AMO Physics

This week we will learn about the structure of atoms. We will start with the simplest model - the Bohr model. Next, we will look at the other quantum numbers describing angular momentum and spin. As atoms aren’t as simple as the Bohr model, we will start adding corrections by discussing the Fine Structure and external effects like the Zeeman Effect. Finally, we will look some basic scattering theory, Fermi’s golden rule, and measuring the states of atoms and emitted photons with absorption spectroscopy.

Goals

• Attach the various quantum numbers for an atom to associated energy contributions

• Understand at a high level where the energy contributions come from.

Deliverables

• Create a table of all the various energy contributions.

An introduction to the Simple Atomic Model

In a full semester quantum mechanics course, you would learn about Schrodinger’s equation, Hamiltonians, and standard quantum mechanics problems to derive energy levels. We will leave this prescription out for a future course, but feel free to peruse the references of this module to learn more.

For now, we care just about the high level physics of our problem. In general, dealing with atoms is extremely complicated! You might have already guessed this - even solving the double pendulum problem analytically in mechanics is hard, and solving the triple pendulum is intractable. Many-body physics is a very open area of physics, as adding more moving bodies, especially in quantum mechanics, scales horribly. If you’re looking for rigorous derivation for all this AMO physics, we will provide some references at the end of this module!

The Bohr model and the Principle (n) quantum number

We start our study with the Hydrogen atom. We do this because this atom is simple, and will act as the foundation for all other structures we will study. A hydrogen atom has a single proton nucleus, and a single orbiting electron. Let’s follow the assumptions Bohr used back in the 1900s. We will assume that the nucleus is huge compared to the electrons. In this case, we can work in the frame of the nucleus and the energy is determined solely by the state of the electron. In this model, the electron will have to live in quantized orbitals. Each of these orbitals will have differing amounts of energies. The resulting energy of the $n$th energy level will be:

$$E_n = \frac{E_1}{n^2}$$

where $E_1$ is the ground state energy, $E_1 = -13.6 eV$. The $n$ number is some measure of “how excited” our atom is, with the energy increasing (getting less negative) as $n$ increases. We call this $n$ the principal quantum number.

Angular Momentum and Spin

You might recall from chemistry classes that we could assign electronics various orbitals. But in those classes, we used the notation 1s, 2s, 2p, and so on. Actually, using just one letter $n$ is not enough to describe our atoms. So what other labels do we need?

An electron orbiting the nucleus should live in some orbital. We use $l$, the angular momentum quantum number, to describe the shape of this orbital. The value of $l$ are constrained to $l \in \{ -n+1,-n+2, \dots, n-2, n-1\}$.

We also have the quantum number $m_l$. This is the magnetic quantum number and describes the orientation of the orbital in space. We should expect different orientations because not all orbitals are perfect spheres! Since they live in 3D space, there are many ways we can orient our atom! The magnetic number is constrained by $m_l \in \{ -l, -l+1, \dots, l-1, l\}$.

We also have $s$, the spin quantum number, and $m_s$, the spin magnetic quantum number. For a single electron, it is known as a spin ½ particle. The magnetic is also constraints by $m_s \in \{ -s, -s+1, \dots, s-1, s\}$, and it tells us the orientation of the spin. For spin ½ particles, $m_s$ can be $\pm ½$, indicating spin up or spin down.

The Fine Structure and other corrections to the energy

In the Bohr model, only the $n$ quantum number changed the energy. But now we have a ton of numbers to describe our atom: $n, l, m_l, s, m_s$. Do these have any effect on the energy?

If we keep working with the Bohr model, no! But of course, we took a bunch of approximations to get there. Calculating the energy of complicated atoms and molecules is an open problem in the field of quantum chemistry. If we didn’t take any approximations, calculating these energies would be impossible. To deal with this, chemists and AMO physicists often start with simple models and add multiple correction as we relax some of the assumptions. Even before we start, the lesson is that atoms are complicated.

The first layer of corrections is known as the Fine Structure. There are three notable corrections: the relativistic correction, spin-orbit coupling, and the Darwin Term. In our discussion, we will discuss the relativistic correction and Spin-Orbit coupling.

The Relativistic Correction

The kinetic energy of the electron would normally take the form $KE = \frac{p^2}{2m_e}$, where $p$ is the magnitude of the momentum of the electron and $m_e$ is the mass of the electrons. If we include the contributions from special relativity, the kinetic energy takes the form

$$KE = \sqrt{p^2c^2 + m_e^2c^4} - m_ec^2$$

If we Taylor expand this, we obtain

$$KE \approx \frac{p^2}{2m_e} - \frac{p^4}{8m_e^3 c^2}.$$

The second term is known as the relativistic correction.

Spin-Orbit Coupling

We previously talked about the $s$ and $l$ quantum numbers corresponding to spin and orbital angular momentum. The operators that describe these are $\pmb{S}$ and $\pmb{L}$ respectively. The spin orbit coupling term is proportional to $\pmb{S} \cdot \pmb{L}$.

We’ve been treating our atom as a stationary proton nucleus with the electron orbiting around it. We can instead work in the frame of reference of the electron. Then, the moving proton generates a current that will create a magnetic field proportional to $\pmb{L}$. This will interact with the magnetic moment of the electron, proportional to $\pmb{S}$.

The important thing here is more in how we solve for this correction. Dealing with a term $\pmb{S} \cdot \pmb{L}$ is really hard in the basis $|n, l, m_l, s, m_s \rangle$. This is because all those numbers work because they correspond to eigenstates of the operators corresponding to each correction and energy term. The strategy for solving this term is to consider the total angular momentum of the electron, given by

$$\pmb{J} = \pmb{L} + \pmb{S}$$

In this case, we can instead use the basis $|n, j, m_j, l, s \rangle$ to track the state of our electron.

The Zeeman Effect

While not necessarily included in the Fine structure, we will now look into the Zeeman Effect. In our experiment, we will put our Rubidium atoms in the presence of a magnetic field.

Recall from your classical E&M courses that the energy of an object with magnetic moment $\pmb{\mu}$ in a magnetic field is

$$E = \pmb{\mu} \cdot \pmb{B}.$$

In our atomic model, the magnetic moment can be written as

$$\pmb{\mu} = -\frac{\mu_B}{\hbar}(g_L \pmb{L} + g_S \pmb{S})$$

Where $\mu_B$ is a constant known as the Bohr Magneton and $g_L$ and $g_S$ factors that tell us how much the angular momentum and spin contribute to the magnetic dipole moment.

From our Fine structure discussion, you might notice that one issue is that if we want to work with the $j$ and $m_j$ numbers, the energy written above depends on $l, m_l, s$, and $m_s$. If the magnetic field magnitude $B$ is weak, we can work in the $|n, j, m_j, l, s\rangle$ basis and rewrite the above contribution in terms of $m_j$. The resulting energy shift for a field in the same direction as the magnetic moment quantization is

$$E_{\text{Zeeman}} = \mu_B g_j B m_j$$

Where $g_j$ tells us the proportionality of the total angular momentum $\pmb{J}$ and the magnetic moment.

In the strong magnetic field limit, the spin-orbit coupling that demanded the use of $\pmb{J}$ is small and can be ignored. This is known as the Paschen-Back Effect.

Fermi's Golden Rule and Scattering Theory

We have learned so much theory about our atoms, but how do we actually extract this information from a real experiment? Scattering theory is one way we can try to solve this! We basically throw things at our atoms and see how they bounce back to extract information. In this case, we are throwing light at our atoms and measuring what is bounced back.

Why does this occur? Let’s talk briefly about some mechanisms for light matter interaction. Let's consider a two level atom with a ground and excited state, designated by $E_g$ and $E_e$ respectively. There are three atom-light interactions we should know:

• Absorption - an atom in the ground state absorbs a photon with energy $E_e-E_g$.
• Stimulated Emission - an atom in the excited state is "encouraged" to emit its photon and drops to the ground state. This requires there to be a photon around the atom to force the stimulated emission.

• Spontaneous Emission - an atom in the excited state spontaneously falls to the ground state and releases a photon. This cannot be explained with classical E&M and the mathematics will not be discussed here. There is no requirement for a photon nearby.

Concept Check: Before moving on, answer the following. What is the difference between spontaneous and stimulated emission, and which do you expect to be used in a laser? Draw some figures so you have a picture in your head! For scattering theory, the important part here is absorption and stimulated emission.

For the case of our atoms, when we shine light on them, we might want to calculate the rate at which the atoms transitions between states. We can actually solve for this rate! With a proper treatment of time dependent perturbation theory, we can calculate the transition rate between two states for an atom interaction with light. This rate is known as Fermi’s Golden Rule, and tells us that the transition rate isn independent of time and depends on the coupling strength between the initial and final states.

Pre-Lab Questions and concept checks

• What are all of our main quantum numbers thus far? Do they contribute to the energy? How? Make a chart that shows all of these levels generically.

• How do we determine what state our electrons are in?

• How practical is this toy atom model? Why is it not exact?

References and additional reading

• Griffiths 3rd Ed Chapter 4 (Hydrogen Atom), Chapter 7 (Time Independent Perturbation Theory, Fine Structure, Zeeman Effect), 8 (Time Dependent Perturbation Theory, Fermi’s Golden Rule)

• Griffiths 3rd Ed Chapter 1, 2 (Introductory Quantum Mechanics)

• Sakurai: Chapter 5 (Zeeman Effect and Fine Structure)

• Foot: Chapter 1, 5 (Zeeman Effect and Fine Structure)

• ChemLibreTexts on Fine and Hyperfine Structure

• MIT OCW 8.06 Lecture on the Relativistic Correction

• MIT OCW 8.06 Lecture on Zeeman Splitting and Fine Structure