In this lecture we will start to put atoms together to build simple molecules. We will first use the Born-Oppenheimer approximation, to eliminate slow processes from the study of the fast electron dynamics. Then, we will study simple mechanisms of binding atoms.

We will discuss the creation of entangled photons and how they can be used for the test of Bell's inequalities.

We will study some basic properties of the laser.

In this second lecture we will finish the discussion of the basic cooking recipes and discuss a few of the consequences like the uncertainty relation, the existance of wave packages and the Ehrenfest theorem.

In this first lecture we will review the foundations of quantum mechanics at the level of a cooking recipe. This will enable us to use them later for the discussion of the atomic structure and interaction between atoms and light.

In the last lecture \cite{electrons} we have seen that we can typically treat complex atomic systems within the central field approximation. \begin{align} \label{eq:lsvsjjhamiltonian} \hat{H} &&=& \underbrace{\sum_i^N \left( \frac{1}{2} \vec{\nabla}^2_{\vec{r}_i} + V_\textrm{cf} (r_i) \right)}_{\hat{H}_0} + \underbrace{\sum^N_{j>i} \left( \frac{1}{r_{ij}} - S(r_i) \right)}_{\hat{H}_1} \end{align}

So we we can can treat atoms through the shell structure known from the atom, but the screening lifts the l degeneracy. For a single outer electron, we have even seen how this screening can be described by the quantum defect.

We would now like to go beyond this simple picture and discuss the following questions:

• How should the residual term $\hat{H}_1$ be taken into account?

• How do we properly take into account the Pauli principle ?

• How can we treat the fine-splitting ?

On the residual coupling

If we ignore the residual coupling, we obtain a spherically symmetric problem, which implies that the angular momentum $\vec{l}_i$ of each electron is conserved. This conservation will be broken by $\hat{H}_{1}$. However, these forces are internal, which implies that the total angular momentum $\vec{L} = \sum_i \vec{l}_i$ is conserved. So we should label the states in the complex Hamiltonian by $\vec{L}$.

The total angular momentum will then set the symmetry of the spatial wavefunction. As already discussed in some detail for the He atom, this has wide-reaching consquence on the spin degree of freedom through exchange interaction.

The Pauli principle and spin

• According to the Pauli principle, each single-particle state can be occupied only by one electron. After distributing all electrons over different single-particle eigenstates (“orbitals”), the resulting state needs to be fully antisymmetrized (Slater determinant).

• There is a simplification for atoms with many electrons: The angular momenta and spins of a complete subshell with n,  l,  {m_−l, ⋯, m_l} add to zero and can be ignored in the further considerations (“shell structure”). Note that this is often broken in molecular binding!

• Alkali atoms are the simplest atoms with shell structure: All but one valence electron add to L = 0, S = 0. The ground state thus has L = 0, S = 1/2.

• For more complex atoms, the valence electrons couple to a total orbital angular momentum L with a given symmetry according to particle exchange.

Let us have a look at two examples for light atoms, starting with :

• 1s² → L = 0, S = 0. The corresponding term is ¹S

• 1s2s → L = 0, {S = 0, S = 1}. The corresponding terms are ¹S and ³S. .

The electronic configuration of is: \begin{align} \underbrace{1s^2 2s^2 2p^6 3s^2}_{L=0,\,S=0} 3p^2 \end{align} Per valence electron we have l = 1 and s = 1/2. So we get L = 0, 1, 2 and S = 0, 1. Here S = 1 means symmetry and S = 0 antisymmetry with respect to particle exchange. In principle we can form the following terms: \begin{align} ^1S,\,^3S,\,^1P,\,^3P,\,^1D,\,^3D \end{align} Which of these terms can be fully antisymmetrized? Here, only the terms ¹S, ³P and ¹D fulfill Pauli’s principle. In general the exchange interaction (seen in the discussion of He), will then lower the energy of the states with high spins.

Optional: Symmetry of the L states

We can construct the following L-states for them: \begin{align} \ket{L=2,M_L=2} &= \ket{\overbrace{1}^{l_1},\overbrace{1}^{m_{l_1}};\overbrace{1}^{l_2},\overbrace{1}^{m_{l_2}}} \label{eq:lstate1} \end{align} \begin{align} \ket{L=1,M_L=0} &= \frac{1}{\sqrt{2}} ( \ket{\overbrace{1}^{m_{l_1}},\overbrace{-1}^{m_{l_2}}} - \ket{-1,1} ) \label{eq:lstate2} \end{align} \begin{align} \ket{L=0,M_L=0} &= \frac{1}{\sqrt{3}}(\ket{\overbrace{1}^{m_{l_1}},\overbrace{-1}^{m_{l_2}}} - \ket{0,0} + \ket{-1,1} ) \label{eq:lstate3} \end{align}

The states and are symmetric and the state is antisymmetric with respect to particle exchange.

We are looking into the quantization of the electromagnetic field. How to find the conjugate momenta and how we can identify the photon as a quantized particle.

After our discussion of extremely simple atoms like hydrogen and helium, we will now discuss the most important properties of more complex atoms. We will see, how we can categorize them and discuss some of the general properties

Dynamical gauge fields are a fundamental concept of high-energy physics. However, learning about them typically takes enormous amounts of time and effort. As such, they are typically a bit mystical to students (including me) of other fields of physics like condensed-matter or AMO. Here, we will give a simple introduction into some of the concepts that might allow for the quantum simulation of these theories with ultracold atomic gases.The reader should know about second quantization and the basics of quantum mechanics as the arguments are based on this formalism.