Merge pull request #14 from fretchen/AMO

Amo

amo/lecture7.md

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+---
+author:
+- Fred Jendrzejewski
+- Selim Jochim
+bibliography:
+- bibliography/converted_to_latex.bib
+date: January 04, 2025
+title: Lecture 7 - Beyond the 'boring' hydrogen atom
+---
+
+In this lecture we will use the hydrogen atom to study static
+perturbations in form of external magnetic fields and relativistic
+effects, leading to the fine structure splitting.
+
+We spend quite some time on the properties of the hydrogen atom in the
+previous lectures [@Jendrzejewski; @atom]. However, we completely
+neglected any effects of quantum-electrodynamics and relativistic
+physics. In this lecture we will study, why this is a good approximation
+for the hydrogen atom and then investigate in a perturbative fashion the
+terms. Most importantly, we will introduce that coupling between the
+orbital angular momentum and the spin of the electron, which leads to
+the fine splitting.
+
+# Perturbation theory
+
+Up to now have studied the hydrogen atom to find its eigensystem and
+then studied how it evolves under the presence of oscillating electric
+fields. This allowed us to understand in more detail the idea of
+eigenstates and then of time-dependent perturbation theory. However, one
+of the most important concepts that can be introduced very nicely on the
+hydrogen atom is stationnary perturbation theory in form of external
+magnetic fields or relativistic corrections. We will remind you of
+perturbation theory here and then apply it to some simple cases.
+
+We can now simply write down the problem as:
+
+$$
+\left(\hat{H}_0+\lambda \hat{W}\right)\left|\psi_m\right\rangle = E_m\left|\psi_m\right\rangle
+$$
+
+$\lambda$ is a very small parameter and $\hat{H}_0$ is
+describing the hydrogen atom system. We will note the eigenvalues and
+eigenstates of this system as:
+
+$$
+
+\hat{H}_0\left|\varphi_n\right\rangle = \epsilon_n \left|\varphi_n\right\rangle
+$$
+
+While, we do not know the exact solution of
+$\left|\psi_m\right\rangle$ and the energy $E_m$, we decide
+to decompose them in the following expansion of the small parameter
+$\lambda$:
+
+$$
+\left|\psi_m\right\rangle = \left|\psi_m^{(0)}\right\rangle + \lambda\left|\psi_m^{(1)}\right\rangle+\lambda^2\left|\psi_m^{(2)}\right\rangle+O(\lambda^3)\\
+E_m = E_m^{(0)} +\lambda E_m^{(1)} + \lambda^2 E_m^{(2)}+O(\lambda^3)\,
+$$
+
+To zeroth order in $\lambda$ we obtain:
+$$
+\hat{H}_0\left|\psi_m^{(0)}\right\rangle = E_m^{(0)}\left|\psi_m^{(0)}\right\rangle
+$$
+
+So it is just the unperturbed system and we can
+identify:
+
+$$
+\left|\psi_m^{(0)}\right\rangle = \left|\varphi_m\right\rangle~~E_m^{(0)} = \epsilon_m
+$$
+
+For the first order we have to solve
+
+$$
+
+(\hat{H}_0-E_m^{(0)}) \left|\psi_m^{(1)}\right\rangle + (\hat{W}-E_m^{(1)})\left|\psi_m^{(0)}\right\rangle= 0\\
+(\hat{H}_0-\epsilon_m) \left|\psi_m^{(1)}\right\rangle + (\hat{W}-E_m^{(1)})\left|\varphi_m\right\rangle= 0
+$$
+
+We can multiply the whole equation by
+$\left\langle\varphi_m\right|$ from the right. As
+$\left\langle\varphi_m\right|\hat{H}_0= \epsilon_m\left\langle\varphi_m\right|$,
+the first term cancels out. Hence, we obtain:
+
+$$
+
+\boxed{E_m^{(1)} = \left\langle\varphi_m\right|\hat{W}\left|\varphi_m\right\rangle}
+$$
+
+We now also need to obtain the correction to the
+eigenstate. For that, we put the solution for the energy into the Ansatz to obain:
+
+$$
+(\hat{H}_0-\epsilon_m) \left|\psi_m^{(1)}\right\rangle + (\hat{W}\left|\varphi_m\right\rangle-\left|\varphi_m\right\rangle\left\langle\varphi_m\right|\hat{W}\left|\varphi_m\right\rangle)= 0
+$$
+
+We can now multiply the whole equation by
+$\left\langle\varphi_i\right|$ from the right and obtain:
+$$
+(\epsilon_i-\epsilon_m)\left\langle\varphi_i\right|\left|\psi_m^{(1)}\right\rangle+\left\langle\varphi_i\right|\hat{W}\left|\varphi_m\right\rangle = 0
+$$
+
+By rewriting the above equation, this directly gives us
+the decompositon of the $\left|\psi_m^{(1)}\right\rangle$
+onto the original eigenstates and have:
+
+$$
+
+\boxed{\left|\psi_m^{(1)}\right\rangle = \sum_{i\neq m} \frac{\left\langle\varphi_i\right|\hat{W}\left|\varphi_m\right\rangle}{(\epsilon_m-\epsilon_i)}\left|\varphi_i\right\rangle}
+$$
+
+And we end the calculation with second order pertubation
+in $\lambda$
+
+$$
+(\hat{H}_0-E_m^{(0)}) \left|\psi_m^{(2)}\right\rangle + (\hat{W}-E_m^{(1)})\left|\psi_m^{(1)}\right\rangle-E_m^{(2)} \left|\psi_m^{(0)}\right\rangle= 0\\
+(\hat{H}_0-\epsilon_m) \left|\psi_m^{(2)}\right\rangle + (\hat{W}-E_m^{(1)})\left|\psi_m^{(1)}\right\rangle-E_m^{(2)} \left|\varphi_m\right\rangle= 0\\
+$$
+
+We can multiply once again whole equation by
+$\left\langle\varphi_m\right|$ from the right, which
+directly drops the first term. The term
+$E_m^{(1)}\left\langle\varphi_m\right|\left|\psi_m^{(1)}\right\rangle$
+drops out as the first order perturbation does not contain a projection
+onto the initial state. So we can write:
+
+$$
+E_m^{(2)}= \left\langle\varphi_m\right|\hat{W}\left|\psi_m^{(1)}\right\rangle
+$$
+
+Plugging in our solution, we obtain:
+
+$$
+\boxed{E_m^{(2)} = \sum_{i\neq m} \frac{|\left\langle\varphi_i\right|\hat{W}\left|\varphi_m\right\rangle|^2}{(\epsilon_m-\epsilon_i)}}
+$$
+
+# Static external magnetic fields
+
+A first beautiful application of perturbation theory is the study of
+static magnetic fields (see Ch 1.9 and Ch. 2.7.1 of [@Hertel_2015] for
+more details). The motion of the electron around the nucleus implies a
+magnetic current
+
+$$
+I = \frac{e}{t} = \frac{ev}{2\pi r}
+$$
+
+and this implies a magnetic moment $M = I A$, with the
+enclosed surface $A=\pi r^2$. It may be rewritten as:
+
+$$
+\vec{M}_L = -\frac{e}{2m_e}\vec{L} =-\frac{\mu_B}{\hbar} \vec{L} \\
+\mu_B = \frac{\hbar e}{2m_e}
+$$
+
+where $\mu_B$ is the **Bohr magneton**. Its potential
+energy in a magnetic field $\vec{B} = B_0 \vec{e}_z$ is then:
+$$
+V_B = -\vec{M}_L\cdot \vec{B}\\
+= \frac{\mu_B}{\hbar} L_z B_0
+$$
+
+Its contribution is directly evaluated from Eq.
+[\[Eq:PerturbFirstOrder\]](#Eq:PerturbFirstOrder){reference-type="eqref"
+reference="Eq:PerturbFirstOrder"} to be:
+
+$$
+E_{Zeeman} = \mu_B m B_0
+$$
+
+This is the Zeeman splitting of the different magnetic
+substates. It is visualized in Fig. [1](#982283){reference-type="ref"
+reference="982283"}.
+
+![The Zeeman effect in the hydrogen atom.
+](figures/Bildschirmfoto-2018-10-31-um-08-07-29/Bildschirmfoto-2018-10-31-um-08-07-29){#982283
+width="0.70\\columnwidth"}
+
+# Trapping with electric or magnetic fields
+
+We have now investigated the structure of the hydrogen atom and seen how
+its energy gets shifted in external magnetic fields. We can combine this
+understanding to study conservative traps for atoms and ions. Neutral
+atoms experience the external field:
+
+$$
+E_{mag}(x,y) = \mu_B m B_0(x,y)
+$$
+
+For ions on the other hand we have fully charged
+particles. So they simply experience the external electric field
+directly:
+
+$$
+E_{el}(x,y) = -q E(x,y)
+$$
+
+Trapping atoms and ions has to be done under very good vacuum such that
+they are well isolate from the enviromnent and high precision
+experiments can be performed.
+
+However, the trap construction is not trivial given Maxwells equation
+$\text{div} \vec{E} = 0$ and $\text{div} \vec{B} = 0$. So, the
+experimentalists have to play some tricks with oscillating fields. We
+will not derive in detail how a resulting **Paul trap** works, but the
+[linked video](https://youtu.be/Xb-zpM0UOzk) gives a very nice
+impression of the idea behind it. A sketch is presented in Fig.
+[2](#149591){reference-type="ref" reference="149591"}.
+
+![The upper stage shows the phases of The two phases of the oscillating
+electric field of a Paul trap. Taken
+from [wikipedia](https://en.wikipedia.org/wiki/Quadrupole_ion_trap).
+Below we can see a linear ion (Paul) trap containing six calcium 40
+ions. Taken
+from [here](https://quantumoptics.at/en/research/lintrap.html) .
+](figures/IonTraps-01/IonTraps-01){#149591 width="0.70\\columnwidth"}
+
+This work on trapping ions dates back to the middle of the last century
+(!!!) and was recognized by the[ Nobel prize in
+1989](https://www.nobelprize.org/prizes/physics/1989/summary/) for
+Wolfgang Paul [@Paul_1990] and Hans Dehmelt [@Dehmelt_1990]. They shared
+the prize with Norman Ramsey, who developped extremely precise
+spectroscopic methods, now known as Ramsey spectroscopy [@Ramsey_1990].
+
+For atoms we can play similiar games with magnetic traps. Again we have
+to solve the problem of the zero magnetic fields. Widely used
+configurations are the Ioffe-Pritchard trap, where quadrupole fields are
+superposed with a bias field [@Pritchard_1983], or TOP-traps
+[@Petrich_1995].
+
+Ion traps are now the basis of ionic quantum computers [@ions] and
+magnetic traps paved the way for quantum simulators with cold atoms
+[@Jendrzejewskia].
+
+# What we missed from the Dirac equation
+
+Until now we have completely neglected relativistic effects, i.e. we
+should have really solved the Dirac equation instead of the Schrödinger
+equation. However, this is is major task, which we will not undertake
+here. But what were the main approximations ?
+
+1.  We neglected the existance of the electron spin.
+
+2.  We did not take into account the relativistic effects.
+
+So, how does relativity affect the hydrogen spectrum? In a first step,
+we should actually introduce the magnetic moment of the spin:
+$$
+\vec{M}_S = -g_e \mu_B \frac{\vec{S}}{\hbar}
+$$
+
+The spin of the electron is $1/2$, making it a fermion
+and the *g factor of the electron* reads
+
+$$
+g_e \approx 2.0023
+$$
+
+Further discussions of the g-factor might be found in
+Chapter 6.6 of [@Hertel_2015].
+
+## Amplitude of the relativistic effects
+
+We saw in lecture 5 [@Jendrzejewski] and 6 [@Jendrzejewskib], that the
+energy levels of hydrogenlike atoms are given by:
+
+$$
+
+E_n = \frac{Z^2 R_{y,\infty}}{n^2}\\
+R_{y,\infty} = \frac{m_e e^4}{32 \pi^2 \epsilon_0^2 \hbar^2}
+$$
+
+We can now use the fine-structure constant, which
+measures the coupling strength of the electric charges to the
+electromagnetic field:
+
+$$
+
+\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c}\\
+= \frac{1}{137.035999139(31)}
+$$
+
+We can now rewrite Eq.
+[\[Eq:EnergyHydrogen\]](#Eq:EnergyHydrogen){reference-type="eqref"
+reference="Eq:EnergyHydrogen"} as:
+
+$$
+E_n = \frac{1}{2} \underbrace{m_e c^2}_{\text{rest mass energy}} Z^2 \alpha^2 \frac{1}{n^2}
+$$
+
+Here, $m_e c^2\approx \SI{511}{\kilo eV}$ is the rest
+mass energy of the electron. $E_n \approx \SI{10}{eV}$ on the other hand
+is the energy of the bound state and therefore in the order of the
+kinetic energy of the electron. As long as it is much smaller than the
+rest-mass of the electron, we can neglect the relativistic effects. A
+few observations:
+
+-   Relativistic effects are most pronounced for deeply bound states of
+    small quantum number $n$.
+
+-   Relativistic effects effects will become important once
+    $(Z\alpha)\approx 1$, so they will play a major role in heavy
+    nuclei.
+
+For the hydrogen atom we can thus treat the relativistic effects in a
+perturbative approach.But the most important consequence of the
+relativistic terms is actually the existance of the electron spin.
+
+## The relativistic mass and Darwin term
+
+1.  "Relativistic mass": The relativistic relation between energy and
+    momentum reads:
+
+$$
+    E_\text{rel} = \sqrt{(mc^2)^2+(\vec{p}c)^2}\\
+    \approx mc^2 + \frac{p^2}{2m}- \frac{\vec{p}^{\,4}}{8m^3c^2} + \cdots
+$$
+
+
+
+    The first two terms of the expansion are the
+    nonrelativistic limit and the third term is the first correction.
+    Therefore, the corresponding Hamiltonian is:
+
+$$
+    \hat{H}_\text{rm} = - \frac{\hat{\vec{p}}^{\,4}}{8m^3c^2}.
+$$
+
+2.  Darwin term: If $r=0$, the potential $V(r)$ diverges to $-\infty$.
+    We get:
+
+$$
+    \hat{H}_\text{Darwin} = \frac{\pi \hbar^2}{2m^2c^2}\left( \frac{Ze^2}{4\pi\epsilon_0}\right) \delta(\hat{\vec{r}})
+$$
+
+If we perform a first correction to the energy of the eigenstates
+$\left\langle n,l,m\right\rangle$ by calculating
+
+$$
+\left\langle n,l,m|\hat{H'|n,l,m}\right\rangle,
+$$
+
+we find that it works perfectly for case (1) and (2)
+which is due to degeneracy. $\hat{H}_\text{rm}$ and
+$\hat{H}_\text{Darwin}$ commute with all observables forming the
+complete set of commuting observables (CSCO) for $\hat{H}_0$
+
+$$
+\hat{H}_0,\hat{\vec{L}}^2, \hat{L}_z,
+$$
+
+with states described by
+$\left|n,l,m\right\rangle$.

pages/amo/@id/+onBeforePrerenderStart.js

@@ -0,0 +1,13 @@
+export { onBeforePrerenderStart };
+
+import blogs from "../../../amo/blogs.json";
+
+async function onBeforePrerenderStart() {
+  const blogURLs = blogs.map((blog, index) => {
+    const blogURL = `/amo/${index}`;
+    return blogURL;
+  });
+  console.log("blogURLs");
+  console.log(blogURLs);
+  return blogURLs;
+}

pages/blog/@id/+onBeforePrerenderStart.js

@@ -0,0 +1,13 @@
+export { onBeforePrerenderStart };
+
+import blogs from "../../../blog/blogs.json";
+
+async function onBeforePrerenderStart() {
+  const blogURLs = blogs.map((blog, index) => {
+    const blogURL = `/blog/${index}`;
+    return blogURL;
+  });
+  console.log("blogURLs");
+  console.log(blogURLs);
+  return blogURLs;
+}