goals this lecture you will be able to:

1. Explain the adiabatic theorem.
2. 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 eigenstate if 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 $L$ under a gravitational force $g$ is $T = 2\pi \sqrt{L/g}$. 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 $L$ very slowly, we would preserve the oscillation of the mass, and indeed the period, as a function of time, would now be $T(t) = 2\pi \sqrt{L(t)/g}$.

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 $L$. The wave function then has the form

$$\psi_i(x) = \sqrt{\frac{2}{L}} \sin \left( \frac{\pi x}{L} \right).$$

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 $\hat{H}(0)$ to a final form $\hat{H}(T)$, then if the particle was in the $n^{\text{th}}$ eigenstate of the initial Hamiltonian to start with, it will be carried to the $n^{\text{th}}$ eigenstate of the final Hamiltonian. All the while, as the wall is moving from say $L$ to $2L$, 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 $n^{\text{th}}$ eigenstate of one system compared to the $n^{\text{th}}$ 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')))

Once Loop Reflect

Adiabatic change in potential

On the other hand, when the change in the potential is smooth, the adiabatic theorem tells us that we can expect the ground state to remain in the ground state even when the wall moves, as long as it moves slowly enough.

from adiabatic import make_adiabatic_potential_animation

L = 200
def pot_slow(i, t):
    width = L/1.7 + t/600 * (L - L/1.7)

    return np.tanh((i - width + 30) / 5) + 1 - np.tanh((i - 30) / 5) + 1

anim = make_adiabatic_potential_animation(potential=pot_slow)
display(HTML(anim.to_jshtml(default_mode='loop')))

Once Loop Reflect

6.1.4. Summary

Take-home message

Adiabatic theorem states that a system will remain on its instantaneous eigenstate if:

• The external perturbation acts slowly enough.
• There's always a non-zero gap with other states.