# Quantum Physics II Midterm Take-Home Exams
###### tags: `Quantum Physics II`

## 1(a) 21-cm line & astronomy
serve as an essential tool to model our Galaxy

### Prediction ([Astro. f. Phys.]() Sec. 6.5 pp.175)
- In 1945, van de Hulst predicted that the interstellar hydrogen gas would emit radiation at the radio wavelength of 21cm.
- Spins of the proton and the electron combine together to form a total spin, which can be parallel (slightly higher energy, $F=1$) or antiparrallel (lower energy, $F=0$).
- Transition between the two states evolve absorption and emission of radiation with wavelength 21 cm.
- forbidden atomic line? not easy to confirm in lab
- [Forbidden mechanism](https://en.wikipedia.org/wiki/Forbidden_mechanism)
- There is a huge amount of hydrogen with low density pervading interstellar space; so it is unlikely that the atom should be collided and resulting in de-exciting.

### Confirmation ([Astro. f. Phys.]() Sec. 6.5 pp.175)
- by Ewen and Purcell (1951) https://www.nature.com/articles/168356a0.pdf and
by Muller and Oort (1951)
## H I clouds
### Application 1: galactic model
#### Theory

#### Result

### Application 2: H I clouds ([Astro. f. Phys.]() Sec. 6.6 pp.181)
The 21-cm line is the most important diagnostic tool for studying HI clouds

1.
2. spin temperature:

## 1(b) 21-cm line & cosmology

## §9.2.2 Physical characteristics and kinematics 物理特性和運動學(pp. 266)
We discussed in §6.5 that the emission at the 21-cm line helped in mapping the distribution and kinematics of the ISM in our Galaxy. The ISMs of external spiral galaxies can also be studied by analysing the emission at the 21-cm line. If the galaxy is moving with respect to us, then we will of course find this line Doppler shifted. Additionally, in the case of a rotating disk, we expect the ISM to be moving towards us on one side of the galaxy and moving away from us on the other side (unless the line of sight is exactly perpendicular to the disk). The Doppler shifts of the 21-cm line should accordingly be different on the opposite sides of the spiral galaxy. This is indeed seen and one can use this variation of Doppler shift to determine how the circular speed $v_c$ of the ISM varies with distance from the centre of the galaxy. Figure 9.6 shows the contours of constant Doppler shift in a spiral galaxy superposed on the optical image of the galaxy. The contour lines go well beyond the optical image, since the 21-cm emission of a typical spiral galaxy usually comes from a region much larger than the optical image. This implies that a spiral galaxy does not end at the edge of its optical image (primarily due to stars) and the disk of non-luminous ISM must be extending well beyond where stars are found.
我們在第6.5節中討論了21公分發射線有助於繪製我們銀河系中 ISM 的分佈圖和運動圖。也可以==通過分析21公分線處的發射來研究外部旋渦星系的 ISM==。如果銀河系相對於我們在運動，那麼我們當然會發現這條線出現都普勒頻移。另外，對於旋轉的圓盤，我們期待看到ISM在星系的一側向我們移動，而在另一側遠離我們（除非視線與圓盤完全垂直）。因此，21公分線的都普勒頻移在旋渦星系的相反兩側應該會不同。我們確實會看到這點，而且我們可以使用都普勒頻移的變化來測定 ISM 的圓周速度 $v_c$ 如何隨距星系中心的距離而變化。圖 9.6 顯示了疊加在星系光學影像上的螺旋星系中，都普勒頻移量固定的輪廓。等量線遠遠超出光學影像的範圍，因為典型螺旋星系的21公分發射線通常來自比光學影像大得多的區域。這意味著螺旋星系不會在其光學影像（主要是恆星發出的光）的邊緣處結束，並且不發光的 ISM 盤一定會延伸到遠超出恆星的位置。

From a figure like Figure 9.6, one can determine how the rotation speed $v_c$ varies with the distance from the centre inside a galaxy. A plot of the circular speed $v_c$ as a function of the radius of a galaxy is known as a **rotation curve**. Before presenting observationally determined rotation curves, let us first discuss what we expect on theoretical grounds. If $v_c$ is the circular speed at a radial distance $r$ from the centre, then equating the centrifugal force with the gravitational force gives $$\dfrac{v_c^2}{r}=\dfrac{GM(r)}{r^2},\tag{9.5}$$ where $M(r)$ is the mass within the radius $r$. We should point out that (9.5) is strictly valid only for a spherically symmetric distribution of matter. In the case of a spiral galaxy, we expect (9.5) to give only an approximate qualitative idea of how $v_c$ varies with r. If we take $M(r)\propto r^3$ in the central region of the galaxy, as we would expect in the case of a uniform spherical distribution, then it follows from (9.5) that $$v_c\propto r\tag{9.6}$$ in the central region of the galaxy. If most of the mass is confined within a certain region, then the circular speed beyond that region, on the other hand, must be given by $$v_c=\sqrt{\dfrac{GM_\text{total}}{r}}\tag{9.7}$$ where $M_\text{total}$ is the total mass. In other words, we expect $v_c$ to fall as $r^{−1/2}$ in the outer regions of the galaxy.
從圖 9.6 所示的圖，可以確定轉速 $v_c$ 如何隨著距銀河系中心的距離而變化。圓周速度 $v_c$ 與星系半徑的函數關係圖稱為**旋轉曲線**。在介紹觀察確定的旋轉曲線之前，讓我們首先討論我們期望從理論上獲得什麼。如果 $v_c$ 是距中心的徑向距離 $r$ 處的圓周速度，則將離心力與重力相等即可得出$$\dfrac{v_c^2}{r}=\dfrac{GM(r)}{r^2}，\tag{9.5}$$ 其中 $M(r)$ 是半徑 $r$ 內的質量。我們應該指出，(9.5) 式僅對物質的球對稱分布嚴格有效。在旋渦星系的情況下，我們期望 (9.5) 式僅給出 $v_c$ 如何隨 $r$ 變化的近似定性概念。 如果我們在銀河系的中心區域取 $M(r)\propto r^3$（如我們在均勻球狀分佈的情況下所期望的），則從 (9.5) 式得出在星系的中心區域有 $$v_c\propto r\tag{9.6}$$ 如果大部分質量被限制在某個區域內，則另一方面，超出該區域的圓周速度必須由下式給出：$$v_c=\sqrt{\dfrac{GM_\text{total}}{r}}\tag{9.7}$$ 其中 $M_\text{total}$ 是總質量。換句話說，我們預期在星系的外部 $v_c$ 是以 $r^{−1/2}$ 的方式遞減。


Now we show in Figure 9.7 the rotation curves of several spiral galaxies determined from the Doppler shift of the 21-cm line. It seems that $v_c$ rises in the central regions of galaxies roughly as we expect from (9.6). However, the rotation curves become asymptotically flat and the values of $v_c$ thereafter remain nearly constant with increasing radial distance. We do not see a fall in $v_c$ as suggested by (9.7). This came as a very big surprise to astronomers when rotation curves of a few spiral galaxies were determined for the first time (Rubin and Ford, 1970; Huchtmeier, 1975; Roberts and Whitehurst, 1975; Rubin, Ford and Thonnard, 1978). Since the 21-cm emission is detected from regions of galaxies beyond the visible disk, it could be ascertained that the ISM keeps on going in circular orbits with constant $v_c$ well beyond the regions emitting visible light in the galaxies.
現在，我們在圖 9.7 中顯示了由21公分線的都普勒頻移測定的幾個旋渦星系的旋轉曲線。在銀河系中心區域，$v_c$ 的上升看起來大致與我們由 (9.6) 式所預期的一樣。但是，旋轉曲線漸趨平緩，並且此後的 $v_c$ 值隨著徑向距離的增加而保持幾乎恆定。 我們沒有看到 (9.7) 式預測的 $v_c$ 下降。當第一次測定幾個旋渦星系的自轉曲線時，這使天文學家感到非常驚訝（Rubin和Ford，1970； Huchtmeier，1975； Roberts和Whitehurst，1975； Rubin，Ford和Thonnard，1978）。由於從可見圓盤以外的星系區域檢測到21公分發射線，因此可以確定 ISM 繼續以恆定的 $v_c$ 繼續沿圓形軌道行進，遠遠超出了在星系中發射可見光的區域。

What stops $v_c$ from falling as r−1/2 as suggested in (9.7)? The most plaus- ible suggestion is that mass distribution continues beyond the visible stellar disk of the galaxy and even beyond the regions from where we receive 21-cm emission. That is why (9.7) based on the assumption that we are at the outer periphery of mass distribution is not applicable. It follows from (9.5) that $M(r)\propto r$ in a region where $v_c$ is constant. So it is difficult to estimate the total mass of a galaxy if we are not able to detect the fall-off of $v_c$ at larger distances. It appears that the total mass of a typical spiral galaxy is at least a few times the total mass of stars emitting light. In other words, most of the matter in a spiral galaxy does not emit light and is usually referred to as **dark matter**.
是什麼造成 $v_c$ 不像 (9.7) 式預測的一樣，以 $r^{−1/2}$ 遞減？最合理的假設是，==質量分佈持續超出星系可見的恆星盤，甚至超出我們接收21公分發射線的區域==。既然 (9.7) 式是基於 $r$ 是位在質量分佈外圍的假設，因此 (9.7) 式不適用。從 (9.5) 式可知，$v_c$ 恆定的區域內的 $M(r)\propto r$。因此，如果我們無法檢測到較遠處的 $v_c$ 衰減，則很難估計星系的總質量。典型的螺旋星系的總質量看起來至少是發光的恆星的總質量的幾倍。換句話說，旋渦星系中的大多數物質都不發光，通常被稱為**暗物質**。

Determining the nature of dark matter is one of the major challenges of modern astronomy. One important component of dark matter is obviously the ISM which exists in the form of a disk extending beyond the disk of stars. Since we do not see a fall-off of vc till the edge of the region where atomic hydrogen (emitting the 21-cm line) is found, it is obvious that there must be matter even beyond this region and this matter is not atomic hydrogen. We have no information about the nature of this matter or its distribution. Does this dark matter lie in the disk beyond the disk of neutral hydrogen or does it form a halo around the galaxy? We do not know the full answer (see the discussion of gravitational lensing in §13.3.2).
確定暗物質的性質是現代天文學的主要挑戰之一。暗物質的一個重要組成部分顯然是 ISM，它以盤狀形式存在，並延伸超過恆星盤。由於直到發現（發射出21公分的）中性氫區域的邊緣都看不到 $v_c$ 的衰減，所以很明顯，即使在該區域之外也必須有物質，並且該物質不是氫原子。我們對這種物質的性質和它的分布還一無所知。這個暗物質是否位於中性氫圓盤之外的圓盤中，還是在銀河周圍形成一團暈？我們不知道完整的答案（請參閱第13.3.2節中有關引力透鏡的討論）。

### §11.8.2 The intergalactic medium 星系際介質
Apart from the gas in clusters of galaxies, is there matter in regions of space between clusters and outside of galaxies? Even if there is matter in the intergalactic space, the question is how we can detect it. The emission from the intergalactic medium lying outside galaxy clusters has not been detected in any band of the electromagnetic spectrum. The only other way of checking the existence of the intergalactic medium is to look for absorption lines in the spectra of objects lying very faraway. Since quasars are the most faraway objects which are bright enough to obtain spectra from, looking for absorption lines in the spectra of quasars is the best way of searching for the intergalactic medium.
除了星系團中的氣體外，在星團之間和星系外的空間是否還存在物質？即使星際空間中存在物質，問題仍然是：我們如何發現它？在電磁波譜的任何頻帶中都未檢測到來自位於銀河團簇外部的星系間介質的發射。檢查星際介質是否存在的唯一其他方法，是在距離很遠的物體光譜中尋找吸收線。由於類星體是最遙遠的物體，其亮度足以從中獲取光譜，因此在==類星體的光譜中尋找吸收線是搜索星系間介質的最佳方法。==

Let us consider the [Lyman-α](https://en.wikipedia.org/wiki/Lyman-alpha_line) absorption line caused by the transition $1s→2p$ in a hydrogen atom. If an [absorbing system is mainly made up of neutral hydrogen atoms](https://en.wikipedia.org/wiki/H_I_region), then we expect this line to be one of the strongest absorption lines. The rest wavelength of this line is $\lambda_{L_\alpha}= 1216 Å$ . Suppose a quasar is at [redshift](https://en.wikipedia.org/wiki/Redshift) $z_\text{em}$. Since quasars typically have broad emission lines, we expect a broad emission line at the redshifted wavelength $(1+z_{\text{em}})\lambda_{L_\alpha}$ of the Lyman-α line. If there is some absorbing material on the line of sight lying at some intermediate redshift $z_\text{abs}$ (obviously we expect $0 <z_{\text{abs}} < z_{\text{em}}$, then we expect absorption at wavelength $(1+z_{\text{abs}})\lambda_{L_\alpha}$. If there is neutral hydrogen gas all along the line of sight, then we would expect to see an absorption trough from $\lambda_{L_\alpha}$ to $(1+z_{\text{abs}})\lambda_{L_\alpha}$ in the spectrum of the quasar corresponding to the full run of possible values of $z_\text{abs}$. The presence or absence of such an absorption trough in the spectrum of a distant quasar would give us an estimate of the amount of neutral hydrogen gas over the line of sight (Gunn and Peterson, 1965).
讓我們考慮由氫原子中的 $1s→2p$ 躍遷引起的[萊曼 α](https://zh.wikipedia.org/wiki/%E4%BE%86%E6%9B%BC%E7%B3%BB) 吸收線。如果[吸收系統主要由中性氫原子](https://zh.wikipedia.org/wiki/%E4%B8%AD%E6%80%A7%E6%B0%A2%E5%8C%BA)組成，那麼我們希望這條線是最強的吸收線之一。該線的其餘波長為 $\lambda_{L_\alpha}= 1216 Å$。假設一個類星體處於紅移 $z_\text{em}$。由於類星體通常具有較寬的發射線，因此我們期望在萊曼 α 線的紅移波長 $(1+z_{\text{em}})\lambda_{L_\alpha}$ 處具有較寬的發射線。 如果視線中有一些吸收材料位於一些中間的[紅移](https://zh.wikipedia.org/wiki/%E7%B4%85%E7%A7%BB) $z_\text{abs}$（顯然，我們期望 $0 <z_{\text{abs}} < z_{\text{em}}$），那麼我們期望在波長 $(1+z_{\text{abs}})\lambda_{L_\alpha}$ 處吸收。如果在視線中始終存在中性氫氣，那麼我們將期望在類星體的光譜中看到從 $\lambda_{L_\alpha}$ 到 $(1+z_{\text{abs}})\lambda_{L_\alpha}$ 的吸收槽，對應於整個 $z_\text{abs}$ 可能值。在遙遠的類星體的光譜中是否存在這種吸收槽，將使我們對視線內的中性氫氣量有一個估計（Gunn、Peterson，1965）。

Figure 11.6 shows the spectrum of a quasar at redshift $z_\text{em}=2.60$, for which the Lyman-α emission line is at $4380Å$ . We, however, do not see a continuous absorption trough from $1216Å$ to $4380Å$. Instead of a trough, we find a large number of narrowly spaced absorption lines. These absorption lines are collectively referred to as the **Lyman-α forest. This implies that we do not have a uniform distribution of neutral hydrogen gas along the line of sight. There must be many clouds of neutral hydrogen lying on the path at different redshifts, which are causing the absorption lines. In the particular spectrum shown in Figure 11.6, there is a prominent absorption feature at $3650Å$ (corresponding to redshift $z_\text{abs}=2.0$) where the radiation seems to fall almost to zero intensity. There must be a very large cloud at this redshift $z_\text{abs}=2.0$. Such large dips in the spectra indicating the presence of large hydrogen clouds are found very often in the spectra of many distant quasars.
圖 11.6 顯示了紅移 $z_\text{em}=2.60$ 時的類星體的光譜，其萊曼 α 發射線為 $4380Å$。但是，我們看不到從 $1216Å$ 到 $4380Å$ 的連續吸收谷。我們找到了許多窄間距的吸收線，而不是低谷。這些吸收線統稱為[**萊曼 α 森林**](https://zh.wikipedia.org/wiki/%E8%90%8A%E6%9B%BC%CE%B1%E6%A3%AE%E6%9E%97)。這意味著我們沿視線沒有均勻分佈的中性氫氣。在路徑上一定有許多中性氫雲出現不同的紅移處，這會導致吸收線。在圖 11.6 所示的特定光譜中，在 $3650Å$ 處有一個突出的吸收特徵（對應於紅移 $z_\text{abs}=2.0$），輻射似乎下降到了零強度。在此紅移 $z_\text{abs}=2.0$ 時一定有非常大一團雲。==在許多遙遠的類星體的光譜中經常發現光譜中的如此大的下降，表明存在大團氫氣雲。==

Readers wishing to know how to analyse these features in quasar spectra quantitatively may consult Peebles (1993, §23). Here we summarize the main conclusions qualitatively. The absence of an absorption trough, which is often referred to as the [Gunn–Peterson test](https://en.wikipedia.org/wiki/Gunn%E2%80%93Peterson_trough) (Gunn and Peterson, 1965), shows that there is very little neutral hydrogen gas outside the clouds and one quantitatively finds that the number density of hydrogen atoms has to be less than about $10^{-6}\text{ m}^{-3}$. For the sake of comparison, remember that the density of X-ray emitting gas in galaxy clusters is of order $10^{3}$ particles $\text{m}^{-3}$. The large hydrogen clouds producing prominent dips in the quasar spectra (like the dip at $3650Å$ in Figure 11.6) are estimated to have masses comparable to the mass of a typical galaxy. The most obvious possibility is that these are galaxies in the making. The smaller clouds producing the absorption lines of the Lyman-α forest, however, have much smaller masses of the order of a few hundred $M_⊙$. Careful analysis of observational data shows that these smaller clouds are most abundant at redshifts $z ≈ 2–3$ and become much less abundant at lower redshifts.
希望了解如何定量分析類星體光譜中這些特徵的讀者可以參閱Peebles（1993，§23）。在這裡，我們定性總結了主要結論。 沒有吸收槽（通常被稱為[**耿恩-彼得森檢驗**](https://zh.wikipedia.org/wiki/%E8%80%BF%E6%81%A9-%E5%BD%BC%E5%BE%97%E6%A3%AE%E6%A7%BD)）（Gunn、Peterson，1965）表明，在雲層之外幾乎沒有中性氫氣，並且可以定量地發現氫原子數量密度必須小於約 $10^{-6}\text{ m}^{-3}$。為了做比較，請記得星系團中發射X射線的氣體的密度約為 $10^{3}$ 個粒子/$\text{m}^{3}$。估計在類星體光譜中產生顯著下降的大型氫雲（例如圖 11.6 中 $3650Å$ 處的下降）具有與典型星系質量相當的質量。最明顯的可能性是這些是正在形成的星系。然而，產生萊曼 α 森林吸收線的較小的雲團的質量要小得多，約為數百 $M_⊙$。對觀測數據的仔細分析表明，這些較小的雲在紅移$z≈2–3$ 時最豐富，而在較低紅移時變得不那麼豐富。

The spectra of distant quasars like the one shown in Figure 11.6 make it clear that neutral hydrogen is mainly found inside isolated clouds. There is very little neutral hydrogen outside these clouds. But does this mean that there is no material outside the clouds and space is really empty in those regions? A more plausible assumption is that there is hydrogen outside the clouds, but it has been ionized and hence is not producing the Lyman-α absorption line. As pointed out in §11.7, matter was ionized before $z ≈ 1100$. Then neutral atoms formed, leading to the matter-radiation [decoupling](https://en.wikipedia.org/wiki/Decoupling_(cosmology)). When the first stars, galaxies and quasars started forming, the ionizing radiation from these objects presumably ionized the intergalactic medium again. This is called the reionization. The absence of neutral hydrogen atoms between the distant quasars and us (apart from the Lyman-α clouds) is believed to be a consequence of this reionization. However, if light started from a very distant quasar before the reionization, then the light path would initially pass through space filled with neutral hydrogen and we would expect to see a Gunn–Peterson trough in the spectrum at the lower-wavelength side of the redshifted Lyman-α line. There are indications that quasars with redshifts larger than $z ≈ 6$ show such troughs in their spectra (Becker *et al.*, 2001).
如圖 11.6 所示，遙遠的類星體的光譜清楚地表明，==中性氫主要存在於孤立的雲中。這些雲之外幾乎沒有中性氫==。但這是否意味著在雲層之外沒有任何物質，並且這些區域中的空間真的是空的？一個更合理的假設是，氫存在於雲層外部，但已被電離，因此不會產生萊曼 α 吸收譜線。如第 11.7 節所述，物質在 $z≈1100$ 之前被電離。然後形成中性原子，導致物質輻射[退耦](https://zh.wikipedia.org/zh-tw/%E9%80%80%E8%80%A6_(%E5%AE%87%E5%AE%99%E5%AD%A6))。==當第一代恆星、星系和類星體開始形成時，這些物體的電離輻射據推測會再次電離星系間介質==。這稱為[**再電離**](https://zh.wikipedia.org/wiki/%E5%86%8D%E9%9B%BB%E9%9B%A2)。人們認為，在遠離的類星體和我們之間（除了萊曼 α 雲之外）沒有中性氫原子是這種再電離的結果。但是，如果光在再電離之前從非常遙遠的類星體發出，那麼光路將首先通過充滿中性氫的空間，並且我們期望在萊曼 α 線中、紅移的波長較低的光譜中看到一個耿恩-彼得森槽。有跡象表明，==紅移大於 $z≈6$ 的類星體在其光譜中顯示出這種低谷==（Becker *et al.*，2001）。

Altogether, we get a picture of the Universe at redshifts $z\sim 2–6$ which is very different from the present-day Universe. Already some quasars have formed and ionized the intergalactic medium. Embedded in this ionized medium, there are clouds of neutral hydrogen (presumably their interiors are shielded from ionizing photons due to higher densities) with masses of order a few hundred $M_⊙$. There are also more massive clouds which appear like galaxies in the making.
總而言之，我們獲得了在紅移 $z\sim 2–6$ 處的宇宙圖，這與當今的宇宙有很大的不同。已經有一些類星體形成並電離了星際介質。埋在這種電離介質中的是質量為幾百個 $M_⊙$ 的中性氫雲（由於較高的密度，它們的內部被屏蔽了電離光子）。還有更多的大質量雲團正在形成，就像星系。