diff --git a/amo/lecture7.md b/amo/lecture7.md new file mode 100644 index 0000000..1ed5411 --- /dev/null +++ b/amo/lecture7.md @@ -0,0 +1,353 @@ +--- +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 +$\left|n,l,m\right\rangle$. diff --git a/pages/amo/@id/+onBeforePrerenderStart.js b/pages/amo/@id/+onBeforePrerenderStart.js new file mode 100644 index 0000000..a1f7bfd --- /dev/null +++ b/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/${index}`; + return blogURL; + }); + console.log("blogURLs"); + console.log(blogURLs); + return blogURLs; +} diff --git a/pages/blog/@id/+onBeforePrerenderStart.js b/pages/blog/@id/+onBeforePrerenderStart.js new file mode 100644 index 0000000..fe30a41 --- /dev/null +++ b/pages/blog/@id/+onBeforePrerenderStart.js @@ -0,0 +1,13 @@ +export { onBeforePrerenderStart }; + +import blogs from "../../../blog/blogs.json"; + +async function onBeforePrerenderStart() { + const blogURLs = blogs.map((blog, index) => { + const blogURL = `/blog/${index}`; + return blogURL; + }); + console.log("blogURLs"); + console.log(blogURLs); + return blogURLs; +} diff --git a/public/amo/lecture3_pic1.png b/public/amo/lecture3_pic1.png new file mode 100644 index 0000000..9d6eb98 Binary files /dev/null and b/public/amo/lecture3_pic1.png differ diff --git a/public/amo/lecture3_pic2.png b/public/amo/lecture3_pic2.png new file mode 100644 index 0000000..4e87853 Binary files /dev/null and b/public/amo/lecture3_pic2.png differ diff --git a/public/amo/lecture3_pic3.png b/public/amo/lecture3_pic3.png new file mode 100644 index 0000000..bb52b8e Binary files /dev/null and b/public/amo/lecture3_pic3.png differ diff --git a/public/amo/lecture3_pic4.png b/public/amo/lecture3_pic4.png new file mode 100644 index 0000000..2089961 Binary files /dev/null and b/public/amo/lecture3_pic4.png differ diff --git a/public/amo/lecture4_pic1.png b/public/amo/lecture4_pic1.png new file mode 100644 index 0000000..224be51 Binary files /dev/null and b/public/amo/lecture4_pic1.png differ diff --git a/public/amo/lecture4_pic2.png b/public/amo/lecture4_pic2.png new file mode 100644 index 0000000..58033e3 Binary files /dev/null and b/public/amo/lecture4_pic2.png differ diff --git a/public/amo/lecture4_pic3.png b/public/amo/lecture4_pic3.png new file mode 100644 index 0000000..e05ece0 Binary files /dev/null and b/public/amo/lecture4_pic3.png differ diff --git a/public/amo/lecture5_pic1.png b/public/amo/lecture5_pic1.png new file mode 100644 index 0000000..008f58e Binary files /dev/null and b/public/amo/lecture5_pic1.png differ diff --git a/public/amo/lecture6_pic1.png b/public/amo/lecture6_pic1.png new file mode 100644 index 0000000..1054603 Binary files /dev/null and b/public/amo/lecture6_pic1.png differ diff --git a/public/amo/lecture6_pic2.png b/public/amo/lecture6_pic2.png new file mode 100644 index 0000000..8d6d6b0 Binary files /dev/null and b/public/amo/lecture6_pic2.png differ diff --git a/public/amo/lecture6_pic3.png b/public/amo/lecture6_pic3.png new file mode 100644 index 0000000..796b742 Binary files /dev/null and b/public/amo/lecture6_pic3.png differ diff --git a/public/amo/lecture6_pic4.png b/public/amo/lecture6_pic4.png new file mode 100644 index 0000000..0991384 Binary files /dev/null and b/public/amo/lecture6_pic4.png differ diff --git a/public/amo/lecture6_pic5.png b/public/amo/lecture6_pic5.png new file mode 100644 index 0000000..dea88d7 Binary files /dev/null and b/public/amo/lecture6_pic5.png differ diff --git a/public/amo/lecture6_pic6.svg b/public/amo/lecture6_pic6.svg new file mode 100644 index 0000000..ef8a969 --- /dev/null +++ b/public/amo/lecture6_pic6.svg @@ -0,0 +1,233 @@ + +image/svg+xmlthe basic orbitals +coupling of and +coupling of and + \ No newline at end of file diff --git a/public/amo/lecture6_pic7.svg b/public/amo/lecture6_pic7.svg new file mode 100644 index 0000000..6e13e4b Binary files /dev/null and b/public/amo/lecture6_pic7.svg differ diff --git a/utils/cleanMd.ts b/utils/cleanMd.ts index 10ed016..31ae489 100644 --- a/utils/cleanMd.ts +++ b/utils/cleanMd.ts @@ -6,12 +6,9 @@ import fs from "fs"; export const removeMath = (fileDirectory: string = "./blog") => { const blogFiles = fs.readdirSync(fileDirectory); - console.log(blogFiles); // go through each file remove \begin{aligned} and \end{aligned} blogFiles.forEach((file) => { - console.log(file); - // check that the file is a markdown file with ending .md or .mdx if (!file.endsWith(".md") && !file.endsWith(".mdx")) { return;