---
# System prepended metadata

title: 'Quantum Mechanics II, Mid-Term Take-Home Examination'
tags: [Quantum Physics II]

---

# Quantum Mechanics II, Mid-Term Take-Home Examination
###### tags: `Quantum Physics II`
> [name=何信佑 B05202054] [time=Tue, May 5, 2020]

https://hackmd.io/@ulynx/Syp91s6YL


::: warning
 (50 points, 10 pages maximum) 

e have talked about the 21-centimeter line due to hyperfine splitting in hydrogen atom at the end of Chapter 7 in the textbook. It has become a very important in astrophysics. 

Please write about why it is so important in **(a) astronomy** (15 points) and **(b) cosmology** (35 points). 
The [attachment 1](https://reurl.cc/b5YEk3) is a good starting point for (a) and the [wikipedia of “hydrogen line”](https://en.wikipedia.org/wiki/Hydrogen_line) is a good starting point for (b).
:::
# 1(a) 21-cm line in Astronomy
## Introduction
The [**21-centimeter line**](https://en.wikipedia.org/wiki/Hydrogen_line), or sometimes called **hydrogen line** is the specific spectral line of atomic hydrogen involving the transition between two [**hyperfine**](https://en.wikipedia.org/wiki/Hyperfine_structure) levels in the ground state. The 21-cm line is obviously named after its wavelength $21~\text{cm}$, which corresponds a frequency of $\nu=1420~\text{MHz}$.^[https://en.wikipedia.org/wiki/Hyperfine_structure] 

Normally an emission line with such a wavelength is [forbidden](https://en.wikipedia.org/wiki/Forbidden_line), as its transition rate is extremely small, only $2.9\times 10^{−15}~\text{s}^{−1}$ and the mean lifetime of its excited state reaches 10 million years.[^21cm] That's why it is unlikely to discover 21-cm line in a laboratory on Earth. However, it is a dominant background radiation as important as cosmic microwave background (CMB).

To understand the hyperfine structure, one needs to take [**spin-spin coupling**](https://en.wikipedia.org/wiki/Angular_momentum_coupling) into consideration. In quantum mechanics, since the hyperfine Hamiltonian $\hat{H_\text{hf}}$ due to spin-spin coupling is far less than Bohr Hamiltonian $\hat{H}_\text{Bohr}$, the former is regarded as a pertubation of the latter. In this case, neither total angular momentum quantum number of electron $J=L+S$ nor nucleas spin quantum number $I$ can be a "good" quantum number for $\hat{H_\text{hf}}$, but their sum $F=I+J$ turns out to be a good one, and hence we will see a splitting of ground state into two states: **triplet** and **singlet**. The former refers to the **parallel** case where total spin number $F=1$, while the latter to the **antiparallel** case where $F=0$. In addition, the  triplet is slightly higher than the unpurtubed ground state, and the singlet is lower.

## Theory^[Griffiths, p.398-399]
The Bohr model of hydrogen atom has already given the exact solution of wave functions, with ground state energy being $-13.6~\text{eV}$. But that is not quite the whole story. We know that either electron or proton possesses "intrinsic angular momenta" called **spins**, $\mathbf{S}_e$ and $\mathbf{S}_p$, which result in magnetic dipole moments $\boldsymbol\mu_e$ and $\boldsymbol\mu_p$. Besides, one of the magnetic dipole moment $\boldsymbol\mu_1$ sets up an internal magnetic field $\mathbf{B}_{\mu_1}$ in which the other magnetic dipole moment $\boldsymbol\mu_2$ sees a potential energy $U=-\boldsymbol\mu_2\boldsymbol\cdot\mathbf{B}_{\mu_1}$.

It can be derived that the energy is $$-\boldsymbol\mu_e\cdot\mathbf{B}_{\mu_p}=\dfrac{\mu_0g_pe^2}{8\pi m_pm_e}\dfrac{3(\mathbf{S}_p\cdot\hat{\mathbf{r}})(\mathbf{S}_e\cdot\hat{\mathbf{r}})-\mathbf{S}_p\cdot\mathbf{S}_e}{r^3} + \dfrac{\mu_0g_pe^2}{3m_pm_e}\mathbf{S}_p\cdot\mathbf{S}_e\delta^3(\mathbf{r}),\tag{1}$$ where $g_p=5.59$ is the Landé g-factor for proton. With aid of perturbation theory , the energy correction to the first order is given by (Eq. 7.91 in the textbook) $$\begin{align}E^1_\text{hf}=&\dfrac{\mu_0g_pe^2}{8\pi m_pm_e}\left<\dfrac{3(\mathbf{S}_p\cdot\hat{\mathbf{r}})(\mathbf{S}_e\cdot\hat{\mathbf{r}})-\mathbf{S}_p\cdot\mathbf{S}_e}{r^3}\right>\\&+\dfrac{\mu_0g_pe^2}{3m_pm_e}\left<\mathbf{S}_p\cdot\mathbf{S}_e\right>\delta^3(\mathbf{0})。\end{align}$$

The first term vanishes if spherical symmetry exists ($\ell=0$). For hydrogen, the ground state wave function gives $|\psi_{100}(\mathbf{0})|^2=1/\left(\pi a^3\right)$. Besides, note that proton and electron are both spin-1/2 particles, so that $S_e^2=S_p^2=\tfrac{3}{4}\hbar^2$, and $S^2=2\hbar^2$ for triplet or $S^2=0$ for singlet; thus, with $\mathbf{S}_p \cdot\mathbf{S}_e=\tfrac{1}{2}\left(S^2-S^2_e-S_p^2\right)$, we have (Eq. 7.96 in the textbook) $$E^1_\text{hf}=\dfrac{4g_p\hbar^4}{3 m_pm_e^2c^2a^4}\left\{\begin{array}{ll}+\frac{1}{4},&\text{(triplet)};&\\-\frac{3}{4},&\text{(singlet)};&\end{array}\right.$$ where we adjusted the coefficient by using Bohr radius $a=4\pi\epsilon_0\hbar^2/m_e e^2=4\pi \hbar^2/m_e  \mu_0c^2e^2$.

The energy between triplet and singlet is (Eq. 7.97 in the textbook) $$\Delta E=\dfrac{4g_p\hbar^4}{3 m_pm_e^2c^2a^4}=5.88\times 10^{-6}~\text{eV}$$

## Discovery
In 1945, H. C. van de Hulst predicted that the interstellar hydrogen gas would emit radiation at the radio wavelength of $21~\text{cm}$.[^1] The idea is based on hypothesis that there is a huge amount of hydrogen pervading interstellar space. The density there is so loose that it is unlikely that atoms should collide with each other and be de-excited. But not until 1951 did van de Hulst's prediction confirmed by Ewen and Purcell at Harvard University[^E&P] and independently by Muller and Oort as well.[^1] Since that time, 21-cm emission line has come to play a vital role in the study of the Milky Way Galaxy, because it readily penetrates the clouds of interstellar dust particles that obstruct optical observations deep into the galactic centre.^[https://www.britannica.com/science/21-centimetre-radiation]

## Application in Astronomy
### ISM 
 Although it appears barely nothing exists in the space between stars, there is actually an abundant amount of matter called **interstellar medium** spreading all over the galaxy and interacting with radiation. The interstellar medium is characterized by gas, dust, or figuratively-called, cloud, mostly composed of hydrogen. The matter can be in form of atom, ion, or molecule.

Among several types of ISM, the **neutral atomic hydrogen** is also referred to as H$\small\text{I}$, which can absorb and emit 21-cm line and typically has  density of order $10^{-6}$-$10^{-8}$ particle per cubic meter and temperature of order $80~\text{K}$.^[Choudruri, p.181]

### The Structure of Milky way
> The Milky Way’s structure is fairly typical of a large spiral system. This structure can be viewed as consisting of six separate parts: (1) a nucleus, (2) a central bulge, (3) a disk (both a thin and a thick disk), (4) spiral arms, (5) a spherical component, and (6) a massive halo. Some of these components blend into each other. [^BrMW]

![](https://cdn.britannica.com/17/94917-004-E96B406D/views-Milky-Way-Galaxy.jpg)

But how do we know the Milky Way is indeed a galaxy? And how do we know it's a spiral one?

As a matter of fact, it takes a really long time until people realized that the Milky Way is not the only galaxy in the universe.^["Milky Way Galaxy: Facts About Our Galactic Home". Space.com. Archived from the original on March 21, 2017. Retrieved April 8, 2017.] After the Great Debate between Shapley and Curtis in 1920, Edwin Hubble finally persuaded of the other independent galaxies.^[https://en.wikipedia.org/wiki/Great_Debate_(astronomy)] We can simply look through a telescope to observe the structure of the Andromeda Galaxy, whereas it was onced a tough task to determine what the sturcture of our own glaxay is like. Luckily, knowing that ISM permeates throughout the Milky Way, we are able to measure the spectrum the 21-cm line in each direction in order to obtain abundance of H$\small\text{I}$ zone and to the delineate the structure of the Milky Way. 

There is still one thing to notice. Usually when we deal with the spectra of astronomical objects, the **Doppler effects** should be taken into accounts, since the entire or part of the object is either radially approaching or receding from Earth. What's more, the Doppler effects in ISM emission is rather essential, for Doppler shifts can be converted into the circular speed of each part of the galaxy. 



![](https://i.imgur.com/eYQfavv.png)^[Choudhuri, p.176]
![](https://i.imgur.com/el5xnNA.png)[^BrMW]


Thanks to 21-centimeter in H$\small\text{I}$ zone, the spiral arms of the Milky Way have been discovered one by one. For example, the [**Near 3 kpc Arm**](https://en.wikipedia.org/wiki/Near_3_kpc_Arm) (also called **Expanding 3 kpc Arm** or simply **3 kpc Arm**) is discovered in the 1950s by astronomer van Woerden and collaborators.

Observed (normal lines) and extrapolated (dotted lines) structure of the spiral arms of the Milky Way, viewed from "north" of the galaxy – stars generally move clockwise in this view. The gray lines radiating from the Sun's position (upper center) list the three-letter abbreviations of the corresponding constellations
![](https://upload.wikimedia.org/wikipedia/commons/1/1d/Milky_Way_Arms.svg =400x)

### External Galaxy ISM and Dark Matter
One of the significant topic in galactic astronomy is to describe the physical characteristics and kinematics of galaxies. Again one can detect the Doppler shift of 21-cm line of an external galaxy to confirm that the circular speed $v_c$ of its ISM does vary with distance from the center of the galaxy. And one can plot the [**rotation curve**](https://en.wikipedia.org/wiki/Galaxy_rotation_curve) that expresses the circular speed $v_c$ as a function of the radius $r$ of a galaxy. Using the Newtonian mechanical relation that the centripetal force equals the gravitational force, we have $v^2/r=GM(r)/r^2$, where is the mass within the radius $r$. In the case of a spiral galaxy, a large portion of mass distribution is expected to be spherically symmetric within some central region such that $M(r)\propto r^3$, so the is $v_c\propto r$; whereas the circular speed of the rest portion is given by $v_c=\sqrt{GM_\text{total}/r}$, which suggests a falling as $r^{-1/2}$.

![](https://upload.wikimedia.org/wikipedia/commons/c/cd/Rotation_curve_of_spiral_galaxy_Messier_33_%28Triangulum%29.png)
Contrary to such prediction, the observational data shows that, for each galaxy, the rotation curve rises up as increasing radial distance $r$ initially and thereafter flatten asymptotically instead of dropping. If the equation of motion is valid, then the plausible hypothesis is that mass distribution extends beyond the visible stellar disk of the galaxy and even beyond the 21-cm regions. We can be calculate the appropriate mass distribution required to fit the actual rotation curve. It turns out that total mass of a typical spiral galaxy is at least a few times the total mass of luminous part of galaxy. This elicits a postulation regarding the existence of a massive, invisible mysterious form of matter, called **dark matter**. 

If dark matter were exists, then it would account for approximately $85\%$. Because of of the matter in the universeecause of its Determining the nature of dark matter is one of the major challenges of modern astronomy.

# 1(b) 21-cm line in Cosmology
## Reionization
In cosmology, the history of Universe is divided into **epochs**, during which some physical phenomena are pervasive or some physical mechanisms are dominant. Immediately after after the Big Bang, the universe was a hot, dense plasma of photons, other leptons, and quarks, so it is called the **Quark epoch**. 

Then there was a peroid when baryonic matter in the universe became sort of hot  plasma under extremely high temperature. The situation is described as *opaque* because electromagnetic radiation suffered from Thomson scattering by free electrons, as the mean free path of each photon was so short that it would soon encounter a next electron.^[https://en.wikipedia.org/wiki/Recombination_(cosmology)] 

With the expansion keeping on, the Universe underwent a sudden drop in free electron density and is said to have turned *transparent*.^[https://en.wikipedia.org/wiki/Decoupling_(cosmology)] -- the charged electrons and protons first became bound to form electrically neutral hydrogen atoms.  The process is called [**recombination**](https://en.wikipedia.org/wiki/Recombination_(cosmology)) and took place at around 370,000 years after Big Bang (at a redshift of $z = 1100$^[Choudhuri, p.341]); the era is thus called the **Epoch of Recombination**. Meanwhile, **photon decoupling** occured; because it is very inefficient for hydrogen to direct recombinations to the ground state (lowest energy), those hydrogen atoms tended to form in higher energy states, followed by quick electron  transition to low energy states with photons emited.^[https://en.wikipedia.org/wiki/Decoupling_(cosmology)] At that time, the ionised plasma gave way to neutral atoms, which did not scatter the photons but allowed them to travel freely to a long distance; despite this, there were no light source such as stars and galaxies, and hence there came a lapse of **Dark Ages**.^[https://en.wikipedia.org/wiki/Chronology_of_the_universe#Dark_Ages] ^[http://astronomy.swin.edu.au/cosmos/E/Epoch+Of+Recombination]


![](https://upload.wikimedia.org/wikipedia/commons/2/29/Reion_diagram.jpg)

The Epoch of Recombination is followed by the **[Epoch of Reionization](https://en.wikipedia.org/wiki/Reionization)** (EoR), which featured a situation in which the gas in the Universe went from being almost completely neutral to a state in which it became almost completely ionized.^[http://www.lofar.org/astronomy/eor-ksp/epoch-reionization.html] 

There are
1. **early reionization** ($z\approx 13$): consistent with the discovery by the [Wilkinson Microwave Anisotropy Probe](https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe) (WMAP) satellite of a large optical depth to Thomson scattering.

## Application in Cosmology
### Intergalactic medium (IGM)
In much the same way as ISM is named, the  intergalactic medium (IGM) refers to the matter in regions of space between clusters or outside of galaxies. Since we never find the spectral lines directed emitted from IGM, the only method to detect IGM to look for absorption lines in the spectra from a faraway object. A type of **[active galactic nucleus](https://en.wikipedia.org/wiki/Active_galactic_nucleus)** (AGN), known as **quasar**, serves as a very suitable candidate becauce it is extremely far away but outstandingly bright.^[Choudhuri, p.348]

Suppose that a quasar located at redshift $z_\text{em}$ is emitting a Lyman-α line resulted from the electron transition from $1s$ to $2p$ in a hydrogen atom, with an ultra-violet wavelength $\lambda_{L_\alpha}=1216Å$, so that the broad emission line of a typical quasar is expected to be $(1+z_\text{em})\lambda_{L_\alpha}$. If the wavelength $(1+z_\text{abs})\lambda_{L_\alpha}$ is absorbed by some material, say H$\small\text{II}$ region, lying at $z_\text{abs}$, then we would expect to see an absorption line there.

![](https://i.imgur.com/9wEzX9T.png)
The spectrum of a quasar at redshift $z_\text{em}=2.60$, for which the Lyman-α emission line is at $4380Å$ is showned in the above figure^[Choudhuri, p.348], where many narrowly-spaced absorption lines (called [**Lyman-α forest**](https://en.wikipedia.org/wiki/Lyman-alpha_forest)) are found from $1216Å$ to $4380Å$ and a prominent through of absorption is centered at $3650Å$ (corresponding to redshift $z_\text{abs}=2.0$). The data suggests there is an large number of small clouds of neutral hydrogen lying on the line of sight at different distances and also there is an enormous cloud. More and more observational results demonstrate the existence of intergalactic H$\small\text{II}$ zones inside isolated clouds; and hydrogen outside these clouds is ionized somehow, so it does not produce any Lyman-α absorption line.^[Choudhuri, p.349] 

Over a redshift of about $z=6$, the Lyman-α forest turns into a [**Gunn–Peterson trough**](https://en.wikipedia.org/wiki/Gunn%E2%80%93Peterson_trough) which indicates that there is so much neutral hydrogen in the intergalactic medium.^[https://en.wikipedia.org/wiki/Lyman-alpha_forest] 

By mapping the intensity of redshifted 21 centimeter radiation it can, in principle, provide a very precise picture of the matter power spectrum in the period after recombination. 
### 
it can provide a picture of how the universe was reionized, as neutral hydrogen which has been ionized by radiation from stars or quasars will appear as holes in the 21 cm background.

---
::: warning
 (50 points, 10 pages maximum) 
In Section 7.4.3 of the textbook, we talked about when the Zeeman effect and fine structure correction were equally important, then how we could put them together in the perturbed Hamiltonian and diagonal it. Now suppose the Zeeman effect is even smaller, so it is about
the same size as the hyperfine structure correction (see eq. 7.93 for the case
of hydrogen atom), then please follow the similar procedure in Section 7.4.3 to find the energy spectrum (i.e. the Eigen-energies). 
- **(a)** Please do it for the $5S$ level of $^{87}\text{Rb}$ which has a nuclear spin $= 3/2$. Plot your result as a frequency shift to the Bohr spectrum as a function of the external $B$ field as shown in [the
wikipedia page of “Zeeman effect”](https://en.wikipedia.org/wiki/Zeeman_effect). (20 points) 
- **(b)** Discuss the applications of the Zeeman effect in astrophysics (10 points) and laser cooling (20 points).
:::

# 2(a) Zeeman effect of Rubidium-87  
The [**Zeeman effect**](https://en.wikipedia.org/wiki/Zeeman_effect) refers to the phenomenon: when an atom is placed in a uniform external magnatic field $\mathbf{B}_\text{ext}$, its energy levels are shifted and thus some of its spectral lines split into several components. Ducth physicist Pieter Zeeman was the first to report these splitting properties in his seminal paper of 1897.[^smf2]
![](https://upload.wikimedia.org/wikipedia/commons/3/32/ZeemanEffect.GIF =200x)
Photograph of the splitting of the $\text{Na}$ $D$ doublet under the influence of an external magnetic field, as reported by Zeeman (1897) [^smf2]
## Theory
The Zeeman Hamiltonian $$\hat{H'_\text{Z}}=-(\boldsymbol\mu_\ell+\boldsymbol\mu_s)\cdot\mathbf{B}_\text{ext},$$ which caused by the magnetic moments due to an external magnatic field $\mathbf{B}_\text{ext}$, can be treated by time-independent perturbation theory; there are different cases depending on the relative magnitude of $\mathbf{B}_\text{ext}$ with respect to the internal field of atom. 

## Rubidium-87
### 1. Definition 
#### Angular Momentum Operator
- An angular momentum operator $\hat{\mathbf{K}}$ satisfies the following properties:
    - $\hat{K^2}\left|\psi\right>=K(K+1)\hbar^2\left|\psi\right>,\quad K=0,1,2,\ldots$ for some vector $\left|\psi\right>$.
    - $\hat{K}_z\left|\psi\right>=m_K\hbar\left|\psi\right>,\quad m_K=-K,-K+1,\ldots,K-1,K$ for some vector $\left|\psi\right>$.
    - $[\hat{K}_p,\hat{K}_q]=i\epsilon_{pqr}\hat{K}_r,\quad p,q,r\in\{x,y,z\}$
- Define the following angular momentum operators:
    - $\mathbf{L}$: electron orbital angular momentum,
    - $\mathbf{S}$: electron spin angular momentum,
    - $\mathbf{I}$: nucleus spin angular momentum,
    - $\mathbf{J}=\mathbf{L}+\mathbf{S}$: total electron angular momentum,
    - $\mathbf{F}=\mathbf{J}+\mathbf{I}$: total atomic angular momentum.

For $^{87}\text{Rb}$, we have:
- atomic number $Z=37$,
- total nucleons $Z+N=87$,
- total electron angular momentum quantum number $J=1/3$,
- eigenvalues of angular momentum at ground state,
    - $L=0$, $m_L=0$
    - $S=1/2$, $m_S=-1/2,\ 1/2$
    - $J=1/2$, $m_J=-1/2,\  1/2$
    - $I=3/2$, $m_I=-3/2,\ -1/2,\ 1/2,\ 3/2$
    - $F=1$ or $F=2$,
        - $m_F=-1,0,1$ or
        - $m_F=-2,-1,0,1,2$  
- effective nuclear charge for $5s$ electron $Z'=4.985$^[Clementi, E.; Raimondi, D. L.; Reinhardt, W. P. (1967). "Atomic Screening Constants from SCF Functions. II. Atoms with 37 to 86 Electrons". Journal of Chemical Physics. 47: 1300–1307. Bibcode:1967JChPh..47.1300C. doi:10.1063/1.1712084.]

#### Spectroscopic Notation 
$(N)^{2s+1}L_j$ See [explanation](https://quantummechanics.ucsd.edu/ph130a/130_notes/node315.html)
::: spoiler
D-line
1. $\text{D}_2$ line: $5^2\text{S}_{1/2}\to5^2\text{S}_{3/2}$ transition
    - much more relevance to current quantum and atom optics experiments
    -  it has a cycling transition that is used for cooling and trapping $^{87}\text{Rb}$
3. $\text{D}_1$ line: $5^2\text{S}_{1/2}\to5^2\text{S}_{1/2}$ transition
:::
### 2. Hamiltonian
#### Hyperfine structure
From [multipole expansion](https://en.wikipedia.org/wiki/Multipole_expansion)
$\hat{H'_\text{hf}}=A_\text{hp}(\mathbf{I}\boldsymbol\cdot\mathbf{J})$
$E^1_\text{hf}=\langle\hat{H'_\text{hf}}\rangle=\dfrac{1}{2}A_\text{hp}\underbrace{[F(F+1)-I(I+1)-J(J+1)]}_{K}$
with $A_\text{hp}=h \cdot 407.25(63)~\text{MHz}$ is the magnetic dipole constant for $5^2\text{S}_{1/2}$
#### External Magnetic Field
$\mathbf{B}=B_k\hat{\mathbf{k}}$ is along the z-direction
$\begin{align}\hat{H'_\text{Z}}&=\dfrac{\mu_B}{\hbar}\left(g_S\mathbf{S}+g_L\mathbf{L}+g_I\mathbf{I}
\right)\boldsymbol\cdot \mathbf{B}\\&=\dfrac{\mu_B}{\hbar}\left(g_SS_z+g_LL_z+g_I I_z
\right) B_z,\end{align}$ 
- $g_S=2.002 319 304 3622(15)$: electron spin $g$-factor
- $g_L=1-(m_e/m_\text{nuc})=0.99999369$ : electron orbital $g$-factor
    - $m_\text{nuc}$: nuclear mass
- $g_I = −0.000 995 141 4(10)$: nuclear $g$-factor 

### 3. Perturbation

#### Weak field ($H_\text{Z}\ll H_\text{fs}$)
- good quantum numbers: $n,\ \ell,\ J,\ m_J$.
- unpurtubed: $\hat{H}_\text{Bohr}+\hat{H}_\text{fs}$
- Perturbation: $\hat{H'}=\dfrac{\mu_B}{\hbar}\left(g_JJ_z+g_I I_z\right) B_z$, with $$g_J\approx1+\dfrac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}$$
#### Very Weak field ($H_\text{Z}\ll H_\text{hf}$)
- good quantum numbers: $n,\ \ell,\ F,\ m_F$.
- unpurtubed: $\hat{H}_\text{Bohr}+\hat{H}_\text{fs}+\hat{H}_\text{hf}$
- Perturbation: $\hat{H'}=\hat{H'_\text{Z}}$, with $$g_F\approx g_J+\dfrac{F(F+1)+I(I+1)-J(J+1)}{2F(F+1)}$$
- energy correction: $E^1=\mu_Bg_F m_F B_z$.
#### Strong-field ($H_\text{Z}\gg H_\text{hf}$)
- good quantum numbers: $J,\ m_J,\ I,\ m_I$.
- unpurtubed: $\hat{H}_\text{Bohr}+\dfrac{\mu_B}{\hbar}\left(g_JJ_z+g_I I_z \right) B_z$
- Perturbation: $\hat{H'}=\hat{H'_\text{hf}}$.
#### Intermediate field
For intermediate fields, the energy shift is more difficult to calculate, and in general one must numerically diagonalize $\hat{H'_\text{hf}}+\hat{H'_\text{Z}}$.
A notable exception is the **Breit-Rabi formula** , which applies to the groundstate manifold of the D transition: $$E_{\left|J=1/2~m_J~I~m_I\right>}=-\dfrac{\Delta E_\text{hf}}{2(2I+1)}+g_I\mu_BmB\pm\dfrac{\Delta E_\text{hf}}{2}\left(1+\dfrac{4mx}{2I+1}+x^2\right)^{1/2}$$

![](https://i.imgur.com/rJjQkUP.png)
![](https://i.imgur.com/6OgPih5.png)

### Result
The measured 87Rb ground-state
hyperfine splitting is $\nu_{^{87}\text{Rb}} = 6834 682610.90429(9)~\text{Hz}$.^[ S. Bize, Y. Sortais, M. S. Santos, C. Mandache, A. Clairon, and C. Salomon, “High-accuracy measurem. https://iopscience.iop.org/article/10.1209/epl/i1999-00203-9/pdf]
### Relevant work

- **Gist <i class="fa fa-github"></i>**: Daniel C. Elton - [Breit_Rabi_Zeeman_Plotter](https://github.com/delton137/Breit_Rabi_Zeeman_Plotter/blob/master/Breit_Rabi_Zeeman_plotter.m)
    ![](https://i.imgur.com/hd9j9Ay.png =300x)

- https://demonstrations.wolfram.com/BreitRabiDiagram/
# 2(b) Zeeman Effect in Astronomy

Researchers in the past might probably wonder whether there is magnetic field around the Sun or even a distant star or not. Clues are given by the Zeeman effect. 

Normally the atoms in a star's atmosphere will absorb certain frequencies of energy in the electromagnetic spectrum, producing characteristic dark absorption lines in the spectrum.[^smf] 

When an atom is immersed in an external magnetic field $B$, individual atomic levels (with energy $E_0$, total angular momentum $J$, and Landé $g$-factor) are split into $(2J + 1)$ substates, characterised by their magnetic quantum number $M ~~(−J\leq M\leq+J)$. The energies of these atomic states are given by: $$E(M) = E_0gm\omega L,$$ where $\omega_L = eB/(2m_ec)$ is the **Larmor frequency**. (Alternatively, one can refer to the weak-field Zeeman energy correction $E^1_\text{Z}=\mu_Bg_JB_\text{ext}m_j$, [Eq. 7.79] in the Griffiths' textbook.)

A group of closely spaced spectral lines is called a **Zeeman component** if when they merge as a single line when $B$ in absence and split into multiple when $B$ in presence. As far as LS coupling is concerned, we may distinct two different types of lines according to their properties.[^smf2]
1. $\boldsymbol\pi$ **components**: Those lines resulting from transitions in which M does not change (i.e. $\Delta M = 0$) are spread symmetrically about the zero-field wavelength $\lambda_0 = hc/E_0$ of the line. 
2. $\boldsymbol\sigma$ **components**: Those lines resulting from transitions in which $\Delta M = \pm 1$ have wavelengths to the red (+) and blue (−) of the zero-field wavelength.

Wavelength separation between the centroids of the $\pi$ and $\sigma$ component groups can be
calculated: $$\Delta \lambda=\dfrac{eN\bar{g}}{4\pi m_e c^2}\equiv\Delta \lambda_Z Bg,$$ where $\bar g$ is the effective Land´e factor of the transition and $\Delta\lambda_Z$ is called the [**Lorentz unit**](https://encyclopedia2.thefreedictionary.com/Lorentz+unit).

### Solar Magnetic Field
Taking the Sun for example, when we observe a solar spectrum, we find that a single spectral line outside a sunspot splits into three components inside the sunspot. Thus, by the means of the Zeeman effect, astronomers can measure the strength of the solar magnetic field and find it to be as strong as $3000~\text{gauss}$. In comparison, geomagnetic field is less than $1~\text{gauss}$.^[http://prc.nao.ac.jp/extra/uos/en/no06/]

### Stellar Magnetic Field

### Surface Activity
**Starspots** are regions of intense magnetic activity on the surface of a star. (On the Sun they are termed **sunspots**.) These form a visible component of magnetic flux tubes that are formed within a star's convection zone. 

# 2(c\) Zeeman Effect

## Doppler cooling
### Principle
For an ensemble of particles, their thermodynamic temperature is proportional to the variance in their velocity.[^23] In this sense, to lower the temperature of an ensemble is, we need to slow down the speed of paritcles. So if we can manipulate the momentum of objects, say, a beam of atoms, we can cool them from room temperature or above to to near absolute zero, i.e. a few kelvins. To do so, we may consider **laser cooling techniques**. The physics lying behind is simple: we can alter the momentum of an atom with a photon under Doppler effect. 

According to the principles of Doppler cooling, an atom modelled as a **two-level** atom can be cooled using a laser by the following processes. If it moves in a specific direction and encounters a counter-propagating laser beam resonant with its transition, it is very likely to absorb a photon.[^23] 
### Process [^23]
1. **Absorption**: according to momentum conservation, the absorption of photon gives the atom a "kick" along the incident direction.
2. **Exication**: the absorption of photon also brings the atom to its excited state; this state is, however, unstable so the atom will decay back to its ground state very soon (after a time on the order of nanoseconds).
3. **Reemission**: the photon will be reemitted in a random direction, and the atom will again increase its speed. 
4. **Recurring**: when steps 1 to 3 is repeatedly applied to an atom, the averagely effects are:
    - ==the absorption process decreases the speed== of the atom always in the same direction (as the absorbed photon comes from the same direction), whereas 
    - ==the emission process does not lead to any change in the speed==  because the emission direction is random. 
5. **Consequence**: the atom is being effectively slowed down by the laser beam.
### Applications
1.  [**optical molasses**](https://en.wikipedia.org/wiki/Optical_molasses) technique: This process itself forms a part of the magneto-optical trap but it can be used independently;
2. spectroscopy and
3. metrology, where cooling allows narrower spectroscopic features. For example, all of the best atomic clock technologies involve Doppler cooling at some point.

### Nobel Prize

By the way, laser cooling technique is thought of as a significant complishment in quantum optics. the Nobel Prize in Physics 1997 was awarded jointly to Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips, who developed laser methods to cool gases to the $\text{µK}$ temperature range and to keep the chilled atoms floating or captured in different kinds of "atom traps".^[https://www.nobelprize.org/prizes/physics/1997/press-release/]

## Zeeman Slower
There remains a problem in the basic scheme of Doppler cooling. The resonance of the atom is rather narrow (on the order of a few megaHertz), and after having decreased its momentum by a few recoil momenta, it is no longer in resonance with the pump beam because in its frame, the frequency of the laser has shifted. To tackle this problem, the Zeeman slower applies the Zeeman effect that implies magnetic field can change the resonance frequency of an atom.^[https://en.wikipedia.org/wiki/Zeeman_slower]

The average acceleration (due to many photon absorption events over time) of an atom with mass, $M$, a cycling transition with frequency, $\omega =ck+\delta$, and linewidth, $\gamma$, that is in the presence of a laser beam that has wavenumber, $k$, and intensity $I=s_{0}I_{s}$ (where $I_{s}=\hbar c\gamma k^{3}/12\pi$ is the **saturation intensity** of the laser) is


[^21cm]: 
[^E&P]: Ewan, H. I.; Purcell, E. M. (September 1951). "Observation of a line in the galactic radio spectrum". Nature. 168 (4270): 356. Bibcode:1951Natur.168..356E. doi:10.1038/168356a0.
[^1]: Choudhuri, p.175
[^87]:https://steck.us/alkalidata/rubidium87numbers.pdf
[^23]: https://en.wikipedia.org/wiki/Zeeman_slower

[^BrMW]: https://www.britannica.com/place/Milky-Way-Galaxy/The-structure-and-dynamics-of-the-Milky-Way-Galaxy

[^smf]: https://en.wikipedia.org/wiki/Stellar_magnetic_field
[^smf2]: Wade, Gregg A. (July 8–13, 2004). "Stellar Magnetic Fields: The view from the ground and from space". The A-star Puzzle: Proceedings IAU Symposium No. 224. Cambridge, England: Cambridge University Press. pp. 235–243.