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交换作用 | 狄拉克指出，如果只是想了解交換作用的核心特點，那就並不需要考慮式（9）右側的頭兩項積分，此時問題可以簡化為只引入一個與電子自旋相關的耦合能量項：
:
此時，哈密頓量中原本針對波函數 _Φa_ 和 _Φb_ 的交換相互作用就可以簡化成電子自旋 ${\vec {s}}_{a}$ 和 ${\vec {s}}_{b}$ 間的相互作用。該模型被稱作經典海森堡模型。在早期文獻中，也被稱作海森堡 - 狄拉克模型：
:
_J_ab 和式 (6) 中的 _J_ex 不是同一個量。_J_ab 稱作「交換常數」，它是交換積分、重疊積分和庫侖積分的函數：
:
不過，對於正交軌道（滿足 _S_ = 0 的軌道），比如同一個原子中的不同軌道而言，仍有 _J_ab = _J_ex。
### 交換作用的效果
如果 _Jab_ 取正值，那麼交換作用將會導致電子自旋傾向於平行排列。這種效應是符合價鍵理論、定域電子占主導的體系（例如原子晶體）中鐵磁性的首要成因。但對於巡遊電子較多的固體，該模型存在著較大的缺陷（詳見後述）。反之，若 _Jab_ 取負值，則電子自旋將傾向於反平行排列，可能會引發反鐵磁性。_Jab_ 的符號是由 _J_ex 和 _CS_2 的相對大小決定的。
它可從自旋三重態和單重態的能量差中直接得出：
:
雖然交換作用的效應體現為磁學特性，但它的產生本質上和磁性無關。交換作用本身只是庫侖排斥和泡利不相容原理的產物。一般而言，兩個原子之間直接的磁矩 - 磁矩相互作用比這弱得多，以至於基本可以忽略。對核間距較大的分子體系而言，交換能級分裂項的求解相當困難。不過對於單電子的氫分子離子而言，能級分裂的解析表達式已經求解出來了（詳見對應詞條連結中的參考文獻）。
一般而言，交換作用是短程的相互作用（只有電子云重疊較明顯時交換積分才有較大的幅值），通常僅體現在相同原子中的不同軌道上的電子之間（原子內交換作用），或體現在最近鄰原子的電子之間（**直接交換作用**）。不過通過中間原子或離子的介導，也會有更長程的交換作用出現。這種現象可見於雙交換作用與超交換作用等更複雜的交換作用中。
## 固體中的直接交換作用
在晶體中，海森堡哈密頓量可以推廣到對所有的原子對 (_i_,_j_) 求和：
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] |
交换作用 | 因為每個原子在求和中被計算了兩次，所以引入了 1/2 這個因子。需要留意的是，這裡的 _J_ 是交換常數 _J_ab ，不是交換積分 _J_ex 。在晶體中，交換積分 _J_ex 和另一個叫做「交換剛度」（記作 _A_）的物理量有關。它是鐵磁體的一種屬性。交換剛度和交換積分的關係取決於晶體結構。
對簡單立方晶體，
:
對體心立方晶體，
:
對面心立方晶體，
:
式 (14) 的形式與伊辛模型類似。兩者的差異在於，伊辛模型使用的是自旋角動量的純量積 _SijSji_。在伊辛模型中，體系能量的定義是：
:
該模型於 1920 年由威廉・楞次（Wilhelm Lenz）提出，他的博士生恩斯特・伊辛（Ernst Ising）在 1925 年對一維的情形進行了求解。
### 海森堡哈密頓量和定域電子模型的局限
因為海森堡的哈密頓量假定了交換作用中的電子是依照海特勒 - 倫敦（Heitler-London）模型（即化學鍵的價鍵理論）所描述的方式定域分布的，所以它對於預測絕緣性良好的窄帶離子化合物和原子晶體的磁學特性是一個不錯的模型。然而，在導電性較好的金屬晶體中，對磁性起主要貢獻的電子恰是巡遊載流子（典型的例子是鐵、鈷、鎳）。這種體系中，海森堡模型給出的交換作用強度要麼太小，要麼連符號也是錯誤的。（例如，從實驗中測得的居里溫度 _T_C ≈ 2⟨_J_⟩/3_k_B 可以得到交換作用強度的平均值 ⟨_J_⟩，計算結果的偏小或反號即是相對於此而言。）因此海森堡模型無法解釋這些體系中的鐵磁性。對於這些金屬體系中的電子波函數，離域模型，即洪特 - 馬利肯 - 布洛赫（Hund-Mulliken-Bloch）的分子軌道 / 能帶模型更接近實際的情形。 | [
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交换作用 | 相應地，在金屬體系中，處理巡遊電子磁性的斯托納（Stoner）模型表現更好。斯托納模型中，鐵磁體裡每個原子的平均自旋磁矩（以玻爾磁子為單位）由自旋多數態和自旋少數態的電子數之差除以原子數目給出，因而斯托納模型中允許出現非整數的平均自旋磁矩。另外，對於鐵磁體而言， $\mu _{S}=-g\mu _{\rm {B}}[S(S+1)]^{1/2}$ （_g_ = 2.0023 ≈ 2）這個式子一般會高估原子中的總自旋磁角動量。例如，按照斯托納模型的預測，金屬鎳中每個鎳原子平均具有 0.54 μB 的淨磁矩，這和實驗上根據飽和磁感應強度、密度和摩爾質量推出的數值 0.61 μB 相當接近。與此同時，對於立方晶體場中孤立的鎳原子（電子排布為 3_d_84_s_2）而言，每個鎳原子擁有兩個未成對電子，因而有 ${\vec {S}}=1$ 。根據前述公式可以算出 $\mu _{S}=2.83\mu _{\rm {B}}$ ，同時一般實驗上可以測量到的數值為 ${\vec {\mu }}_{S}=g\mu _{\rm {B}}{\vec {S}}=2\mu _{\rm {B}}$ ，顯然高於金屬鎳中的原子平均磁矩。
通常說來，對於金屬晶體而言，價層 _s_ 和 _p_ 軌道上的電子最好按照離域電子處理，而 4_f_ 、5_f_ 、3_d_ 、4_d_ 電子則介於定域與離域之間。怎樣處理這些電子，取決於體系中原子核間距的具體數值。對離域電子和定域電子都對磁性產生貢獻的體系（常見於含稀土元素的化合物），RKKY（Ruderman–Kittel–Kasuya–Yosida）相互作用是目前較為通用的模型。
## 參見
* 雙交換作用
* 超交換作用
* 泡利不相容原理
* 斯萊特行列式
* 交換對稱性
* Holstein–Herring 方法
* 自旋交換作用
* 多極交換作用
* 反對稱交換
## 參考文獻
## 外部連結
* Exchange Mechanisms （頁面存檔備份，存於網際網路檔案館） in E. Pavarini, E. Koch, F. Anders, and M. Jarrell: Correlated Electrons: From Models to Materials, Jülich 2012, ISBN 978-3-89336-796-2 | [
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交换作用 | * Exchange Interaction and Energy （頁面存檔備份，存於網際網路檔案館）
* Exchange Interaction and Exchange Anisotropy （頁面存檔備份，存於網際網路檔案館） | [
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巴拿赫-阿勞格魯定理 | 泛函分析和鄰近數學分支中，**巴拿赫 - 阿勞格魯定理**或**阿勞格魯定理**（英語：Banach–Alaoglu theorem 或 Alaoglu's theorem）斷言，任意賦範向量空間的連續對偶空間中，閉單位球在弱 \* 拓撲中為緊。常見證明將弱 \* 拓撲中的單位球看成一系列緊集之積的閉子集。根據吉洪諾夫定理，該些緊集的積拓撲空間仍為緊，故該球亦然。
定理在量子力學方面有應用。系統的可觀測量是某個 C\* 代數中的自伴算子，而量子態則是該代數上的線性泛函。此框架下，定理可以推出，每個量子態皆是純態的凸線性組合。
## 歷史
納里奇（Narici）與貝肯斯坦（Beckenstein）書中，稱阿勞格魯定理為「非常重要的結果 —— 也許是關於弱 \* 拓撲唯一（_the_）最重要的事 —— 迴響傳遍泛函分析。」1912 年，赫利（Helly）證明，閉區間上連續函數的空間 $C([a,b])$ ，其連續對偶空間的單位球，為弱 \* 可數緊。1932 年，斯特凡・巴拿赫證明，任何可分賦範向量空間的連續對偶中，閉單位球必為弱 \* 序列緊（他僅考慮了序列緊）。
一般情況的證明，是由列奧尼達・阿勞格魯於 1940 年發表。納里奇與貝肯斯坦書中，引述 Pietsch [2007] 指，至少有 12 個數學家可以主張自己證明此定理或某個重要前身。
**布爾巴基 - 阿勞格魯定理**（英語：Bourbaki–Alaoglu theorem）是尼古拉・布爾巴基將原定理推廣到局部凸空間的對偶拓撲的結果。此定理亦稱為**巴拿赫 - 阿勞格魯定理**或**弱 \* 緊定理**（英語：weak-\* compactness theorem），也常簡稱為**阿勞格魯定理**（英語：Alaoglu theorem）。
## 敍述
### 一般敍述
對於域 $\mathbb {K} $ 上的向量空間 $X$ ，以 $X^{\#}$ 表示其代數對偶（所有線性泛函組成的空間）。兩者由雙線性求值映射 $\left\langle \cdot ,\cdot \right\rangle :X\times X^{\#}\to \mathbb {K} $ 所聯繫，該映射由
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巴拿赫-阿勞格魯定理 | 定義。所以，三元組 $\left\langle X,X^{\#}\right\rangle $ （兩個空間及一個映射）組成對偶系，稱為**典範對偶系**。
若 $X$ 進一步具有拓撲，即為拓撲向量空間（TVS），則可分辨其上的函數連續與否，並定義其連續對偶 $X^{\prime }$ 為代數對偶 $X^{\#}$ 中，連續泛函組成的子集。以 $\sigma \left(X^{\#},X\right)$ 表示 $X^{\#}$ 上的弱 \* 拓撲。類似有 $\sigma \left(X^{\prime },X\right)$ 是 $X^{\prime }$ 上的弱 \* 拓撲。
弱 \* 拓撲又稱逐點收斂拓撲，因為給定映射 $f$ 和一網映射 $f_{\bullet }=\left(f_{i}\right)_{i\in I}$ ，網 $f_{\bullet }$ 在弱 \* 拓撲中收斂至 $f$ ，當且僅當對定義域中每點 $x$ ，函數值組成的網 $\left(f_{i}(x)\right)_{i\in I}$ 收斂到 $f(x)$ 。
>
> **阿勞格魯定理**
>
> 設 $X$ 為任意拓撲向量空間（無需豪斯多夫或局部凸）， $X^{\prime }$ 為其連續對偶，則對於 $X$ 中原點的任何鄰域 $U$ （ $0\in U\subseteq X$ ），其極集
>
>
> : $U^{\circ }=\left\{x^{\prime }\in X^{\prime }~:~\sup _{x\in U}\left|x^{\prime }(x)\right|\leq 1\right\}\subseteq X',$
>
> 在 $X^{\prime }$ 上的弱 \* 拓撲 $\sigma \left(X^{\prime },X\right)$ 中，必為緊集。
>
> 此外， $U^{\circ }$ 亦是 $U$ 相對於典範對偶系 $\left\langle X,X^{\#}\right\rangle $ 的極集，在拓撲空間 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 同樣為緊。
>
>
### 賦範特例 | [
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巴拿赫-阿勞格魯定理 | 若 $X$ 為賦範向量空間，則原點鄰域的極集，在對偶空間中為閉，且其範數有上界。特別地，若 $U$ 為 $X$ 的開（或閉）單位球，則 $U$ 的極集為連續對偶空間 $X^{\prime }$ 的閉單位球（對偶空間配備平常的對偶範數）。此時，定理化為以下特例：
>
> **巴拿赫 - 阿勞格魯定理**
>
> 若 $X$ 為賦範空間，則連續對偶空間 $X^{\prime }$ 中，算子範數的閉單位球，為弱 \* 拓撲中的緊集。
>
>
當 $X$ 的連續對偶 $X^{\prime }$ 是無窮維賦範空間時， $X^{\prime }$ 中的閉單位球，不可能是平常範數拓撲的緊集。原因是，範數拓撲的閉單位球為緊，當且僅當空間為有限維（見 F・里斯定理）。此定理顯示出，在同一個向量空間上，考慮不同的拓撲，到㡳有何用。
但注意，巴拿赫 - 阿勞格魯定理並不推出弱 \* 拓撲為局部緊，因為僅知閉單位球在強拓撲中為原點的鄰域，在弱 \* 拓撲中則不一定。弱 \* 拓撲中，單位球的內部可能為空，除非空間為有限維。實際上，韋伊證明，局部緊的豪斯多夫拓撲向量空間必為有限維。
## 證明
### 對偶理論證明
記 $X$ 的基域為 $\mathbb {K} $ ，此處為實數域 $\mathbb {R} $ 或複數域 $\mathbb {C} $ 。證明會用到極集、對偶系、連續線性算子的基本性質，可參見該些條目，以下亦會簡單提及。 | [
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巴拿赫-阿勞格魯定理 | 先列舉一些常見定義和性質。當代數對偶 $X^{\#}$ 配備弱 \* 拓撲 $\sigma \left(X^{\#},X\right)$ 時，為一個豪斯多夫局部凸拓撲向量空間，記為 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 。空間 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 總是完備，但連續對偶 $\left(X^{\prime },\sigma \left(X^{\prime },X\right)\right)$ 則不一定，此即證明需牽涉 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 的原因。具體而言，本證明用到的性質是：完備豪斯多夫空間的子集為緊，當且僅當其為閉，且完全有界。注意 $X^{\prime }$ 從 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 繼承的子空間拓撲，等於弱 \* 拓撲 $\sigma \left(X^{\prime },X\right)$ 。為驗證此事，只需檢查對每個 $x^{\prime }\in X^{\prime }$ ， $X^{\prime }$ 中的網在其中一個拓撲中收斂到 $x^{\prime }$ ，當且僅當在另一個拓撲中亦然（因為兩個拓撲結構相等，當且僅當其具有的收斂網完全一樣）。
三元組 $\left\langle X,X^{\prime }\right\rangle $ 也是對偶對（有雙線性映射 $(x,f)\mapsto f(x)$ ），但與 $\left\langle X,X^{\#}\right\rangle $ 不同，前者一般而言未必是對偶系。以下定義極集時，會註明是對於何種對偶而言。
設 $U$ 為 $X$ 原點的鄰域，又設：
* $U^{\circ }=\left\{f\in X^{\prime }~:~\sup _{u\in U}|f(u)|\leq 1\right\}$ 為 $U$ 相對 $\left\langle X,X^{\prime }\right\rangle $ 的極集； | [
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巴拿赫-阿勞格魯定理 | * $U^{\circ \circ }=\left\{x\in X~:~\sup _{f\in U^{\circ }}|f(x)|\leq 1\right\}$ 為 $U$ 相對 $\left\langle X,X^{\prime }\right\rangle $ 的二重極集；
* $U^{\#}=\left\{f\in X^{\#}~:~\sup _{u\in U}|f(u)|\leq 1\right\}$ 為 $U$ 相對 $\left\langle X,X^{\#}\right\rangle $ 的極集。
極集的基本性質有 $U^{\circ \circ \circ }\subseteq U^{\circ }$ 。
下證巴拿赫 - 阿勞格魯定理，分若干步：
1. 先證 $U^{\#}$ 在拓撲 $\sigma \left(X^{\#},X\right)$ 中為 $X^{\#}$ 的閉子集：設 $f\in X^{\#}$ ，又假設 $f_{\bullet }=\left(f_{i}\right)_{i\in I}$ 為 $U^{\#}$ 中的網，在 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 中收斂到 $f$ 。欲證 $f\in U^{\#}$ ，即 $|f(u)|\leq 1$ 對任意 $u\in U$ 皆成立。因為在純量域 $\mathbb {K} $ 中， $f_{i}(u)\to f(u)$ ，而每個值 $f_{i}(u)$ 皆屬於（ $\mathbb {K} $ 的）閉子集 $\left\{s\in \mathbb {K} :|s|\leq 1\right\}$ ，故網的極限 $f(u)$ 亦必在該子集中。於是 $|f(u)|\leq 1$ 。 | [
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巴拿赫-阿勞格魯定理 | 2. 其次，欲證 $U^{\#}=U^{\circ }$ ，以推出 $U^{\circ }$ 既是 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 的閉子集，亦是 $\left(X^{\prime },\sigma \left(X^{\prime },X\right)\right)$ 的閉子集：有包含關係 $U^{\circ }\subseteq U^{\#}$ ，因為連續線性泛函尤其是線性泛函。反之，欲證 $\,U^{\#}\subseteq U^{\circ }$ ，設 $f\in U^{\#}$ 滿足 $\;\sup _{u\in U}|f(u)|\leq 1$ ，換言之線性泛函 $f$ 在鄰域 $U$ 上有界，而泛函有界等價於連續，故 $f\in X^{\prime }$ ，從而 $f\in U^{\circ }$ ，即所求證。用第 1 步，結合交集 $U^{\#}\cap X^{\prime }=U^{\circ }\cap X^{\prime }=U^{\circ }$ 在 $X^{\prime }$ 的子空間拓撲中為閉，推得 $U^{\circ }$ 為閉。
3. 欲證 $U^{\circ }$ 對 $X^{\prime }$ 的 $\sigma \left(X^{\prime },X\right)$ 拓撲而言是完全有界子集：由二重極集定理， $U\subseteq U^{\circ \circ }$ ，又因為鄰域 $U$ 為 $X$ 中的吸收集， $U^{\circ \circ }$ 亦同。可以證明，此結論推出 $U^{\circ }$ 是 $X^{\prime }$ 對 $\sigma \left(X^{\prime },X\right)$ 而言的有界子集。由於 $X$ 分辨 $X^{\prime }$ 各點， $X^{\prime }$ 的子集在 $\sigma \left(X^{\prime },X\right)$ 意義下有界，當且僅當在同樣意義下完全有界。所以，尤其有 $U^{\circ }$ 在 $\sigma \left(X^{\prime },X\right)$ 意義下完全有界。 | [
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巴拿赫-阿勞格魯定理 | 4. 欲證 $U^{\circ }$ 亦為 $X^{\#}$ 在 $\sigma \left(X^{\#},X\right)$ 拓撲下的完全有界子集：已知 $X^{\prime }$ 上， $\sigma \left(X^{\prime },X\right)$ 拓撲等於 $X^{\prime }$ 從 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 繼承的子空間拓撲，結合第 3 步與「完全有界」的定義，即推出 $U^{\circ }$ 為 $X^{\#}$ 在 $\sigma \left(X^{\#},X\right)$ 拓撲下的完全有界子集。
5. 最後，欲證 $U^{\circ }$ 為 $X^{\prime }$ 在 $\sigma \left(X^{\prime },X\right)$ 拓撲下的緊子集：因為 $\left(X^{\#},\sigma \left(X^{\#},X\right)\right)$ 為完備拓撲向量空間，又 $U^{\circ }$ 為其閉（第 2 步）而完全有界（第 4 步）的子集，所以 $U^{\circ }$ 為緊。定理證畢。
### 較初等的證明
以下證明，僅用到集合論、點集拓撲、泛函分析的基本概念。拓撲方面，需要熟悉使用拓撲空間中的網、積拓撲、兩者與逐點收斂的關聯（為方便起見，證明中也會給出部分細節）。同時也要瞭解，線性泛函為連續，當且僅當其在原點的某個鄰域上有界（見次線性泛函）。
設向量空間 $X$ 的基域為 $\mathbb {K} $ ，為實數系 $\mathbb {R} $ 或複數系 $\mathbb {C} $ 兩者之一。對任意實數 $r$ ，以
: $B_{r}:=\{s\in \mathbb {K} :|s|\leq r\}$
表示以原點為球心，半徑為 $r$ 的閉球。在 $\mathbb {K} $ 中，此為緊的閉集。
#### 極集的等價表示
由於 $U$ 是 $X$ 中原點的鄰域，可知 $U$ 亦是 $X$ 的吸收集，即對每個 $x\in X$ ，皆有正實數 $r_{x}>0$ 使 $x\in r_{x}U:=\left\{r_{x}u:u\in U\right\}$ 。以 | [
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巴拿赫-阿勞格魯定理 | : $U^{\#}:=\left\{f\in X^{\#}~:~\sup _{u\in U}|f(u)|\leq 1\right\}~=~\left\{f\in X^{\#}~:~f(U)\subseteq B_{1}\right\}$
表示 $U$ 相對典範對偶系 $\left\langle X,X^{\#}\right\rangle $ 的極集。將證明，此極集 $U^{\#}$ ，與定理提到， $U$ 相對 $\left\langle X,X^{\prime }\right\rangle $ 的極集 $U^{\circ }$ ，兩者相等。
$U^{\circ }\subseteq U^{\#}$ 成立，是因為連續線性泛函按定義必是線性泛函。反之，欲證 $U^{\#}\subseteq U^{\circ }$ ，設 $f\in U^{\#}$ 滿足 $\sup _{u\in U}|f(u)|\leq 1$ ，即線性泛函 $f$ 在鄰域 $U$ 上有界。所以 $f$ 是連續線性算子（換言之 $f\in X^{\prime }$ ），從而有 $f\in U^{\circ }$ ，即所求證。
至此，已證明 $U^{\circ }=U^{\#}$ ，餘下的證明中，需理解笛卡兒積 $\prod _{x\in X}\mathbb {K} $ 與所有 $X\to \mathbb {K} $ 的映射構成的空間 $\mathbb {K} ^{X}$ 等同。仍需證明以下兩個命題：
1. $U^{\circ }$ 為 $\mathbb {K} ^{X}$ 的閉子集。
* 此處 $\mathbb {K} ^{X}=\prod _{x\in X}\mathbb {K} $ 配備的是逐點收斂拓撲，等同於積拓撲。
2. $U^{\circ }\subseteq \prod _{x\in X}B_{r_{x}}.$
* 其中 $B_{r_{x}}\subseteq \mathbb {K} $ 表示以原點 $0$ 為球心， $r_{x}$ 為半徑的閉球。本證明開始時，對每個 $x\in X$ ， 已定義 $r_{x}$ 為使 $x\in r_{x}U$ 的任意一個實數 $r_{x}>0$ 。特別地，對於 $u\in U$ ，可以選 $r_{u}:=1$ 。 | [
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巴拿赫-阿勞格魯定理 | 以上命題推出， $U^{\circ }$ 為 $\prod _{x\in X}B_{r_{x}}$ 的閉子集，而由吉洪諾夫定理，該積空間為緊（因為每個閉球 $B_{r_{x}}$ 皆為緊）。因為緊空間的閉子集仍為緊，所以有 $U^{\circ }$ 為緊集，從而證畢巴拿赫 - 阿勞格魯定理的主要結論。
#### 極集為閉
以下證明前述命題 1。代數對偶 $X^{\#}$ 總是積空間 $\mathbb {K} ^{X}=\prod _{x\in X}\mathbb {K} $ 的閉子集。要證明 $U^{\circ }$ 在 $\mathbb {K} ^{X}$ 中為閉，祇需證明集合
: ${\widetilde {U}}^{\circ }~:=~\left\{f\in \mathbb {K} ^{X}~:~\sup _{u\in U}|f(u)|\leq 1\right\}$
是 $\mathbb {K} ^{X}$ 的閉子集，因為若有此結論，則 ${\widetilde {U}}^{\circ }\cap X^{\#}=U^{\#}=U^{\circ }$ 是 $\mathbb {K} ^{X}$ 中兩閉集之交，故亦為閉集。
設 $f\in \mathbb {K} ^{X}$ ，又設 $f_{\bullet }=\left(f_{i}\right)_{i\in I}$ 為 ${\widetilde {U}}^{\circ }$ 中的網，在 $\mathbb {K} ^{X}$ 中收斂到 $f$ 。需要證明 $f\in {\widetilde {U}}^{\circ }$ 。換言之，要證對每個 $u\in U$ ， $|f(u)|\leq 1$ （或等價寫成 $f(u)\in B_{1}$ ）。由於在純量域 $\mathbb {K} $ 中， $\left(f_{i}(u)\right)_{i\in I}\to f(u)$ ，且每項 $f_{i}(u)$ 皆屬於 $\mathbb {K} $ 中的閉子集 $B_{1}=\left\{s\in \mathbb {K} :|s|\leq 1\right\}$ ，此網的極限 $f(u)$ 亦必屬於該閉集，即 $f(u)\in B_{1}$ 。證畢命題 1。
上述證明可以推廣，以論證以下命題：
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巴拿赫-阿勞格魯定理 | > 設 $U\subseteq X$ 為任意集合， $B\subseteq Y$ 為拓撲空間 $Y$ 的閉子集，則在 $Y^{X}$ 的逐點收斂拓撲中， $P_{U}:=\left\{f\in Y^{X}~:~f(U)\subseteq B\right\}$ 為閉子集。
>
>
命題 1 為其特殊情況，取 $Y:=\mathbb {K} $ 和 $B:=B_{1}$ 便得。
#### 極集包含於緊空間之積
以下證明前述命題 2。對任意 $z\in X$ ，以 $\Pr {}_{z}:\prod _{x\in X}\mathbb {K} \to \mathbb {K} $ 表示到第 $z$ 個坐標的投影。欲證 $U^{\circ }\subseteq \prod _{x\in X}B_{r_{x}}$ 。換言之，欲對每個 $x\in X$ ，證明 $\Pr {}_{x}(U^{\circ })\subseteq B_{r_{x}}$ 。
於是選定 $x\in X$ ，設 $f\in U^{\circ }$ ；要證 $\Pr {}_{x}(f):=f(x)\in B_{r_{x}}$ 。由 $r_{x}$ 的定義， $x\in r_{x}U$ ，故 $\,u_{x}:=\left({\frac {1}{r_{x}}}\right)x\in U$ 。因為 $f\in U^{\circ }=U^{\#}$ ，線性泛函 $f$ 滿足 $\;\sup _{u\in U}|f(u)|\leq 1$ ，所以由 $u_{x}\in U$ ，可知
: ${\frac {1}{r_{x}}}|f(x)|=\left|{\frac {1}{r_{x}}}f(x)\right|=\left|f\left({\frac {1}{r_{x}}}x\right)\right|=\left|f\left(u_{x}\right)\right|\leq \sup _{u\in U}|f(u)|\leq 1.$
所以 $|f(x)|\leq r_{x}$ ，即 $f(x)\in B_{r_{x}}$ ，證畢命題 2。
## 序列版本
巴拿赫 - 阿勞格魯定理有個特殊情況，對可分空間使用，並將「緊」換成「序列緊」。此時定理斷言：
>
> 可分賦範向量空間的對偶中，閉單位球在弱 \* 拓撲下是序列緊。
>
>
### 可度量 | [
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巴拿赫-阿勞格魯定理 | 實際上，可分空間的對偶的閉單位球上，弱 \* 拓撲可度量，故緊與序列緊等價。
明確而言，設 $X$ 為可分賦範向量空間，而 $B$ 為連續對偶 $X^{\prime }$ 中的閉單位球。根據 $X$ 可分的定義，有某個可數稠密子集，列舉為 $x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }$ 。則下式定義一個度量：對於 $x,y\in B$ ，
: $\rho (x,y)=\sum _{n=1}^{\infty }\,2^{-n}\,{\frac {\left|\langle x-y,x_{n}\rangle \right|}{1+\left|\langle x-y,x_{n}\rangle \right|,}}$
其中 $\langle \cdot ,\cdot \rangle $ 表示 $X^{\prime }$ 與 $X$ 的對偶匹配，即將後一個元素代入到前一個元素求值。此度量下， $B$ 為序列緊之事，用類似阿爾澤拉 - 阿斯科利定理的對角線證法，即可證明。
由於證明本質為構造性（而非如一般情況，用到非構造性的選擇公理），在偏微分方程學中，有時使用序列巴拿赫 - 阿勞格魯定理，構造偏微分方程或變分問題的解。舉例，若有某個可分賦範空間 $X$ ，其對偶上有泛函 $F:X^{\prime }\to \mathbb {R} $ ，欲求最小值，則常見策略是先構造序列 $x_{1},x_{2},\ldots \in X^{\prime }$ ，使 $F$ 的泛函值趨向下確界，然後訴諸序列巴拿赫 - 阿勞格魯定理，取出子序列 $(x_{n_{k}})_{k}$ ，在弱 \* 拓撲下收斂到極限 $x$ ，並確定 $x$ 使 $F$ 取最小值。最後一步通常要求 $F$ 在弱 \* 拓撲下為（序列）下半連續。
考慮另一個例子，設 $X=C_{0}(\mathbb {R} )$ 為實軸上，在無窮遠處消失的連續函數組成的空間，則由里斯 - 馬可夫表示定理， $X^{\prime }$ 為實軸上全體有限拉東測度的空間。此時序列巴拿赫 - 阿勞格魯定理等價於赫利選擇定理。
### 證明
下證序列版本的巴拿赫 - 阿勞格魯定理。
對每個 $x\in X$ ，設
: $D_{x}=\{z\in \mathbb {C} :|z|\leq \|x\|\},$
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巴拿赫-阿勞格魯定理 | : $D=\prod _{x\in X}D_{x}.$
因為 $D_{x}$ 是複平面的緊子集， $D$ 在積拓撲中亦為緊（根據吉洪諾夫定理）。
$X^{\prime }$ 中的閉單位球 $B_{1}\left(X^{\prime }\right)$ ，可以自然地看成 $D$ 的子空間：考慮映射
: $f\in B_{1}\left(X^{\prime }\right)\mapsto (f(x))_{x\in X}\in D,$
其為單射，且對於 $B_{1}\left(X^{\prime }\right)$ 的弱 \* 拓撲和 $D$ 的積拓撲而言，是連續映射。在像集上，映射的逆也連續。
欲完成定理的證明，只需證明映射的像為閉集。給定網 $D$ 中的網
: $\left(f_{\alpha }(x)\right)_{x\in X}\to \left(\lambda _{x}\right)_{x\in X},$
等式 $g(x)=\lambda _{x}$ 定義的泛函 $g$ ，也在 $B_{1}(X^{\prime })$ 中。定理證畢。
## 推論
### 賦範空間
假設 $X$ 為賦範空間，則其連續對偶空間 $X^{\prime }$ 具有對偶範數。
* $X^{\prime }$ 中的閉單位球為弱 \* 緊。相比之下，若 $X^{\prime }$ 為無窮維，則其閉單位球在範數拓撲中必不為緊（F・里斯定理）。
* 某巴拿赫空間自反，當且僅當其閉單位球在弱拓撲 $\sigma \left(X,X^{\prime }\right)$ 下為緊。
* 若 $X$ 為自反巴拿赫空間，則 $X$ 中每個有界序列，都有弱收斂子列。（此為對 $X$ 某個弱可度量子空間應用巴拿赫 - 阿勞格魯定理的結果。更簡潔而言，是應用埃伯萊恩 - 什穆良定理。）舉例，設 $X$ 為 _L_p 空間 $L^{p}(\mu )$ ，其中 $1<p<\infty $ 。設 $(f_{n})_{n}$ 為 $X$ 中函數組成的有界序列。則存在子列 $(f_{n_{k}})_{k}$ ，且有 $f\in X$ 使得對於 $X^{\prime }=L^{q}(\mu )$ 中的任意函數 $g$ 成立，其中 ${\frac {1}{p}}+{\frac {1}{q}}=1$ 。對於 $p=1$ ，沒有相應的結論，因為 $L^{1}(\mu )$ 不自反。 | [
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巴拿赫-阿勞格魯定理 | ### 希爾伯特空間
* 任意希爾伯特空間中，閉有界集必然弱相對緊，即其在弱拓撲的閉包為弱緊，故每個有界網必有弱收斂子網（希爾伯特空間皆自反）。
* 由哈恩－巴拿赫定理，範數拓撲中的閉凸集，在弱拓撲中也是閉集，故希爾伯特空間或自反巴拿赫空間中，凸有界集的範數閉包必為弱緊。
* 設 $H$ 為希爾伯特空間， $B(H)$ 為其上有界算子的空間，則 $B(H)$ 可以配備以下兩種不同的拓撲：一則超弱拓撲，即 $B(H)$ 作為跡類算子空間 $B_{1}(H)$ 的對偶所具備的弱 \* 拓撲；二則弱算子拓撲，是使形如 $T\mapsto \langle Tx,y\rangle $ 的映射皆連續的最弱的拓撲，此拓撲比超弱拓撲更弱。此定義下， $B(H)$ 中的閉有界子集，關於弱算子拓撲為相對緊。所以，算子的有界序列必有某個弱極限點。其推論是， $B(H)$ 配備弱算子拓撲或超弱拓撲時，滿足海涅 - 博雷爾性質。
## 與選擇公理的關係
通常，會用到吉洪諾夫定理來證明巴拿赫 - 阿勞格魯定理，所以要依賴於 ZFC 公理系統，尤其是選擇公理。主流泛函分析中，許多結果皆依賴選擇公理。然而，本定理在可分空間的情況（見 § 序列版本）並不依賴選擇公理，該情況下有構造性證明。對於不可分的情況，超濾子引理比選擇公理嚴格弱，但亦足以證明巴拿赫 - 阿勞格魯定理。反之，巴拿赫 - 阿勞格魯定理也推出超濾子引理，所以兩者等價。
## 參見
* 畢曉普 - 菲爾普斯定理
* 巴拿赫 - 馬祖爾定理
* Δ 收斂
* 埃伯萊恩 - 什穆良定理
* 古德斯汀定理
* 詹姆斯定理
* 克列因 - 米爾曼定理
* 馬祖爾引理
* 拓撲向量空間 —— 有「連續」概念的向量空間
## 註
## 參考資料
* Köthe, Gottfried. Topological Vector Spaces I [拓撲向量空間一]. New York: Springer-Verlag. 1969 （英語）. 見 §20.9。
* Meise, Reinhold; Vogt, Dietmar. Introduction to Functional Analysis [泛函分析導論]. Oxford: Clarendon Press. 1997. ISBN 0-19-851485-9 （英語）. 見 Theorem 23.5，p. 264。 | [
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] |
巴拿赫-阿勞格魯定理 | * Narici, Lawrence; Beckenstein, Edward. Topological Vector Spaces [拓撲向量空間]. Pure and applied mathematics Second. Boca Raton, FL: CRC Press. 2011. ISBN 978-1584888666. OCLC 144216834 （英語）.
* Rudin, Walter. Functional Analysis [泛函分析]. International Series in Pure and Applied Mathematics **8** Second. New York, NY: McGraw-Hill Science/Engineering/Math. 1991. ISBN 978-0-07-054236-5. OCLC 21163277 （英語）. 見 Theorem 3.15，p. 68。
* Schaefer, Helmut H.; Wolff, Manfred P. Topological Vector Spaces [拓撲向量空間]. GTM **8** Second. New York, NY: Springer New York Imprint Springer. 1999. ISBN 978-1-4612-7155-0. OCLC 840278135 （英語）.
* Schechter, Eric. Handbook of Analysis and its Foundations [分析及其基礎手冊]. San Diego: Academic Press. 1997. doi:10.1016/B978-0-12-622760-4.X5000-6 （英語）.
* Trèves, François. Topological Vector Spaces, Distributions and Kernels [拓撲向量空間、分佈、核]. Mineola, N.Y.: Dover Publications. 2006 [1967]. ISBN 978-0-486-45352-1. OCLC 853623322 （英語）. | [
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附加质量 | 在流體力學中，**附加質量**或**虛擬質量**是指系統額外增加的一部分慣性值，因為做加減速運動的物體會帶動其周邊的部分流體一起運動。在實際計算中，可以將它簡化理解為一定體積的流體」粘附「在物體上，與物體同步運動，儘管事實上這部分流體的運動狀態並非完全一致。
**附加質量係數**指附加質量與等效置換的流體質量（即流體密度乘以等效體積）之比，是一個無量綱量值。通常，附加質量是一個二階張量，它將流體加速度矢量與作用在物體上的合力矢量關聯在一起。
## 背景
弗里德里希・貝塞爾 (Friedrich Bessel) 於 1828 年提出了附加質量的概念，用於描述鐘擺在流體中的運動。這種鐘擺的周期相對於它在真空中的周期增加了（即使考慮了浮力效應），表明物體周圍的流體增加了系統的有效質量。
附加質量的概念可以說是物理學中重整化的首個案例。 這個概念類似於用經典物理學方法來模擬量子力學中的准粒子。但需要注意，它不應與相對論中的質量增加相混淆。
人們常常錯誤地認為附加質量是由流體的動量引起的，事實並非如此。不妨設想一個裝滿流體的巨大盒子，其中流體的動量在每個時刻都為零。附加質量實際上由准動量決定：附加質量與物體的加速度的乘積等於流體准動量對時間的導數。
## 附加質量力
因浸沒在流體中的物體的相對速度發生變化而產生的非定常力由兩部分組成：附加質量力和巴塞特力。
力的起源是水下物體加速做功導致流體獲得動能的趨勢。
可以證明，對於浸沒在無粘性、不可壓縮流體中的球體，附加質量力為
: : $\mathbf {F} ={\frac {\rho _{\mathrm {c} }V_{\mathrm {p} }}{2}}\left({\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}-{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\right),$
其中粗體符號表示向量， $\mathbf {u} $ 是流體流速， $\mathbf {v} $ 是球體速度， $\rho _{\mathrm {c} }$ 是流體的質量密度（連續相）， $V_{\mathrm {p} }$ 是球體的體積，D/D_t_ 表示物質導數。
當我們分析球體的動量方程時，可以輕易推導出「附加質量」的概念。 | [
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附加质量 | : : $m_{\mathrm {p} }{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=\sum \mathbf {F} +{\frac {\rho _{\mathrm {c} }V_{\mathrm {p} }}{2}}\left({\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}-{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\right),$
$\sum \mathbf {F} $ 是球體上所有其他力項的總和，例如重力、壓力梯度、阻力、升力、巴塞特力等。
將球體速度的導數從方程的右側移動到左側，得到
: : $\left(m_{\mathrm {p} }+{\frac {\rho _{\mathrm {c} }V_{\mathrm {p} }}{2}}\right){\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=\sum \mathbf {F} +{\frac {\rho _{\mathrm {c} }V_{\mathrm {p} }}{2}}{\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}},$
所以球體被加速，就好像它增加了相當於自身一半體積的流體的質量，並且由於流體的運動，等式右邊多了一項額外的作用力。
## 應用
通過將有效質量視為自身質量與附加質量的總和，可將附加質量合併到大多數物理方程中。這個總和通常被稱為「虛擬質量」。
物體的附加質量可以基於牛頓第二定律簡寫成以下形式
: $F=m\,a$ 變成 $F=(m+m_{\text{added}})\,a.$
可以證明一個球體（半徑為 $r$ ）的附加質量 是 ${\tfrac {2}{3}}\pi r^{3}\rho _{\text{fluid}}$ ，它等於球體體積的一半與流體密度的乘積_。_對於一般物體，附加質量是一個張量（稱為誘導質量張量），其分量取決於物體的運動方向。附加質量張量中的元素並非都具有維度質量，有些元素是質量 × 長度，有些是質量 × 長度 2 。 | [
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附加质量 | 所有在流體中非勻速運動的物體都會受到附加質量的影響，但由於附加質量取決於流體的密度，因此處於相對低密度流體中的緻密物體，這種影響通常被忽略。對於流體密度與物體密度相當或大於物體密度的情況，附加質量甚至可能大於物體自身質量，忽略它會給計算帶來重大誤差。
例如，在水中上升的球形氣泡的質量為 ${\tfrac {4}{3}}\pi r^{3}\rho _{\text{air}}$ 但額外的質量 ${\tfrac {2}{3}}\pi r^{3}\rho _{\text{water}}.$ 由於水的密度大約是空氣的 800 倍（在 RTP 下），因此在這種情況下附加質量大約是氣泡自身質量的 400 倍。
### 海洋工程學
這些原則也適用於船舶、潛艇和海上平台。在船舶工業中，附加質量被稱為流體動力附加質量。在船舶設計中，在進行適航性分析時必須考慮加速附加質量所需的能量。對於船舶來說，附加質量很容易達到船舶質量的 1/4 或 1/3，因此除了摩擦阻力和興波阻力之外，它顯著增加了船舶的慣性力。
對於在水中自由下沉的某些特定形狀物體，與下沉體相關的流體動力學附加質量可能比物體的質量大得多。例如，當下沉的物體存在一個大的平面，其法向量指向運動方向（向下）時，就會發生這種情況。當這樣的物體突然減速時（例如，由於與海床的碰撞），大量的動能被釋放。
在近海工業中，不同幾何形狀的水動力附加質量是大量研究的主題。這些研究通常需要作為海底墜落物體風險評估的因子（研究重點是量化墜落物體對海底基礎設施的影響風險）。由於流體附加質量在撞擊瞬間占下沉物體總質量的很大一部分，因此它會顯著影響海底保護結構的設計載荷。
臨近邊界（或另一個物體）會影響流體動力附加質量。這意味著附加質量取決於對象的幾何形狀及其與邊界的距離。對於浮體（例如船舶），這意味著水的深度會改變浮體的響應（由于波浪作用），但這種影響在深水中幾乎不存在。對水動力附加質量的影響程度取決於物體的幾何形狀和位置以及邊界的形狀（例如，碼頭、海堤、艙壁或海床）。
在邊界附近自由下沉的物體相關的水動力附加質量類似於浮體的情形。一般而言，流體動力學附加質量隨著邊界與實體之間的距離的減小而增加。在規劃海底安裝或預測淺水條件下浮體的運動時，此特性很重要。
### 航空學
對於航空器（除了比空氣輕的氣球和飛艇），通常不會考慮附加質量，因為空氣的密度非常小。
## 參考文獻
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] |
附加质量 | * Basset force for describing the effect of the body's relative motion history on the viscous forces in a Stokes flow
* Basset–Boussinesq–Oseen equation for the description of the motion of – and forces on – a particle moving in an unsteady flow at low Reynolds numbers
* Darwin drift for the relation between added mass and the Darwin drift volume
* Keulegan–Carpenter number for a dimensionless parameter giving the relative importance of the drag force to inertia in wave loading
* Morison equation for an empirical force model in wave loading, involving added mass and drag
## 外部連結
* MIT OpenCourse Ware （頁面存檔備份，存於網際網路檔案館）
* Naval Civil Engineering Laboratory
* Det Norske Veritas DNV-RP-H103 Modelling And Analysis Of Marine Operations （頁面存檔備份，存於網際網路檔案館） | [
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] |
量子機器學習 | **量子機器學習**，是將量子算法整合到機器學習程序中。該術語最常見的用法是指用於分析量子計算機上執行的經典數據的機器學習算法，即量子增強機器學習。常規機器學習算法被用來計算海量數據，而量子機器學習利用量子位和量子運算或專門的量子系統來提高算法在程序中完成的計算速度和數據存儲。在實際操作中，量子機器學習會混合常規機器學習，先用常規計算機執行機器學習程序，然後將無法通過常規計算機完成的子程序交由量子計算機完成。這些子程序可能比較複雜，在量子計算機上執行會有著更顯著的速度提升。此外，量子算法可以用來分析量子態而不僅僅局限於常規數據。
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反事实确定性 | 在量子力學中，**反事實確定性（英語：counterfactual definiteness，簡稱** CFD**）** 是指「有意義地」談論尚未進行的測量的結果的確定性的能力（即假設物體的存在和物體的屬性的能力，即使它們尚未被測量）。術語「反事實確定性」被用於物理計算的討論，尤其是與量子糾纏現象以及貝爾不等式相關的討論。在此類討論中，「有意義地」意味著能夠在統計計算中將這些未測量的結果與已測量的結果同等對待。反事實確定性是與物理系統的物理和數學模型直接相關的一個（有時是假設但未說明的）方面，而不是關於未測量結果的含義的哲學問題。
「反事實」可能作為名詞出現在物理討論中。在這種情況下的意思是「一個可以被測量但因故沒有被測量的值」。
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光子能 | **光子能**是單個光子攜帶的能量。 能量大小直接正比於光子的電磁頻率，因此相當於與波長成反比。光子頻率越高，能量就越高。等價地，光子波長越長，能量就越低。
光子能可用任何能量單位表示。 而在這些單位中間，常用於描述光子能的是電子伏特（eV）和焦耳（以及它的倍數，如微焦耳）。由於一焦耳等於 6.24 × 1018 eV，更大的單位在描述具有更高能量和頻率光子的能量，例如伽馬射線時可能更有用，而不常用於描述較低能量光子，如電磁波譜中射頻範圍的光子。
## 公式
### 物理學
光子能直接正比例於頻率。
其中
* $E$ 表示能量（焦耳）
* $h$ 表示普朗克常數：6.62607015 × 10−34 (m2kgs−1)
* $f$ 表示頻率（赫茲）
這個方程式也叫普朗克 - 愛因斯坦關係式。
另一種形式如下，
其中
* $E$ 表示光子能量（焦耳）
* _$\lambda $_ 表示光子的波長（米）
* $f$ 表示頻率（赫茲）
* $c$ 表示真空中的光速：每秒 299792458 米
* $h$ 表示普朗克常數：6.62607015 × 10−34 (m2kgs−1)
1 Hz 頻率下的光子能等於 6.62607015 × 10−34 J
也等於 4.135667697 × 10−15 eV（電子伏特）
### 電子伏特
能量常用電子伏特計量。
若光子能以電子伏特計、 波長以微米計，則方程大致為
: $E{\text{ (eV)}}={\frac {1.2398}{\lambda {\text{ (μm)}}}}$
此方程僅在波長以毫米計算時正確。
波長 1 μm（接近紅外輻射）時，光子能約為 1.2398 eV。
### 在化學、量子物理和光學工程中
請看
其中
* $E$ 表示光子能（焦耳）
* $h$ 表示普朗克常數：6.62607015 × 10−34 (m2kgs−1)
* 希臘字母 _ν_（nu）表示光子的頻率
## 例子
以 100 MHz 發射信號的調頻廣播站發射的光子含有約 4.1357 × 10−7 eV 能量。這非常少的能量大致是電子質量的 8 × 10−13 倍（通過質能關係轉換）。 | [
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] |
光子能 | 超高能伽馬射線具有 100 GeV 至超過 1 PeV（即 1011 至 1015 電子伏特），或 16 納焦耳至 160 微焦耳的光子能。這相當於 2.42 × 1025 至 2.42 × 1029 Hz 的頻率。
在光合作用中，特定的葉綠素分子在光系統 I 吸收波長 700 nm 的紅色光光子，相當於每個光子含有≈ 2 eV ≈ 3 x 10−19 J ≈ 75 kBT 的能量（這裡 kBT 表示單位熱能）。 要從 CO2 和水分子中生成一分子葡萄糖，最少需要 48 個光子（化學勢差 5 x 10−18 J），最大能量轉換效率 35%。
## 另請參見
* 光子
* 電磁輻射
* 電磁波譜
* 普朗克常數與普朗克單位制
* 普朗克 - 愛因斯坦關係式
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欧洲航天局科学计划 | 歐洲太空總署的**科學計劃**（**Science Programme**）是一項長期的空間科學和太空探索任務規劃，由該機構科學理事會管理，通過制定各十年期航天計劃，資助歐洲航天機構和其它組織所開發、發射和運營的任務。首期「地平線 2000」計劃在 1985 年至 1995 年間推動了包括四項「奠基石任務」— 太陽和太陽圈探測器及星簇 2 號衛星、X 射線多鏡面牛頓衛星、羅塞塔號和赫歇爾空間天文台等在內的八項任務的開發；第二期「地平線 2000+」計劃，在 1995 年至 2005 年期間促進了「蓋亞任務、雷射干涉空間天線開路者號和貝皮可倫坡號的開發；當前的第三期「宇宙願景」計劃，自 2005 年來，已資助了十項任務的開發，其中包括三項旗艦任務：木星冰月探測器、先進高能天體物理學望遠鏡和雷射干涉空間天線。目前正在規劃即將推出的第四期「遠航 2050」計劃。另外 ，在科學計劃中，還有與歐洲以外航天機構和組織的合作，如與美國宇航局的「卡西尼 - 惠更斯號」和中國國家航天局的微笑計劃項目。
## 管理
科學計劃由歐洲太空總署 (ESA) 科學理事會管理，其目標包括提升歐洲在太空科學領域的地位、促進技術創新以及維護歐洲太空基礎設施，如發射服務和太空飛行器運營，為歐空局強制性計劃之一，所有歐空局成員國都必須參與，並按照國民生產淨值的比例繳納一定的會費，以確保該計劃及其任務的長期財務安全。該計劃的制訂過程採用「自下而上」的結構，歐洲科學界可通過諮詢機構引導計劃的方向。這些機構向總幹事和科學部主任提出有關該計劃的建議，並可將建議獨立報告給歐空局的科學計劃委員會（SPC）— 整個計劃的管理機構。該計劃目前的諮詢結構由天文學工作組 (AWG) 和太陽系與探索工作組 (SSEWG) 組成，他們向高級空間科學諮詢委員會 (SSAC) 報告，該委員會則向該機構的董事會報告。諮詢機構的成員資格為期三年，天文學工作組和太陽系與探索工作組的主席也是高級空間科學諮詢委員會的成員。另外，還可設立特設諮詢小組，就某些任務提案或規劃周期的制定提供諮詢意見。 | [
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欧洲航天局科学计划 | 該計劃中的任務是從歐洲科學界成員向歐空局提交的提案中競選評出，在每次競選中，歐空局都會概述提案需要滿足的各類任務評審標準，它們分別是「L」級大型任務、「M」級中型任務、「S」級小型任務和「F」級快速任務，每類任務都有不同的預算上限和實施時間表。參選提案隨後由天文學工作組、太陽系與探索工作組、歐空局工程師和所有相關的特設工作組進行審查，作為「第 0 階段」可行性研究的一部分。對需要開發新技術的任務，會在研發期間由歐洲空間研究與技術中心的並行測試部門進行審查。研究結束後，在「A 階段」最多選出三項入圍提案，並為每項候選任務制定初步設計。然後，科學計劃委員會就哪項提案進入「B」到「F」階段作出最終決定，包括任務中所用太空飛行器的開發、建造、發射和處置。在 A 階段，每項候選任務會分配兩個相互競爭的承建商來建造太空飛行器，並在 B 階段選擇獲選任務的承建商。
## 歷史
### 背景 | [
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欧洲航天局科学计划 | 歐洲太空總署成立於 1975 年 5 月，由歐洲空間研究組織 (ESRO) 和歐洲發射裝置發展組織 (ELDO) 合併而成。1970 年，歐洲空間研究組織的發射計劃管理諮詢委員會 (LPAC) 決定不執行當時認為超出該組織預算和能力的天文學或行星探測任務，這意味著所有大型科學任務必須要與其他政府航天機構和組織進行合作。1980 年，這一政策得到有效改變，時任歐空局科學部主任恩斯特・特倫德倫伯格（Ernst Trendelenburg）和新成立的決策機構 — 科學計劃委員會 (SPC) 決定選擇發射飛越哈雷彗星的「喬托號」偵測任務和「依巴谷」天體測量任務，加上 1983 年 3 月選定的國際紫外線探測天文台，成為三項首次用阿麗亞娜空間運載火箭發射的歐洲探索任務，自此使歐洲擁有了自主的發射服務。除此之外 ，缺乏長期科學任務計劃，以及美國宇航局在國際合作的太陽極地任務（後命名為「尤利西斯號」）預算挫折，刺激了長期科學計劃的發展。通過該計劃，歐空局可在更長時間內獨立於其他機構和組織持續地規劃任務。歐空局科學理事會的決策和諮詢結構在該計劃成立之前就立即發生了改革。20 世紀 70 年代，歐空局科學諮詢委員會（SAC）接替了發射計劃管理諮詢委員會 (LPAC)，就所有科學問題向總幹事提供諮詢，天文工作組（AWG）和太陽系工作組（SSWG）也直接向總幹事報告。20 世紀 80 年代初，歐空局科學諮詢委員會（SAC）被空間科學諮詢委員會 (SSAC) 取代，後者的任務是向科學部主任報告天文學工作組和太陽系工作組的發展情況。此外，前歐空局科學諮詢委員會主席羅傑 - 莫里斯・博內特（Roger-Maurice Bonnet）於 1983 年 5 月接替特倫德倫伯格擔任科學部主任。
### 地平線 2000
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欧洲航天局科学计划 | 1983 年 11 月和 12 月，歐空局根據博內特於 1983 年底向科學計劃委員會提交的社會驅動計劃的構想，首次向歐洲科學界公開徵集任務提案。這次徵集共產生了 68 項提案，其中 30 項屬於天文學領域，34 項為太陽物理學領域，還有 4 項其它領域的概念。並從空間科學諮詢委員會、歐洲核子研究中心、歐洲科學基金會、歐洲南方天文台和國際天文學聯合會等機構召集相關成員，組建了一個由時任荷蘭空間研究所主管約翰・布雷克 (Johan-Bulek) 領導的臨時「調查委員會」，專門審查提交的提案。在整個 1984 年初，調查委員會為一系列任務制定了計劃，這些任務劃分為三類 — 實施時間較長，需要兩年預算的「奠基石」級任務；需一個年度預算的中型任務和只花費半年度預算的小型任務。1984 年，科學計劃的預算為每年 1.3 億歐洲貨幣單位，並建議在 1991 年之前每年增加 7%，屆時預算將固定為每年 2 億歐洲貨幣單位。中型和小型類項目稍後將合併為一個中型類項目，代表花費一半預算的任務。這一類別在內部被稱為「藍色任務」，以它們在公開計劃圖表中被藍框表示而命名。該計劃最初三項奠基石項目都各自針對某一特定的科學領域，參選提案需致力於填補這一領域，而中型任務的目標則有待於與任務提案一起進行競選。選定的奠基石項目是彗星採樣返回任務、X 射線光譜學任務和亞毫米天文學任務。因財務和技術不足而未選定的基礎目標，但調查委員會稱具有地平線 2000 計劃外機會的項目有太陽探測器、火星探測車和二維干涉測量任務。 | [
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欧洲航天局科学计划 | 1984 年 6 月，調查委員會在威尼斯聖喬治・馬焦雷島召開了最後一次會議，會上向時任歐空局總幹事埃里克・奎斯特加德（Erik Quistgaard）和歐洲科學界主要成員提交了「地平線 2000」計劃。該計劃的廣義目標是擴大科學知識，將歐洲建立為空間科學的發展中心，為歐洲科學界提供機會，並促進太空技術產業的創新。會議通過了太陽系工作組提出的第四號奠基石項目 — 日地科學計劃（STSP），該計劃由太陽和日光層觀測站和星團衛星提案組成，成為地平線 2000 計劃下首批選中發射的任務。1985 年，奎斯特加德在羅馬舉行的部長理事會上提出了地平線 2000 計劃並獲得批准。到 1989 年，預算每年只增加 5%，而不是要求到 1991 年達到的 7%，這隻夠為地平線 2000 目標提供約一半的資金。但 1990 年海牙部長理事會批准將 5% 的年增長率延長至 1994 年，這使得所有地平線 2000 任務都能獲得全額資金。
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欧洲航天局科学计划 | 1985 年 6 月，歐空局在靈比舉辦的一次研究會上，X 射線多鏡面任務（XMM）被列為 X 射線光譜學奠基石任務，該任務是一台由 12 架低能望遠鏡和 7 架高能望遠鏡組成的空間天文台。但受客觀條件限制，到 1987 年，任務有效載荷整體削減至 7 台望遠鏡。雖然歐洲 X 射線天文台衛星（EXOSAT）的成功啟發了任務規劃者通過將太空飛行器發射到高偏心軌道來提高任務觀測效率，可使太空飛行器的有效載荷減少到最終設計的三架大型望遠鏡 — 每架望遠鏡的反射面積為 1500 厘米 2。到 1986 年，預計第四號奠基石項目 — 日地科學計劃的成本將超出所分配的 4 億歐洲貨幣單位預算。在 1986 年 2 月的一次會議上，科學計劃委員會被告知有可能取消該奠基石項目，以便轉為在太陽和日光層天文台、星簇衛星和克卜勒火星軌道器提案中選擇一項中型任務，這在空間科學諮詢委員會成員中得到了普遍的支持。而一個月前發生的挑戰者號太空梭災難對該計劃產生了極大的負面影響，因為太陽和日光層天文台原計劃搭載在太空梭上發射升空。儘管如此，通過簡化太陽和日光層天文台，以及將星團衛星減少為三顆，太陽系工作組、空間科學諮詢委員會和科學計劃委員會還是堅持了對日地科學計劃項目的承諾。1986 年 10 月與美國宇航局達成的一項合作協議，進一步降低了任務的成本 — 美國宇航局將提供測試、發射服務和衛星操作，並提供各類科學儀器，同時取消自已的赤道任務，轉而使用搭載了美國探測儀器的第四顆星簇衛星。
第一次中級飛行任務是 1982 年歐空局在地平線 2000 計劃前，從提交的提案中選出的，一艘由一組美國和歐洲科學家提出的搭載在美國卡西尼號太空飛行器上的土衛六探測器，並與美國 - 歐洲聯合的萊曼紫外線天文學衛星和魁煞甚長基線干涉天文台一起成為最終入圍者。歐洲 - 蘇聯聯合的維斯塔小行星多次飛越任務和格拉斯普伽馬射線天文台也參加了評選，但被天文學工作組和太陽系工作組拒絕。挑戰者號失事導致的預算削減迫使美國宇航局撤消了對萊曼和魁煞項目的支持。1988 年 11 月，科學計劃委員會選擇了土衛六探測器，根據會上瑞士天文學家的建議，該探測器更名為惠更斯號，以紀念在 1655 年發現了土衛六的克里斯蒂安・惠更斯。 | [
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欧洲航天局科学计划 | 在 1989 年 6 月的第二次中級任務競選中，美國和歐洲機構聯合提出了國際伽瑪射線天體物理實驗室，一台結合了格拉斯普與美國核天體物理探測器（NAE）功能的伽馬射線天文台。美國核天體物理探測器在當年美國宇航局探索者計劃評選中落敗，但美國宇航局支持這一提案，俄羅斯科學院後來曾提出用質子運載火箭發射它，以換取對該探測器的觀測時間。儘管很擔心美國宇航局對該任務的承諾及資金來源，但在 1993 年 6 月，科學計劃委員會還是選擇了國際伽瑪射線天體物理實驗室項目，並由美國宇航局提供深空網絡服務和一台光譜儀。作為回應，美國宇航局不通過競選，直接將國際伽瑪射線天體物理實驗室項目列為探索者計劃任務。對於這一點，加上對為此次任務所設計光譜儀靈敏度的擔憂，在美國宇航局的諮詢機構中引起了爭議。1994 年 9 月，歐空局和美國航天局以缺乏財政支持為由，決定終止美國航天局在光譜儀方面的合作，法國國家太空研究中心迅速承擔了財政負擔，並主導了新光譜儀的設計和製造。
1993 年 11 月，羅塞塔號和「福斯特號」入選為第三和第四次奠基石任務，後一任務最終被重命名為赫歇爾空間天文台。1996 年 7 月，宇宙背景輻射各向異性衛星和背景各向異性測量（COBRAS/SAMBA）後改名為普朗克衛星，被選為第三次中型任務。截至 2016 年 12 月，地平線 2000 任務中尚有四項（包括三項奠基石任務和一個中型任務）仍在運行。
### 地平線 2000+
「地平線 2000+」是地平線 2000 計劃的延續，為上世紀 90 年代中期制定的 1995 年至 2015 年期間執行的任務計劃。它包括了另外兩項奠基石級任務，即 2013 年發射的星際測繪任務蓋亞號和 2018 年發射的貝皮可倫坡號水星探測任務，以及將於 2015 年發射，以測試未來雷射干涉空間天線（莉莎號）的技術演示衛星雷射干涉空間天線開路者號，
「地平線 2000」和「地平線 2000+」計劃下的所有任務都取得了成功，只有 1996 年發射的星簇衛星因火箭爆炸而被毀，但在 2000 年成功建造並發射了替代的星簇 2 號衛星。
### 宇宙願景 | [
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欧洲航天局科学计划 | 宇宙願景 2015-2025 是歐空局空間科學任務規劃的當前長期計劃，最初的構想和概念徵集發起於 2004 年，隨後在巴黎舉辦了一次研討會，探討在天文學和天體物理學、太陽系探索和基礎物理學等更廣泛領域內更全面地定義宇宙願景的主題。到 2006 年初，圍繞以下 4 項關鍵問題制定了 10 年計劃：
* 行星形成和生命出現的條件為何？
* 太陽系如何運行？
* 宇宙基本物理定律為何？
* 宇宙如何起源以及由何組成？
2007 年 3 月，正式發布了任務構想徵集，共收到 19 項天體物理學、12 項基礎物理學和 19 項太陽系任務建議。
#### 大型
大型（L 級）飛行任務最初打算與其他合作夥伴聯合實施，歐空局設定的費用不超過 9 億歐元。然而，在 2011 年 4 月，很明顯，美國的預算壓力意味著在第一項大型任務上不太可能與美國宇航局取得合作。因此，接下來的選擇被迫推遲，任務範圍重新確定為由歐空局主導，國際參與有限。宇宙願景計劃共選擇了三項大型任務，分別為：計劃於 2023 年發射的木星和木衛三軌道飛行器「果汁號」；2034 年發射 X 射線天文台「雅典娜號」以及 2037 年發射的太空引力波觀測台「莉莎號」。
#### 中型
中級（M 級）任務是相對獨立的項目，成本上限約為 5 億歐元。前兩項中級任務：近距離太陽觀測的太陽物理學軌道飛行器任務和旨在研究暗能量及暗物質的「歐幾里德」可見光 - 近紅外太空望遠鏡任務於 2011 年 10 月選出。2014 年 2 月 19 日，尋找系外行星並測量恆星振盪的「柏拉圖號」一舉擊敗回聲號 (EChO)、閣樓號 (LOFT)、馬可波羅 - R (MarcoPolo-R) 和時空探索者與量子等效原理空間測試 (STE-QUEST) 等空間探測計劃，成為第三項中級任務。2015 年 3 月，在對第四次中級飛行任務提案作初步篩選後，6 月 4 日公布了三項入選進一步研究的候選任務名單：雷神號電漿體天文台、西佩號 X 射線天文台和系外行星大氣紅外遙感大調查衛星愛麗兒號。最終，通過觀測附近系外行星凌日來確定其化學成分和物理條件的愛麗兒號空間天文台，在 2018 年 3 月 20 日被宣布中選第四項中級任務。第五項中級任務角逐於 2021 年 6 月結束，獲勝的展望號金星軌道器將在 2031 發射升空。斯皮卡號遠紅外天文台和忒修斯號伽馬射線望遠鏡是此次參選中的另外兩項提案。
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欧洲航天局科学计划 | 歐空局的小型飛行任務（S 級）預算不超過 5000 萬歐元，2012 年 3 月發布了第一次任務提案徵集公告，獲選的提案需要在 2017 年前完成發射準備，此次征告收到約 70 份意向書。基奧普斯號是一項通過光度測量搜索系外行星的任務，2012 年 10 月被選為第一項 S 級任務，將於 2019 年秋季發射；2015 年 6 月，歐空局與中國科學院聯合開展的地球磁層圈和太陽風交互作用研究項目 — 微笑任務，從 13 項競選提案中脫穎而出，
成為第二項 S 級任務，截至 2021 年 4 月，微笑計劃將定於 2024 年 11 月發射。
#### 快速型
在 2018 年 5 月 16 日的歐空局科學計劃委員會 (SPC) 研討會上，提出了創建一系列特殊機會的快速級（F 級）任務。 從第四號中級任務開始，這些快速級任務將與每次的中級任務一起聯合發射，並將側重於「創新實施」，以擴大任務涵蓋的科學主題範圍。但將快速級任務納入宇宙願景計劃需要增加科學預算。快速級任務從選擇到發射必須在十年時間之內，且重量不超過 1000 公斤。2019 年 6 月，選出了首個快速級任務 - 彗星攔截器。
#### 機會任務
有時，歐空局會資助其它航天機構主持的太空任務。機會任務能讓歐空局科學界以相對較低的成本參與合作夥伴主導的任務，設定的成本上限為 5000 萬歐元。歐空局的機會任務包括對日出號、虹膜號、顯微鏡號、普羅巴 3 號、X 射線成像和光譜學任務（XRISM）、火星太空生物、愛因斯坦探針衛星和火星衛星探測器的資助。對日本宇宙航空研究開發機構的斯皮卡號（宇宙和天體物理學太空紅外望遠鏡）任務的資助雖被評估為宇宙願景中的機會任務，但它不再在該框架內考慮，現已被列為第五號中級任務提案之一。
### 遠航 2050
歐空局科學計劃的下一個十年期規劃是「遠航 2050」，它將涵蓋 2035 年至 2050 年的空間科學任務。該計劃起始於 2018 年 12 月任命的高級委員會和 2019 年 3 月發表的白皮書公告。。
該計劃目前預計有三項大型任務和六到七項中型任務，以及小型和機遇任務。2021 年 6 月 11 日，高級委員會發布了遠航 2050 計劃，並為接下來的三大任務推薦了以下科學主題：
* 巨行星的衛星 — 探測擁有海洋的氣體巨行星衛星的任務。
* 從溫和的系外行星到銀河系 — 描述系外行星特徵或調查銀河系形成歷史的任務。 | [
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] |
欧洲航天局科学计划 | * 早期宇宙的新物理探測器 — 通過宇宙微波背景、引力波或其他基本天體物理現象來研究早期宇宙的任務。
## 任務
**地平線 2000**
* 奠基石 1 – **太陽和日光層觀測站**，1995 年 12 月發射，運行中 – 歐空局 - 美國宇航局為空間天氣預報提供實時數據的聯合太陽觀測任務。
* 奠基石 1 – **星簇衛星**，1996 年 6 月發射，失敗 – 使用四顆相同衛星研究行星磁層的地球觀測任務，發射失敗。
* 重新發射– **星簇 2 號衛星**，2000 年 7 月和 8 月發射，運行中 – 成功完成任務接替。
* 奠基石 2 – **X 射線多鏡面任務**，1999 年 12 月發射，運行中 –一架研究宇宙中所有範圍 X 射線源的 X 射線空間望遠鏡。
* 奠基石 3 – **羅塞塔號**，2004 年 3 月發射，已完成 – 丘留莫夫－格拉西緬科彗星軌道器任務，研究彗星及其演化。
* 奠基石 4 – **赫歇爾空間天文台**，2009 年 5 月發射，已完成 –通用天文學紅外太空觀測任務。
* 中 1 號 – **惠更斯號**，1997 年 10 月發射，已完成 – 「卡西尼 - 惠更斯號」任務的土衛六著陸器組件，首次登陸外太陽系星球。
* 中 2 號 – **國際伽瑪射線天體物理實驗室**，2002 年 10 月發射，運行中 – 伽馬射線空間天文台，也可觀測 X 射線和可見光波長。
* 中 3 號 – **普朗克衛星**，2009 年 5 月發射，已完成 – 繪製宇宙微波背景及其各向異性的宇宙學任務。
**地平線 2000 +**
* 第 1 次任務– **蓋亞號**，2013 年 12 月發射，運行中 – 天體測量任務，測量銀河系中超過十億個天體的位置及距離。
* 第 2 次任務– **雷射干涉空間天線開路者號**，2015 年 12 月發射，已完成 –演示宇宙願景莉莎號引力波觀測站技術的任務。
* 第 3 次任務– **貝皮可倫坡號**，2018 年 10 月發射，運行中 – 歐空局 - 日本宇宙航空研究開發機構聯合水星偵測任務，使用兩艘分別運行的獨特探測器。
**宇宙願景**
* 大 1 號 – **果汁號**，2023 年 8 月發射，2031 年進入軌道。未來 – 木星軌道飛行器任務，重點研究四顆伽利略衛星木衛二、木衛三和木衛四。 | [
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欧洲航天局科学计划 | * 大 2 號 – **雅典娜號**，2034 年發射，未來 – X 射線空間觀測任務，用於接替「X 射線多鏡面任務」望遠鏡。
* 大 3 號 – **莉莎號**，2037 年發射，未來 – 第一次專門的引力波太空觀測任務。
* 中 1 號 – **太陽軌道器**，2020 年發射，運行中 –太陽觀測任務，旨在在 0.28 天文單位的近日點對太陽進行遙感和原位研究。
* 中 2 號 – **歐幾里得衛星**，2023 年初發射，未來 – 重點聚焦暗物質和暗能量的可見光和近紅外太空觀測任務。
* 中 3 號 – **柏拉圖號**，2026 年發射，未來 – 類似太空觀測任務的凌日系外行星巡天衛星，旨在發現和觀測系外行星。
* 中 4 號 – **愛麗兒號**，2029 年發射，未來 – 研究已知系外行星大氣層的紅外太空天文台任務。
* 中 5 號 – **展望號**，2031 年發射，未來 – 金星測繪軌道飛行器任務。
* 小 1 號 – **太陽和日光層觀測站**，2019 年 12 月發射，運行中 –重點研究已知系外行星的空間望遠鏡任務。
* 小 2 號 – **微笑計劃**，2024 年 11 月發射，未來 –歐空局與中國科學院聯合執行的地球觀測任務，研究地球磁層與太陽風之間的交互作用。
* 快 1 號 – **彗星攔截器**，2029 年發射，未來。
### 時間表
## 另請查看
* 歐空局計劃和任務列表
* 太陽系探測器列表
* 空間望遠鏡列表
## 參考文獻
**注釋**
**資料來源**
**引文**
## 外部連結
* 歐空局科學計劃規劃周期 （頁面存檔備份，存於網際網路檔案館）
* 空間科學 （頁面存檔備份，存於網際網路檔案館） | [
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时空探索者与量子等效原理空间测试 | **時空探索者和量子等效原理空間測試**（**STE-QUEST**）是一項提議的衛星任務，旨在高精度測試愛因斯坦等效原理並探索宇宙中新的基礎成分及相互作用。它應該包含一台原子干涉儀。
2011 年，該提案入選歐洲航天局宇宙願景計劃中五個候選項目之一，進入初始評估階段。
2014 年 2 月 19 日，柏拉圖號任務從包括時空探索者和量子等效原理空間測試在內的其他候選提案中勝出。
## 參考資料
## 外部連結
* 宇宙願景網站上的描述 （頁面存檔備份，存於網際網路檔案館）
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] |
系外行星特征观测台 | **系外行星特徵觀測台**（Exoplanet Characterisation Observatory ）簡稱**回聲號**（EChO）是一架擬議的空間望遠鏡項目，作為歐洲航天局宇宙願景參選提案之一，與其他四項提案一起角逐中 3 號探測任務。2014 年 2 月 19 日，柏拉圖號任務力壓包括回聲號在內的其他候選項目，一舉成為該計劃中第三項中型航天任務。
回聲號將是第一項專門調查系外行星大氣的任務，它將以太陽系作為背景來研究這些行星對生命的適宜性。旨在通過提供高解析度、多波長光譜的觀測，以測量已知系外行星中代表性星球的大氣成分、溫度和反照率，並建立它們的內部結構約束模型，從而提升對行星形成和演化的理解。它將環繞拉格朗日點 2 運行，距離地球 150 萬公里，朝著背向太陽的方向運行。
## 斯皮卡回聲設備
繼柏拉圖號中選中 3 號任務後，出現了一項關於在日歐聯合的斯皮卡號遠紅外望遠鏡上添加回聲號儀器的提議。提議的「斯皮卡上的回聲儀器」（SPEChO）是一台光譜儀，其光波長範圍為 5 到 20 微米，將利用凌日光譜觀察系外行星大氣。斯皮卡回聲設備將使斯皮卡號能進行專門的系外行星科學研究，具有恢復回聲號最初科學目標的潛力，該設備也可能對星系探測非常有用。然而，斯皮卡回聲設備的能力與詹姆斯・韋伯空間望遠鏡的中紅外儀重疊，因此，斯皮卡回聲設備相對於其他儀器的優勢已成為是否安裝在斯皮卡號上的一個焦點。
## 中 3 號任務的其他候選項目
* 馬可波羅 - R（MarcoPolo-R）
* 閣樓號（LOFT）
* 時空探索者與量子等效原理空間測試（STE-QUEST）
## 參考文獻
## 外部連結
* 歐空局網站有關回聲號的介紹 （頁面存檔備份，存於網際網路檔案館）
* 歐空局公告文章 （頁面存檔備份，存於網際網路檔案館） | [
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光子气 | 在物理學中，**光子氣體**是一種類似氣體的光子集合，它具有許多與傳統氣體（如氫或氖）相同的特性 —— 包括壓力、溫度和熵。處於平衡狀態的光子氣體最常見的例子是黑體輻射。
光子是玻色子的一種，玻色子遵循玻色 - 愛因斯坦統計並具有整數自旋。一個玻色子氣體由三個態函數所給定，如溫度，體積和粒子數。然而，對於黑體，能量分佈是由光子與物質（通常是容器壁）的交互作用建立的。在這種交互作用中，光子的數量不守恆。因此，黑體光子氣體在熱力學平衡時的化學勢為零。描述黑體狀態所需的狀態變量的數量因此從三個減少到兩個（例如溫度和體積）。
## 黑體光子氣體的熱力學
在大質量粒子的古典理想氣體中，粒子的能量遵守馬克士威 - 波耳茲曼分佈。這種分佈是因為粒子相互碰撞時交換能量（和動量）而建立的。在光子氣體中，也會有平衡分佈，但光子不會相互碰撞（除非在非常極端的條件下，參見雙光子物理學），因此必須通過其他方式建立分佈。建立分佈的最常見方式是通過光子與物質的交互作用。如果光子被系統的牆壁吸收和發射，並且牆壁處於特定溫度，那麼光子的平衡分佈會是該溫度下的黑體輻射分佈。
玻色氣體和具有黑體分佈的光子氣體之間的一個非常重要的區別是系統中的光子數量不守恆。光子可能與壁中的電子碰撞而消失，並將電子激發到更高的能量狀態。此電子可能會經由一系列步驟回到低能階，每個步驟都會釋放一顆光子回到光子氣體中。雖然發射光子的光子能量總和與吸收光子相同，但發射光子的數量會有所不同。可以證明，由於缺乏對系統中光子數量的限制，對於黑體輻射，光子的化學勢必須為零。
可以由量子力學推導出黑體光子氣體的熱力學。推導得出頻譜能量密度 _u_ （每單位頻率間隔與每單位體積的能量），由普朗克定律給出：
: $u(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}~{\frac {1}{e^{\frac {h\nu }{kT}}-1}}$
其中 _h_ 是普朗克常數，_c_ 是光速，_ν_ 是頻率，_k_ 是波茲曼常數，_T_ 是溫度。
對頻率積分並乘以體積 _V_ 可得出光子氣體的內能：
: $U=\left({\frac {8\pi ^{5}k^{4}}{15(hc)^{3}}}\right)VT^{4}$
以及光子數 _N_ ：
: $N=\left({\frac {16\pi k^{3}\zeta (3)}{(hc)^{3}}}\right)VT^{3}$ | [
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光子气 | 其中 $\zeta (n)$ 是黎曼 ζ 函數。對於特定溫度，粒子數 _N_ 以固定方式隨體積變化。
對於相對論量子氣體（光子本質上就是相對論性的）的狀態方程式由下式給出
: $U=3PV$
那麼我們可以結合上面的公式來產生一個看起來很像理想氣體的狀態方程式：
: $PV={\frac {\zeta (4)}{\zeta (3)}}NkT\approx 0.9\,NkT$
下表總結了黑體光子氣體的熱力學狀態函數。其中壓力可以寫成 $P=bT^{4}$ ，與體積無關（ $b$ 是一個常數）。
## 等溫轉換
作為光子氣體的熱力學過程的一個例子，考慮一個帶有可移動活塞的氣缸。圓柱體的內壁是「黑體」，使光子的溫度可以保持在特定溫度。這意味著圓柱體內的空間將包含黑體分佈的光子氣體。這種氣體在沒有由外部引入光子的情況下存在 —— 牆壁將提供光子。假設活塞一直被推入氣缸，因此體積非常小。體積內的光子氣體會施加壓力在活塞上，使其向外移動。為了使轉換過程等溫，必須對活塞施加幾乎相同值的反作用力，使活塞的運動為非常慢。這個力將等於壓力乘以活塞的截面積（ _A_ ) 。過程在恆溫下進行，直到光子氣體的體積達到 _V0_ 。將此力對距離 ( _x_ ) 積分，可得對光子氣體所做的總功。
: $W=-\int _{0}^{x_{0}}P(A\mathrm {d} x)$
其中使用 _V = Ax_。定義
: $b={\frac {8\pi ^{5}k^{4}}{15c^{3}h^{3}}}$
壓力是
: $P(x)={\frac {bT^{4}}{3}}$
所做的總功為
: $W=-{\frac {bT^{4}Ax_{0}}{3}}=-{\frac {bT^{4}V_{0}}{3}}$
所需要的熱量為
: $Q=U-W=H_{0}$
其中 _H0_ 是過程結束時的焓。可以看出，焓是產生光子氣體所需的能量。
## 參見
* 盒中氣體 —— 理想氣體分佈函數的推導
* 波色氣體
* 費米氣體
* 普朗克黑體輻射定律 —— 光子能量隨頻率或波長的分佈
* 斯特凡－波茲曼定律 —— 黑體輻射通量
* 輻射壓
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光子气 | * Baierlein, Ralph. The elusive chemical potential (PDF). American Journal of Physics. April 2001, **69** (4): 423–434 [2021-12-20]. Bibcode:2001AmJPh..69..423B. doi:10.1119/1.1336839. （原始內容存檔 (PDF) 於 2021-04-28）.
* Herrmann, F.; Würfel, P. Light with nonzero chemical potential (PDF). American Journal of Physics. August 2005, **73** (8): 717–723 [2012-06-29]. Bibcode:2005AmJPh..73..717H. doi:10.1119/1.1904623. （原始內容 (PDF) 存檔於 2016-03-04）.
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柏拉图号 | **行星凌日和恆星振盪觀測**（PLAnetary Transits and Oscillations of stars），簡稱**柏拉圖號**（PLATO）是一架由歐洲航天局開發，並將於 2026 年發射的空間望遠鏡。任務目標是在多達一百萬顆恆星中搜尋行星凌日，並發現和描述環繞黃矮星（如我們的太陽、次巨星和紅矮星周圍的石質太陽系外行星。該任務重點尋找位於類太陽恆星周圍適居帶中的類地行星，在那裡可能會存在液態水。它是歐空局宇宙願景計劃中的第三項中級任務，以古希臘著名哲學家柏拉圖之名命名。該任務的第二目標是研究恆星振盪或重大活動，以測量恆星的質量及演化，並實現對行星宿主恆星年齡等特徵的精確描述。
## 歷史
2007 年，一組科學家響應歐空局 2015-2025 年宇宙願景計劃徵召，首次向歐洲航天局（歐空局）提出了柏拉圖號任務概念。2009 年完成項目初評，2010 年 5 月進入規劃階段。在 2010 年 7 月發布任務徵集後，歐空局於 2011 年 2 月為定於 2024 年發射的中級任務（M3 任務）選擇了四項候選提案。2014 年 2 月 19 日，柏拉圖號被宣布為宇宙願景計劃即將實施的第三項中級探索任務，經評審的其他參選概念還包回聲號、閣樓號、馬可波羅 - R 和時空探索者與量子等效原理空間測試等四個候選項目。
2015 年 1 月，歐空局選擇泰雷茲阿萊尼亞宇航、空客國防航天和 OHB System AG （頁面存檔備份，存於網際網路檔案館）三家公司對柏拉圖號系統及子系統展開並行的 B1 階段研究，2016 年可行性研究完成，2017 年 6 月 20 日，歐空局科學計劃正式批准了柏拉圖號項目，這意味著該任務已從藍圖進入到建造階段。在接下來的數月中，將向工業界招標建造太空飛行器平台。
負責柏拉圖任務探測設備和對探索活動作出重要貢獻的聯合方有海克・勞爾 (Heike Rauer) 教授領導的德國航空航天中心 (DLR) 行星研究所；一支來自義大利、瑞士和瑞典的國際團隊完成了望遠鏡光學設備的設計並由義大利國家天文物理研究所帕多瓦天文台的羅伯托・拉加佐尼 (Roberto Ragazzoni) 進行了調試；義大利航天局、瑞士航天局和瑞典國立宇宙委員會對望遠鏡光學單元的開發給予了資助。 | [
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] |
柏拉图号 | 柏拉圖是「行星凌日和恆星振盪觀測」的英文縮略詞，但也是古希臘一位哲學家的名字：柏拉圖（公元前 428-348），曾努力尋求一則計算行星（漂泊的恆星）軌道並能符合哲學家「一致性」和「規律性」所需的定律。
## 目標
該任務目標是探測類地系外行星及太陽型恆星宜居帶，描述確定其宜居性所需的整體特性。為實現這一目標，任務設定了以下目標：
* 發現並描述附近的眾多系外行星系統，精確測定 3% 行星的半徑，10% 恆星的年齡和 10% 行星的質量（後者結合地基徑向速度測量）；
* 探測並描述環太陽型恆星周圍宜居帶中地球般大小的行星和超級地球的特徵；
* 發現並描述眾多系外行星系統，以研究它們的典型結構，以及對宿主恆星和環境的依賴性；
* 測量恆星振盪以研究恆星的內部結構及其隨時間的演化；
* 發現良好的光譜測量目標，以研究系外行星大氣。
柏拉圖號與對流旋轉和行星凌日任務和克卜勒太空望遠鏡的不同之處在於，它將研究相對明亮的恆星（4 到 11 星等），從而能夠更準確地測定行星參數，並通過後續使用地基望遠鏡徑向速度測量，更容易確認行星並測量它們的質量。它的停留時間也將比美國宇航局的凌日系外行星巡天衛星任務要長，使得它對周期較長的行星更為有效。
## 光學設備
柏拉圖的有效載荷基於多望遠鏡方法，總共涉及 26 台相機：分成 4 組的 24 台「普通」相機和 2 台用於拍攝明亮恆星的「快速」相機。24 台「普通」相機以 25 秒的讀出頻率工作，監視亮度低於視星等 8 的恆星。而兩台「快速」相機則以 2.5 秒的頻率工作，以觀察 4 到 8 星等的恆星。相機為使用六鏡頭的折射望遠鏡，每台相機都帶有一個 120 毫米直徑的鏡頭，視場寬度達 1100 度 2。各台相機均配備了由四個 4510 x 4510 像素 CCD 構成的獨立凝視陣列。
24 台「普通攝像機」將排列成四組，每組六台，其視線與 + ZPLM 軸偏移 9.2 度，這種特殊配置將允許測量每個範圍為 2250 度 2 的瞬時視野點，該空間天文台將每年繞平均視線旋轉一圈，對同一天空區域進行連續觀測。
## 發射
該太空觀測台計劃在 2026 年發射到日 - 地拉格朗日點 2。
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] |
柏拉图号 | 每季的測光數據（包括光曲線）和高水平科研結果將在驗證結束後 6 個月，最遲不超過一年內公開發布。數據按季度處理，是因為太空飛行器每旋轉 90 度所需保持的時間。第一季度的觀察數據驗證和通信渠道更新需要半年時間。隨後的各季只需 3 個月。少數恆星（25 萬顆恆星中不超過 2000 顆）將具有專屬狀態，意味著僅柏拉圖任務聯盟成員才能在給定的時間段內訪問這些數據，他們可選擇柏拉圖號前 3 個月內所觀測的各個區域。專屬期限限於地面觀測完成或任務存檔階段結束後（發射日期 + 7.5 年）的 6 個月，以先到者為準。
## 另請查看
* 宇宙願景，歐空局計劃 (2015 年 - 2025 年)
* 空間望遠鏡列表
* 系外行星特性探測衛星，一架 2019 年的測定已知太陽系外行星大小的歐洲太空望遠鏡。
* 凌日系外行星巡天衛星 (TESS) 美國宇航局，於 2018 年發射。
## 參考文獻
## 外部連結
* 官方網站
* 官方畫廊 （頁面存檔備份，存於網際網路檔案館）
* 柏拉圖 2.0 任務 （頁面存檔備份，存於網際網路檔案館）科學論文
* 柏拉圖號能為系外行星天文學做些什麼？ （頁面存檔備份，存於網際網路檔案館） | [
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] |
大型X射线计时观测台 | **大型 X 射線計時觀測台**（Large Observatory for X-ray Timing)，簡稱**閣樓號**(LOFT) 是歐洲太空總署一項擬議的太空任務，最初定於 2022 年左右發射，現提議在 2025 年左右發射。該任務將致力於研究中子星、黑洞和其他緻密星的快速 X 射線變化。閣樓號項目得到了大範圍的國際合作支持，研究人員遍布義大利、瑞士、德國、丹麥、英國、希臘、愛爾蘭、荷蘭、波蘭、捷克共和國、西班牙等大部分歐洲國家，以及巴西、加拿大、以色列、美國和土耳其的幫助。斯隆荷蘭空間研究所則承擔了該項目的首席研究員任務。
該任務被提交歐空局宇宙願景中三號提案評選，並與其他三項任務一起被選中進入初始評估階段。
2014 年 2 月 19 日，柏拉圖號任務擊敗包括閣樓號在內的其它候選項目而一舉中選。儘管如此，閣樓號已提交宇宙願景中計劃將於 2025 年發射的中 4 號任務徵選。
## 任務架構
大型 X 射線計時任務觀測台由兩部儀器組成。
* 大面積探測器（LAD），一台 10 米 2 準直 X 射線探測器；
* 寬景深監視器（WFM），一台編碼掩模廣角 X 射線監視器。
大面積探測器（LAD）在 2-50 千電子伏範圍內的有效面積為～10 米 2（比當前的星載 X 射線探測器，例如羅西 X 射線計時探測器大一個數量級），但仍然適合傳統平台和中小型運載火箭，這要歸功於大面積矽漂移探測器（SDD）的整體設計。
廣角監視器（WFM）是一台掩碼編碼的 X 射線監視器，具有大視場（隨時觀察大面積探測器可探測的約 50% 的天空），它也基於矽漂移探測器技術，其工作能量範圍與大面積探測器相同，即 2-50 千電子伏。
## 目標
閣樓號的主要科學目標是：
* 中子星結構及其狀態方程的測定；
* 強引力場中的物理學研究，如黑洞周圍的吸積盤中；
* 黑洞質量和自旋的直接測量。
除這些課題外，閣樓號還將研究各種天體物理源的 X 射線光譜和變異性，例如磁星、活動星系核、激變變星、X 射線瞬變和伽馬射線暴。
閣樓號的獨特功能，使該儀器肯定能夠在廣泛的天體物理研究領域中提供新的突破。
## 另請查看
* 回聲號
* 馬可波羅 - R
* 時空探索者與量子等效原理空間測試
## 參考文獻
## 外部前兩日
* 為歐空局宇宙願景計劃中下一中級任務挑選的四項候選方案 （頁面存檔備份，存於網際網路檔案館）
* 官方網站 | [
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物理哲学 | 在哲學中，**物理哲學**是關於當代物理中的概念性與解釋性的問題。其中許多問題與理論物理學家的研究領域重疊。物理哲學可大致分為以下三個領域：
* 對量子力學的解釋：主要關於如何形式化地看待量子測量以及該理論對現實的描述。
* 空間與時間的本質：時空是物質，或僅僅是一種關係？同時性是常規的或者相對的？暫時的不對稱性是否可以單純還原為熱力學不對稱性？
* 理論間的關係：不同物理理論之間的關係，如熱力學與統計力學。這領域與科學還原論重疊。
## 時空哲學
時間與空間（時空）的本質是物理哲學的重要主題。
### 時間
時間常被視為一種量綱（無法以其他物理量定義的物理量），因為時間像是一個根本上基礎的概念，無法由更簡單的事物所定義。然而，有如圈量子重力的理論則聲稱時時空是湧現的。圈量子重力的創始人之一 Carlo Rovelli 曾說：「再也不是時空上的場，而只是場上的場。」
時間的標準長度（秒）被定義為銫原子 9,192,631,770 次超精細結構轉移的振動的時長（ISO 31-1）。時間為何及如何表現均被上述定義所蘊含。以時間與其他基礎物理量（空間與質量）進行數學操作即可定義如速度、動量、能量以及場等等概念。
牛頓與伽利略，以及二十世紀前的大多數人都認為時間是普世皆相同的。現代的時間概念是基於愛因斯坦的相對論以及閔可夫斯基的時空之上，其中聲稱時間的流逝在不同的參考系下是不同的。時間也能被量子化，理論上最小的時間長度為普朗克時間。愛因斯坦的廣義相對論以及遙遠退後星系的紅移指出宇宙乃至時空本身是在 138 億年前的大爆炸時開始。愛因斯坦的狹義相對論使得那些對「現在」這個概念有特別形而上意義的時間理論不再那麼可靠，因為依賴於參考系的時間概念似乎不允許一個特別的「目前的時間」。
### 時間旅行
某些理論，尤其是狹義以及廣義相對論，暗示了時空的合適的幾何性質，或是在空間中的某些運動，可能會導致穿越至過去或未來。相關概念概念有如封閉類時曲線。
愛因斯坦的狹義相對論認為時間膨脹可以被解釋為時間旅行。理論聲稱，相對於靜止觀察者，時間對於高速移動的物體流逝更慢：一個移動中的時鐘會看上去跑得更慢；隨著時鐘趨於光速，時鐘的指針將會看起來完全靜止。在有名的雙生子佯謬中該效應被更深入探討。這些效應在實驗上是可觀測到的，愛因斯坦理論中的一重要等式被應用在 GPS 衛星以及其他高科技的日常應用。 | [
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物理哲学 | 廣義相對論導致第二種類似的時間旅行。對於遙遠的觀察者，在深重力井內的時鐘會跑的更慢。一個進出過深重力井的時鐘和靜止在觀察者處的時鐘相比，其所經歷過的時間（指針的秒數）更少。
科學界常認為往過去的時間旅行不可能，因為其違背因果律。
例如，若你穿越到過去並殺掉年輕的你（或是你的祖父，見祖父悖論）將會導致什麼？史蒂芬・霍金指出，沒有出現過未來人，表示了時間旅行很可能不存在。這也是費米悖論的一個變種，只是把外星人替換成未來人。
### 空間
空間是物理中基本物理量之一，意味目前沒有已知的更基本物理量可以定義其。所以有如其他基本物理量（如時間與質量），空間是由測量所定義的。目前，標準的空間長度（米）被定義為光在真空中經由 1/299792458 秒穿越的距離（完全精準）。
在經典物理中，空間是三維的歐氏空間，任何位置可以用三個坐標以及時間參數來表示。狹義以及廣義相對論使用四維時空而非三維空間；目前有許多純理論使用高於四維的空間。
## 量子力學哲學
量子力學的詮釋是當代物理哲學的一大焦點。許多哲學工作是在嘗試理解疊加態：
物體在一個時間內不在一個確定的空間，而是同時「在這」也「在那」。如此激進的看法違背許多形上學的常理。許多當代的量子力學哲學都嘗試理解經驗上十分成功的量子力學形式系統如何闡述世界。
### 艾弗雷特的多世界詮釋
艾弗雷特的多世界詮釋是關於對量子系統的波函數的詮釋。其否認波函數坍縮，並認為疊加態應該被直接看做多重世界中物體的位置，而非是變量的不確定性。有時這被當作科學實在論 —— 科學的目的即為直接描述現實 —— 的結果。
艾弗雷特詮釋的一個問題就在在於機率。艾弗雷特詮釋完全是決定論的，而機率在量子力學中有不可磨滅的重要性。
當代的艾弗雷特擁護者表示以某些決策論的證明可以理解遵循玻恩定則的機率性。
物理學家 Roland Omnés 認為實驗上無法分辨艾弗雷特的看法以及傳統的看法，前者認為波函數退相干成不同的各自存在的世界，後者認為波函數退相干後只有一個真正存在的結果。兩種觀點之間的衝突代表了一個巨大的「鴻溝」。「在我們以理論模型重新建構現實時，現實的種種性質都顯現出來了；除了一個性質：事實的獨特性。」
### 不確定性原理 | [
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物理哲学 | 不確定性原理是一個數學關係，表現出對任何成對共軛物理量的同時測量存在精準度的上限，如位置與動量。在算符的形式中，這個極限是變量之對應算符的交換子的值。不確定性原理回答了一個問題：如果圍繞原子核電子是波，如何測量它的位置？當量子力學在發展時，它被視作以波函數連結經典描述與量子描述的一種關係。
在 1927 年五月，在尼爾斯・玻爾處工作的維爾納・海森堡形式化了不確定性原理，為哥本哈根詮釋奠基了基礎。海森堡曾研究保羅・狄拉克與帕斯夸爾・約爾旦的論文，他發現了在測量等式中基本變量的一個問題。他的分析表示不確定性，或不準確度，總是在同時測量一個粒子的位置與動量時升高。海森堡的結論是這些不準確度不是實驗不夠完美的問題，而是自然的基礎性質，也是量子力學中算符的定義所導致的內在數學性質。
在量子力學中，批評者常將哥本哈根詮釋這一詞彙與海森堡的不確定性原理一起使用甚至混用（如愛因斯坦以及物理學家阿爾弗雷德・朗德）, 他們相信決定論並將波爾 - 海森堡的理論看作威脅。在哥本哈根詮釋中，不確定性原理意味著物理宇宙本身不是決定論的，而是機率的結合。舉例，成千上萬個光子穿過干涉狹縫所形成的圖案（概率分布）可使用量子力學計算，但個別光子的準確路徑無法以任何方式預測，即使是理論上無限準確的測量也不行。
## 物理哲學史
### 亞里斯多德物理學
亞里斯多德物理學將宇宙看作球體。物質，由地、水、火、風四種古典元素所組成。以太中的物體如日月、行星以及星星圍繞著宇宙中心旋轉。
運動被定義為在位置的改變，
也就是空間。
### 牛頓力學
亞里斯多德物理中所隱含的，有關物體在空間中移動的假設被牛頓力學中的第一運動定律所替換。
這包括了月球、一顆蘋果；一切物質皆如此，空氣與水、石頭、甚至火焰，沒有東西天生可以一直加速運動。
絕對空間是三維的歐氏空間，無限而沒有中心。
「靜止」意味著隨著時間流逝，在絕對空間中的同一位置。
空間的拓撲與仿射空間都必須允許等速度直線運動；因此時間與空間要都有確定、穩定的維度。.
### 萊布尼茨
戈特弗里德・萊布尼茨，1646 – 1716，是與牛頓同時代的人物。他對當時的靜力學與動力學頗有貢獻，時常與笛卡爾及牛頓有異議。他基於動能與位能創建了一套新的運動動力學，其中認為空間是相對的，而非牛頓認為的絕對。一個看出萊布尼茨物理思考已趨成熟的著作是 1695 年的 _Specimen Dynamicum_。 | [
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] |
物理哲学 | 直至次原子粒子與量子力學的出現，萊布尼茨認為自然無法完全被還原為靜力學與動力學的猜想都被當作不合理。
早在愛因斯坦前，他就反對牛頓，認為時空不是絕對，而是相對的：
「我的想法是，我說過很多遍，我把空間當作相對的，就如時間。前者作為一種共存，而後者是一種接續。」
### 愛因斯坦有關物理哲學重要性的言論
愛因斯坦對於自己理論的哲學意義很感興趣，他寫道：
> 「我完全同意你認為方法論、科學史以及科學哲學很重要並具有教育意義。如今很多人 —— 甚至專業科學家 —— 都讓我覺得像是他們看過千萬棵樹，但沒見過森林。有歷史、哲學背景相關知識就可以從大部分科學家都持有的，當代科學界的偏見中獨立出來。我認為這種獨立就是一個單純的工匠或專家與真正追求真理的人的區別。」Einstein. letter to Robert A. Thornton, 7 December 1944. _EA_ 61–574.
>
>
又言道：
> 「為什麼一個個富有天賦的自然科學家會開始擔憂知識論的問題？難道他的專長不能用在更有價值的地方嗎？有些同事會說，我也感覺到有些人是如此，他們都有這樣的擔憂。我無法感同身受…… 那些被證實很有用的概念駕馭了我們，使得我們都忘了它們其實也出身平凡，並不是不可改變的。就這樣，他們變成了『思考必要物』，『先驗概念』等等。」
>
>
> 「這樣的錯誤總是使得科學進步的道路艱澀難行。正因如此，我們應當做到能夠熟練分析這些長存的概念並展現出什麼樣的情景下它們才是正當並有用的，以及他們如何從經驗中生成。這樣的話，他們至高無上的權威終將被打破。」Einstein, 1916, "Memorial notice for Ernst Mach," _Physikalische Zeitschrift_ 17: 101–02.
>
>
## 參見
## 參考
## 延伸閱讀
* David Albert, 1994. _Quantum Mechanics and Experience_. Harvard Univ. Press.
* John D. Barrow and Frank J. Tipler, 1986. _The Cosmological Anthropic Principle_. Oxford Univ. Press. | [
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物理哲学 | * Bernard d'Espagnat, 1989. _Reality and the Physicist_. Cambridge Univ. Press. Trans. of _Une incertaine réalité; le monde quantique, la connaissance et la durée_.
* --------, 1995. _Veiled Reality_. Addison-Wesley.
* --------, 2006. _On Physics and Philosophy_. Princeton Univ. Press.
* Roland Omnes, 1994. _The Interpretation of Quantum Mechanics_. Princeton Univ. Press.
* --------, 1999. _Quantum Philosophy_. Princeton Univ. Press.
* Huw Price, 1996. _Time's Arrow and Archimedes's Point_. Oxford Univ. Press.
* Lawrence Sklar, 1992. _Philosophy of Physics_. Westview Press. ISBN 0-8133-0625-6, ISBN 978-0-8133-0625-4
* Victor Stenger, 2000. _Timeless Reality_. Prometheus Books.
* Carl Friedrich von Weizsäcker, 1980. _The Unity of Nature_. Farrar Straus & Giroux.
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] |
物理哲学 | * Encyclopædia Britannica, Philosophy of Physics, David Z. Albert
## 外部連結
* Stanford Encyclopedia of Philosophy:
* "Absolute and Relational Theories of Space and Motion"—Nick Huggett and Carl Hoefer
* "Being and Becoming in Modern Physics"—Steven Savitt
* "Boltzmann's Work in Statistical Physics"—Jos Uffink
* "Conventionality of Simultaneity"—Allen Janis
* "Early Philosophical Interpretations of General Relativity"—Thomas A. Ryckman
* "Everett's Relative-State Formulation of Quantum Mechanics"—Jeffrey A. Barrett
* "Experiments in Physics"—Allan Franklin
* "Holism and Nonseparability in Physics"—Richard Healey
* "Intertheory Relations in Physics"—Robert Batterman
* "Naturalism"—David Papineau
* "Philosophy of Statistical Mechanics"—Lawrence Sklar
* "Physicalism"—Daniel Sojkal
* "Quantum Mechanics"—Jenann Ismael
* "Reichenbach's Common Cause Principle"—Frank Artzenius
* "Structural Realism"—James Ladyman
* "Structuralism in Physics"—Heinz-Juergen Schmidt
* "Supertasks"—JB Manchak and Bryan Roberts | [
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] |
物理哲学 | * "Symmetry and Symmetry Breaking"—Katherine Brading and Elena Castellani
* "Thermodynamic Asymmetry in Time"—Craig Callender
* "Time"—by Ned Markosian
* "Time Machines" —John Earman, Chris Wüthrich, and JB Manchak
* "Uncertainty principle"—Jan Hilgevoord and Jos Uffink
* "The Unity of Science"—Jordi Cat | [
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沃德-高橋恆等式 | 在量子場論中，**沃德 - 高橋恆等式**是一個藉由理論的全域或規範對稱性來聯繫不同關聯方程式的恆等式，其不受重整化影響。
沃德 - 高橋恆等式一開始是由約翰・克萊夫・沃德和高橋康提出，在量子電動力學中用以連結電子的波函數重整化和節點重整化，這保證了在微擾理論的任意階下兩者產生的紫外發散皆會對消。之後也被延伸用於證明在微擾理論中任意階下的戈德斯通定理。
廣義來說，沃德 - 高橋恆等式是量子化版本的諾特定理，將守恆流與連續性對稱作聯繫。在量子場論中這樣的對稱性幾乎都會產生廣義的沃德 - 高橋恆等式，其在量子層級下要求物理量的對稱性。在閱讀如麥可・佩斯金和丹尼爾・施羅德撰寫的課本時，此廣義版本應與原始的沃德 - 高橋恆等式作區分。
以下的討論將使用量子電動力學作例子考慮一阿貝爾理論中的沃德 - 高橋恆等式。在非阿貝爾理論中，如量子色動力學，對應的恆等式被稱作斯拉夫諾夫 - 泰勒恆等式。
## 沃德 - 高橋恆等式
沃德 - 高橋恆等式應用於動量空間中的關聯方程式，其中並無要求全部的外部動量要在殼。令
: : ${\mathcal {M}}(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})=\epsilon _{\mu }(k){\mathcal {M}}^{\mu }(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})$
為一量子電動力學中的關聯方程式（使用愛因斯坦求和約定）。其中包含一帶有動量 $k$ 、極化向量 $\epsilon _{\mu }(k)$ 的外部光子、 $n$ 個動量為 $p_{1}\cdots p_{n}$ 的初始態電子、和 $n$ 個動量為 $q_{1}\cdots q_{n}$ 的末態電子。另外定義 ${\mathcal {M}}_{0}$ 為移除該光子後的關聯方程式，則沃德 - 高橋恆等式給出
: : $k_{\mu }{\mathcal {M}}^{\mu }(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})=e\sum _{i}\left[{\mathcal {M}}_{0}(p_{1}\cdots p_{n};q_{1}\cdots (q_{i}-k)\cdots q_{n})\right.$ | [
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沃德-高橋恆等式 | : : : : : : : : : : : : : : : : $\left.-{\mathcal {M}}_{0}(p_{1}\cdots (p_{i}+k)\cdots p_{n};q_{1}\cdots q_{n})\right]$
其中 $e$ 代表基本電荷（帶負號）。可發現當 ${\mathcal {M}}$ 中每個外部電子皆在殼時，恆等式的右手邊中各項皆會有一外部電子離殼，因此不會對散射矩陣產生貢獻。
## 沃德恆等式
沃德恆等式是沃德 - 高橋恆等式在討論散射矩陣時的特殊形式。其描述物理上可能的散射過程，因此所有外部粒子皆在殼。再次令 ${\mathcal {M}}(k)=\epsilon _{\mu }(k){\mathcal {M}}^{\mu }(k)$ 為某個量子電動力學過程的機率幅，其中含有一帶動量 $k$ 、極化向量 $\epsilon _{\mu }(k)$ 的外部光子。則沃德恆等式給出
: : $k_{\mu }{\mathcal {M}}^{\mu }(k)=0$
物理上，此恆等式代表 ξ 規範中縱向的極化向量是不符合物理的，不應存在於散射矩陣中。
此恆等式可應用在限制量子電動力學中真空極化和電子節點方程式的張量結構。
## 推導
在路徑積分表述中，沃德 - 高橋恆等式源自測度泛函規範變換下的不變性。精確來說，若考慮某規範變換 $\varepsilon $ 產生的影響 $\delta _{\varepsilon }$ （此處僅要求測度泛函在轉換下維持不變）, 則
: $\delta _{\varepsilon }\int \left({\mathcal {F}}e^{iS}\right){\mathcal {D}}\phi =\int \delta _{\varepsilon }\left({\mathcal {F}}e^{iS}\right){\mathcal {D}}\phi =0$
其中 $S$ 是作用量、 ${\mathcal {F}}$ 是由場 $\phi $ 組成的泛函。若該規範變換對應到一全域對稱性，則在忽略邊界項後會有一守恆流 $J$ | [
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] |
沃德-高橋恆等式 | : $\delta _{\varepsilon }S=\int \left(\partial _{\mu }\varepsilon \right)J^{\mu }\mathrm {d} ^{d}x=-\int \varepsilon \partial _{\mu }J^{\mu }\mathrm {d} ^{d}x$
因此，沃德 - 高橋恆等式變為
: $\langle \delta _{\varepsilon }{\mathcal {F}}\rangle -i\int \varepsilon \langle {\mathcal {F}}\partial _{\mu }J^{\mu }\rangle \mathrm {d} ^{d}x=0$
此為諾特定理 $\partial _{\mu }J^{\mu }=0$ 在量子場論下的版本。
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] |
數字算符 | 在量子力學中，**粒子數算符**（英語：particle number operator）是在無法精確觀測粒子數的系統中，用來計算可以量測到的粒子數目。
## 符號
<N>
## 大小
N = 升算符 \* 降算符
見階梯算符、創生及消滅算符
## 引述
## 參考資料 | [
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隧道磁阻 | **隧道磁阻**( **TMR** ) 是發生在**磁隧道結**( **MTJ** ）中的磁阻效應，由一個薄絕緣體及被其隔開的兩個鐵磁體組成的組件。絕緣層足夠薄（通常為幾奈米）的情況下，電子可以從一個鐵磁體隧穿過去另一邊。由於這個過程在古典物理學中不可能實現的，所以隧道磁阻是一種嚴格的量子力學現象。
磁性隧道結通過薄膜技術進行製造。工業規模上的薄膜沉積通過磁控濺射沉積完成；實驗室規模通過分子束外延、脈衝雷射沉積以及電子束物理氣相沉積製備。隧道的結通過光刻法製備。
## 現象學的描述
鐵磁薄膜的兩個磁化方向可以通過外部磁場單獨切換。相對於反平行方向，如果磁化方向平行，電子更可能隧穿過絕緣膜。因此，這種結可以在兩種電阻狀態之間切換，一種具有低電阻，而另一種具有非常高的電阻。
## 歷史
1975 年，法國雷恩大學的 Michel Jullière 通過 Fe / 鍺 - 氧 / 鈷結在 4.2 K 時最初發現。阻值的相對變化在 14％左右，並沒有引起重視。 1991 年日本東北大學的宮崎照宣發現室溫下阻值變化為 2.7%。後來在 1994 年，宮崎照宣在由無定形氧化鋁絕緣體隔開的鐵中測得 18% 的相對阻值變化，而 Jagadeesh Moodera 在 CoFe 和 Co 電極結中測得 11.8%。 當時觀察到的最高相對變化是鋁氧化物絕緣體在室溫下測得的 70%。
自 2000 年以來，結晶氧化鎂（MgO）的隧道勢壘一直在開發之中。 2001 年 Butler 和 Mathon 獨立做出理論預測，以鐵為鐵磁體，並以 MgO 為絕緣體，隧道磁阻可達百分之幾千。 同年，Bowen 等人首次報告基於 MgO 的磁性隧道結 [Fe/MgO/FeCo (001)] 中表現出顯著的 TMR 現象。 2004 年，Parkin 和湯淺新治製造出室溫下超過 200% TMR 的 Fe/MgO/Fe 結。 2008 年，日本東北大學的 S. Ikeda, H. Ohno 小組在 CoFeB/MgO/CoFeB 結中觀察到室溫下高達 604% 效果以及 4.2 K 下超過 1100% 的效果。
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隧道磁阻 | 現代硬碟驅動器的讀取頭原理基於磁隧道結。TMR，或者說磁隧道結，也是新型的非揮發性存儲器 MRAM 的前身。第一代的技術通過在每個位點上創建交叉磁場並在其上寫入數據，但是這種方法的縮放只能限制在 90-130 nm 左右。 目前正在開發兩種第二代技術：熱輔助開關 (TAS) 和自旋轉移扭矩。磁性隧道結也用於傳感應用。例如，TMR 傳感器可以測量現代高精度風向標的角度。
## 物理解釋
相對電阻變化或效應幅度定義為
: $\mathrm {TMR} :={\frac {R_{\mathrm {ap} }-R_{\mathrm {p} }}{R_{\mathrm {p} }}}$
其中 $R_{\mathrm {ap} }$ 是反平行狀態下的電阻，而 $R_{\mathrm {p} }$ 是平行狀態下的電阻。
Jullière 用鐵磁電極自旋極化解釋了 TMR 效應。自旋極化 _P_ 由自旋相關的狀態密度 (DOS) ${\mathcal {D}}$ 在費米能量中計算得到：
$$
P={\frac {{\mathcal {D}}_{\uparrow }(E_{\mathrm {F} })-{\mathcal {D}}_{\downarrow }(E_{\mathrm {F} })}{{\mathcal {D}}_{\uparrow }(E_{\mathrm {F} })+{\mathcal {D}}_{\downarrow }(E_{\mathrm {F} })}}
$$
自旋向上的電子是那些自旋方向平行於外部磁場的電子，而自旋向下的電子則是與外部磁場反平行排列的電子。現在由兩個鐵磁體 _P 1_ 和 _P 2_ 的自旋極化給出相對電阻變化：
$$
\mathrm {TMR} ={\frac {2P_{1}P_{2}}{1-P_{1}P_{2}}}
$$
如果沒有施加電壓到結上，電子會以相等的速率在兩個方向上隧穿。在偏置電壓 _U 下_，電子會優先隧穿到正極。假設在隧穿過程中自旋守恆，電流可以用雙電流模型來描述。取決於結的磁性狀態，總電流分為兩個部分電流，一個用於自旋向上電子，另一個用於自旋向下電子。
定義有兩種可能的反平行狀態。首先可以使用具有不同矯頑力的鐵磁體（通過使用不同的材料或不同的薄膜厚度）。其次，一個鐵磁體可以與反鐵磁體耦合（交換偏置）。在這種情況下，未耦合電極的磁化保持「自由」。 | [
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隧道磁阻 | _如果 P 1_ 和 _P 2_ 等於 1，即如果兩個電極都具有 100% 自旋極化，則 TMR 變為無限大。在這種情況下，磁性隧道結變成了一個開關，在低電阻和無限電阻之間進行磁性切換。擁有這種能力的材料稱為鐵磁半金屬。它們傳導電子的能力是完全自旋極化的。理論預測有許多材料具有此屬性（例如 CrO 2 ，以及各種 Heusler 合金），但實驗確認其屬性一直受到爭論。然而，如果只考慮那些進入傳輸的電子，Bowen 等人測得 La0.7Sr0.3MnO3 和 SrTiO3 之間的界面處高達 99.6%，自旋極化實際上相當於證明了該特性。
TMR 隨著溫度升高和偏置電壓升高而降低。兩者原則上都可以通過磁振子相互作用來理解，以及由於氧空位引起的局部狀態的隧穿（參見下文的對稱濾波部分）。
## 隧道屏障中的對稱濾波
在引入外延氧化鎂（MgO）之前，非晶氧化鋁被用作 MTJ 的隧道勢壘，室溫 TMR 一般在百分之幾十的範圍內。 MgO 屏障將 TMR 提高到數百個百分點。這種大幅增加反映了電極和勢壘電子結構的協同組合，這反過來又反映了結構有序結的實現。實際上，MgO 過濾了具有特定對稱性的電子的隧道傳輸，這些電子在流經體心立方鐵基電極的電流內完全自旋極化。因此，在 MTJ 的電極磁化平行 (P) 狀態下，這種對稱的電子可以支配結電流。相反，在 MTJ 的反平行 (AP) 狀態下，該通道被阻塞，因此具有下一個最有利於傳輸的對稱性的電子主導結電流。由於這些電子相對於更大的勢壘高度隧穿，這導致相當大的 TMR。
除了跨過基於 MgO 的 MTJ 的這些大的 TMR 值之外， 勢壘電子結構對隧穿自旋電子學的影響，已經通過設計出具有特定對稱性的電子的結來間接證實。這首先是通過檢查具有全自旋 (P=+1) 和對稱極化隧道的鑭鍶錳酸鹽半金屬電極的電子如何穿過電偏置的 SrTiO 3 隧道勢壘來實現的。 後來也證明了在樣品生長期間在結界面處插入適當的金屬間隔物的更簡單的實驗發現 。 | [
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隧道磁阻 | 雖然在 2001 年首次提出的理論 預測與 MTJ 的 P 狀態中的 4eV 勢壘高度和 MTJ 的 AP 狀態中的 12eV 相關的大 TMR 值，但實驗表明勢壘高度低至 0.4eV。 如果考慮到 MgO 隧道勢壘中氧空位的局部狀態，這一矛盾就會得到解決。跨越 MgO MTJ 的廣泛的固態隧穿光譜實驗在 2014 年揭示了電子在地面上的保留和氧空位的激發態，這取決於溫度，決定了給定對稱性電子的隧穿勢壘高度，以及從而製作出有效的 TMR 比率及其溫度依賴性。這種低勢壘高度進而能夠實現自旋轉移矩所需的高電流密度，這將在下文進行討論。
## 磁隧道結 (MTJ) 中的自旋轉移矩
自旋轉移矩的影響已在 MTJ 中得到廣泛研究和應用，其中有一個隧道勢壘夾在一組兩個鐵磁電極之間，因此右電極有（自由）磁化，同時假設左電極（具有固定磁化）充當自旋偏振器。然後可以將其固定到磁阻隨機存取存儲器設備中的某些選擇電晶體，或連接到硬碟驅動器應用中的前置放大器。
線性響應的電壓驅動的自旋轉移轉矩矢量可以從轉矩算子的期望值計算得到：
$$
\mathbf {T} =\mathrm {Tr} [{\hat {\mathbf {T} }}{\hat {\rho }}_{\mathrm {neq} }]
$$
其中 ${\hat {\rho }}_{\mathrm {neq} }$ 是穩態輸運的規範不變非平衡密度矩陣，在零溫度極限下，以及在線性響應狀態下， 扭矩算子 ${\hat {\mathbf {T} }}$ 從自旋算子的時間導數可以得到：
$$
{\hat {\mathbf {T} }}={\frac {d{\hat {\mathbf {S} }}}{dt}}=-{\frac {i}{\hbar }}\left[{\frac {\hbar }{2}}{\boldsymbol {\sigma }},{\hat {H}}\right]
$$
使用一維緊束縛哈密頓量的一般形式：
$$
{\hat {H}}={\hat {H}}_{0}-\Delta ({\boldsymbol {\sigma }}\cdot \mathbf {m} )/2
$$
其中總磁化強度（作為宏自旋）沿著單位矢量 $\mathbf {m} $ 和涉及任意經典向量的泡利矩陣性質 $\mathbf {p} ,\mathbf {q} $ , 由
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隧道磁阻 | ({\boldsymbol {\sigma }}\cdot \mathbf {p} )({\boldsymbol {\sigma }}\cdot \mathbf {q} )=\mathbf {p} \cdot \mathbf {q} +i(\mathbf {p} \times \mathbf {q} )\cdot {\boldsymbol {\sigma }}
$$
$$
({\boldsymbol {\sigma }}\cdot \mathbf {p} ){\boldsymbol {\sigma }}=\mathbf {p} +i{\boldsymbol {\sigma }}\times \mathbf {p}
$$
$$
{\boldsymbol {\sigma }}({\boldsymbol {\sigma }}\cdot \mathbf {q} )=\mathbf {q} +i\mathbf {q} \times {\boldsymbol {\sigma }}
$$
可以獲得一個解析表達式 ${\hat {\mathbf {T} }}$ （可以使用緊湊形式表示 $\Delta ,\mathbf {m} $ ，以及泡利自旋矩陣的向量 ${\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})$ )。
一般 MTJ 中的自旋轉移力矩矢量有兩個分量：平行分量和垂直分量：
一個平行分量： $T_{\parallel }={\sqrt {T_{x}^{2}+T_{z}^{2}}}$
和一個垂直分量： $T_{\perp }=T_{y}$
在對稱 MTJ（由具有相同幾何形狀和交換分裂的電極製成）中，自旋轉移矩矢量只有一個有效分量，因為垂直分量消失了：
$T_{\perp }\equiv 0$ .
因此，只有 $T_{\parallel }$ 對比 $\theta $ 需要在正確電極的位置繪製以表徵對稱 MTJ 中的隧道效應，使其可以用於工業規模的生產和表徵。
注意：在這些計算中，有源區域（需要計算延遲格林函數）應由隧道勢壘 + 有限厚度的右側鐵磁層組成（如在實際設備中）。活性區域連接到左鐵磁電極（建模為具有非零塞曼分裂的半無限緊束縛鏈）和右 N 電極（沒有任何塞曼分裂的半無限緊束縛鏈），由相應的自能項。
## 理論與實驗的差異 | [
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隧道磁阻 | 已經可以預測理論隧道的磁阻比。然而，已觀察到的最大僅為 604%。 一種猜想是晶界可能會影響 MgO 勢壘的絕緣性能。然而，掩埋堆疊結構中的薄膜結構很難確定。晶界可以充當通過材料的短路傳導路徑，從而降低器件的電阻。最近，使用新的掃描透射電子顯微鏡技術，FeCoB/MgO/FeCoB MTJ 內的晶界已被原子分辨。這允許對真實薄膜中存在的結構單元進行第一原理密度泛函理論計算。這樣的計算表明，帶隙可以減少多達 45%。
除了晶界之外，硼間隙和氧空位等點缺陷都可能會顯著改變隧道磁阻。最近的理論計算表明，硼間隙會在帶隙中引入缺陷態，可能會進一步降低 TMR 這些理論計算也得到了實驗證據的支持，實驗證據顯示了兩個不同系統之間 MgO 層中硼的性質以及如何 TMR 是不同的。
## 相關
* 量子隧道
* 磁阻
* 巨磁阻 (GMR)
* 自旋轉移力矩
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羅伯特塞西爾帕爾默中學 | **羅伯特塞西爾帕爾默中學** （英語：Robert Cecil Palmer Secondary School，簡稱 **RCPSS**）是一所位於加拿大不列顛哥倫比亞省列治文的公立男女同校中學，為 8-12 年級的學生提供教育。 帕爾默中學是列治文學區下屬的眾多學校之一。 它是列治文唯一一所提供 Pre-AP 獎勵計劃、ACE-IT 管道計劃以及之前的 Palmer Hockey Academy 的學校。 列治文虛擬學校也設在 Palmer 大樓內。
## 歷史
羅伯特塞西爾帕爾默初中於 1959 年 9 月首次開學，招生人數為 769 人。這所學校以羅伯特塞西爾帕爾默（Robert Cecil Palmer）的名字命名，他是列治文的居民，在省法院工作了 22 年。 學校在 1976 年 4 月 10 日遭遇了一場可怕的火災。學校在 1978 年進行了兩次翻修以從火災中恢復過來。 1996 年，列治文學區將所有高中（僅限 11 年級和 12 年級）和初中（僅限 9 年級和 10 年級）轉換為 8 至 12 年級的全科學校。 帕爾默於 2000 年 5 月正式重新開放，成為一所 8 至 12 年級的完整中學。
## 座右銘和顏色
學校的顏色，海軍藍和粉藍色，是 1959 年由學生會選定的。 學校的希臘語座右銘 KUDOS 可以翻譯為「為了勇敢」或「為了榮譽」，它是由第一藝術系設計的學校校徽的一部分。
## 帕爾默格里芬 (Palmer Griffins)
格里芬代表帕爾默家族，是帕爾默家族徽章的一部分。 它是一種神話中的生物，是獅子和鷹的混合體。 獅鷲是智慧、力量和忠誠的象徵。 帕爾默格里芬的名字經常用於帕爾默的運動隊伍。
## 課程和項目
羅伯特塞西爾帕爾默中學採用全年線性系統。 學年分為 2 個學期。 學生需要修讀 8 門課程； 學生上學期參加四節 90 分鐘的課程，第二個學期參加另外四節不同的課程。
### 分級量表
Palmer 使用與不列顛哥倫比亞省教育部製定的相同評分標準。
對於任何低於 49% 的學生，學生將被分配一個「I」（未完成），如果未完成課程要求，則會變成「F」（失敗）。
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] |
羅伯特塞西爾帕爾默中學 | Palmer 通過其選擇性的 Pre-AP / 激勵計劃提供榮譽級別的課程。 該計劃發展學術卓越，並為數學、科學和社會學生的科目提供加速。 激勵計劃還通過領導機會、戶外和體育教育以及社區服務來挑戰學生，目的是建立強大的個人品格。 激勵計劃建立在促進高中生全面教育和培養學習者批判性思維的基礎上。
你需要申請， 然後
截至 2016-2017 年，該計畫提供：
* 英語、數學、社會研究和科學的強化（榮譽）課程
* 8、9、10 年級數學和科學的 2 年加速課程
* 準備為 12 年級的 AP 課程和考試
* 補充獎勵課程的旅行和戶外露營體驗
* 領導和社區服務機會
### ACE IT 職業準備計劃
行業培訓加速學分註冊 (ACE IT) 計劃是高中學徒制的課堂組成部分。 ACE IT 學生是在行業培訓局 (ITA) 註冊的青年學徒。
ACE IT 計劃通常作為學區和大專院校之間的合作夥伴關係提供，並在任一地點進行現場培訓。 成功完成 ACE IT 課程的學生將獲得高中畢業和高等教育證書的學分。 ACE IT 計劃的註冊是免費的。
Palmer 是列治文唯一一所擁有 ACE IT 項目的管道項目的學校。
### 獨立研究計劃
獨立指導研究 (IDS) 與里士滿虛擬學校合作，在老師的監督下，讓學生開始自己選擇的科目學習並獲得畢業學分。 學生必須提交 IDS 申請，並且之前已經表現出獨立的工作能力和良好的職業道德才能註冊該課程。
## 學生社
Palmer 擁有大量學生經營的俱樂部 / 社，包括但不限於：
* 藝術俱樂部
* 舞蹈隊
* 話劇社
* Griffins that Give
* 新聞社
* 量子連續體社
* 辯論社
* 圖書館「監視器」
* 年鑑社
* Destination Imagination Club
* 同伴輔導社
* 研究生委員會
* 數學社
* 電腦遊戲俱樂部
* 急救社
* 生態團隊
* SOGI+ Club
### 全球網絡
學校設有 Global Network。 Global Network 是一個人道主義服務社，由 Palmer 學生組成，他們為當地和全球社區提供服務。 通過本地和全球項目，全球網絡的目標是提高人們對世界各地發生的問題的認識，此外還創建一個平台來討論和幫助解決這些問題。 為此，俱樂部全年舉辦各種活動和籌款活動，所有收益都用於支持事業或活動。
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羅伯特塞西爾帕爾默中學 | 學校定期參加不列顛哥倫比亞省內的科學競賽，包括年度 UBC 物理奧林匹克和昆特蘭科學挑戰賽。 學校在 2015 年 SD38 冰棒橋搭建大賽中獲得第二名。
### JAX Club and JAXOLA
最近，以深澤比爾為首的 JAX 俱樂部（前身為日本交流俱樂部）已由 Thomas Kemper 蘇打水以一個名為 JAXOLA 的獨家品牌供應，以促進俱樂部和 Palmer 的興趣。 | [
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Α輻射 | **α 粒子**（英語：Alpha particle），也被稱作 **α 射線**或 **α 輻射**，是由兩個質子及兩個中子組成的粒子，其與氦 - 4 的原子核相同。它以第一個希臘字母 α 命名，可以寫作 α 或 α2+，通常在 α 衰變中產生。因為 α 粒子與氦原子核相同，有時也會寫成 He2+
或 4
2He2+
以代表 + 2 價（失去兩個電子）的氦離子。當 α 粒子從環境中得到電子，它就會變成一般（電中性）的氦原子 4
2He。
α 粒子的自旋為 0。由於標準 α 放射性衰變的機制，α 粒子一般會有大約 5 MeV 的動能以及相當於 4% 光速的速度。（見下方「# 能量與吸收」）。它是具高游離性的粒子輻射，通常（當其由放射性 α 衰變產生時）有較小的穿透深度（可以被數公分的大氣層或皮膚阻擋）。
然而，由三元分裂產生的所謂長射程 α 粒子能量及穿透深度都是原本的三倍。比起核分裂過程所產生的氦原子核，組成 10~12% 宇宙射線的氦原子核通常也有較大的能量，能夠穿透人體及數公尺的固體屏障，而少部分由粒子加速器製造的高能氦原子核也具有相同的能量。
## 名稱
有些科學作家會把「雙游離氦原子核（He2+
）」和「α 粒子」當作相同的術語，但這個命名法並非定義良好。和 β、γ 粒子 / 射線一樣，粒子的名字都隱含了它的製造過程及所含能量，但這些都沒有被嚴格地應用。 因此，當所指的是恆星的核融合反應（例如 3 氦過程），甚至是宇宙射線的一部分時，"α 粒子" 可以適當地作為術語使用，比 α 衰變所產生的還要高能的 α 是一種不常發生的核分裂 -三元分裂的常見產物。然而，由粒子加速器（迴旋加速器、同步加速器等）所製造的氦原子核比較不能夠被稱為「α 粒子」。
## α 粒子的來源
### α 衰變
α 粒子廣為人知的來源為重原子（>106 u）的 α 衰變。當原子在 α 衰變中放出 α 粒子，原子的質量數會因為失去 α 粒子的四個核子而減少四，原子序會因為失去兩個質子而減少二，該原子會變成新的元素。例如：鈾衰變成釷、鐳衰變成氡即為這類由 α 衰變進行的核轉換。 | [
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Α輻射 | α 粒子容易從所有具高放射性的原子放出，如鈾、釷、錒、鐳以及超鈾元素。與其他衰變不同的是，α 衰變必須要有一個最小的原子核以完成這一過程。迄今為止發現能夠協助此過程的最小原子核為鈹 - 8 以及碲元素中最輕的核種，其質量數介於 104 到 109 之間。α 衰變有時會把原子核留在激發態，而原子核會放出 γ 射線以釋出多餘的能量。
#### α 衰變的反應機制
和 β 衰變相比，α 衰變中包含的基本交互作用是電磁力與核力的平衡。帶正電的原子核與 α 粒子之間的庫侖靜電力促使 α 衰變發生，而核力則會抑制它發生。在古典物理學中，α 粒子並沒有足夠的能量脫離原子核內部強力形成的位能井。
然而，量子穿隧效應使 α 粒子能夠在能量不足的情況克服核力。物質波的性質允許 α 粒子在距離原子核較遠的區域停留，此時電磁力能夠抵銷核力。因此 α 粒子能夠逃脫。
### 三元分裂
核分裂過程中若發生相對稀有（數百分之一）的三元分裂，會產生特別高能的 α 粒子。在此過程中，比起一般過程的兩個，它會生成三個高能粒子，其中最小的粒子有 90% 的機率為 α 粒子，由於此類 α 粒子的能量高達 16 MeV，它們被稱為長射程 α 粒子，能量比起 α 衰變產生的 α 粒子還高出許多。中子誘發核分裂（發生在核反應爐的核反應）或是可裂變物質及可裂變錒系元素的核種（也就是可進行核分裂的重原子）進行自發核分裂（一種放射性衰變）時都可能發生三元分裂。無論是誘發或自發分裂，較重的原子核使長射程 α 粒子能夠有比起 α 衰變產生的 α 粒子有更高的能量。
### 加速器
高能氦原子核（氦離子）可以從迴旋加速器、同步加速器等粒子加速器製造。傳統上我們一般不會將其稱為 α 粒子。
### 太陽的核反應
氦原子核可能參與星體的核反應，歷史上我們偶爾會將其視為 α 反應（參見 3 氦過程）。
### 宇宙射線
組成宇宙射線 10～12％的超高能氦原子核有時會被當成 α 粒子，宇宙射線生成的機制目前仍爭論不休。
## 能量與吸收
α 衰變放出的 α 粒子的能量有少部分和放射過程的半衰期有關，蓋格 - 納托法則表明半衰期時間在位數上的差異對能量的影響小於 50%。 | [
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Α輻射 | 放出的 α 粒子所擁有的能量不同，雖然較大的原子核會放出較高能量的 α 粒子，但大部分 α 粒子的能量會介於 3~7 MeV （兆電子伏特），其上下限對應到放射半衰期極短和極長的核種放出的 α 粒子。能量與波長的關係使其可以用 α 光譜測定法識別特定核種。
擁有 5 MeV 動能的典型 α 粒子其速度為 15,000 km/s，相當於光速的 5%。對單一粒子來說這個能量相當高，但高質量代表它比起其他輻射（例如 β 粒子、中子輻射）來說速度較小
因為它的帶電及質量，α 粒子可以輕易地被材料吸收，它也只能在空氣中移動數公分，它可以被衛生紙或人類的皮膚吸收，通常可以穿透皮膚 40 微米，相當於數個細胞的深度。
## 對生物的影響
由於 α 粒子的吸收範圍短且無法穿透皮膚表層，除非直接攝入或吸入會放射 α 粒子的物質，不然 α 粒子一般來說不會對生命造成危害。因為其高質量及強吸收，若放出 α 粒子的放射性同位素進入人體（吞入、攝入或注射，1950 年前二氧化釷懸浮液曾被當作 X 光的顯影劑使用），它是游離輻射中游離性最強、最具破壞性的，劑量過大可能引起任何種類的輻射中毒。估計同劑量 α 粒子對染色體的影響比 β、γ 輻射高出 10 到 1000 倍，平均為 20 倍。一項與體內暴露在鈽和鈾的 α 輻射的歐洲核工人有關的研究發現當相對生物效應為 20 時，其致癌率（以肺癌為例）與外部暴露在 γ 輻射一樣，也就是說，攝入一單位的 α 粒子，其危險性與暴露在 20 單位的 γ 輻射一樣。強 α 放射物釙 - 210（1 mg 的 210Po 每秒放出的 α 粒子與 4.215 g 的 226Ra 相同）被懷疑與肺癌和膀胱癌有關（菸草的成分之一）。釙 - 210 曾被用來殺死俄羅斯反政府人士，2006 年被用於毒害前俄羅斯聯邦調查局中校亞歷山大・瓦爾傑洛維奇・利特維年科。
α 粒子放射性同位素被吞入時，它的危險度會遠勝於預計的半衰期與衰變速率，因為其高相對生物效應會對生物造成傷害。比起同等活性的 β 或 γ 放射性同位素，α 粒子的危險性在平均上高出 20 倍，若攝入 α 放射物則會提高到 1000 倍。
## α 粒子的歷史
在 1899 年，物理學家拉賽福（在加拿大蒙特婁的麥基爾大學工作）與維拉爾（在巴黎工作）以穿透性與在磁場中的偏折把輻射分類成三種，拉賽福命名為 α、β 與 γ。拉賽福將 α 定義為那些對一般物體穿透力最低的輻射。 | [
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Α輻射 | 拉賽福的工作也包含了測量 α 粒子的質荷比，讓他有了 α 粒子是 + 2 價氦離子的猜想（之後發現是氦原子核）。1907 年，拉賽福和托馬斯終於證明 α 粒子確實是氦離子。他們讓 α 粒子粒子穿過一個真空管的玻璃薄壁，之後管中捕捉到大量的氦離子，他們讓管中產生電火花，之後由光譜得知是氦離子，並證實 α 粒子就是氦離子。
因為 α 粒子能自然產生，又帶有能夠參與核反應的高能量，因此對 α 粒子的研究可以帶來早期的核物理學知識。拉賽福用溴化鐳放出的 α 粒子推斷湯木生的梅子布丁模型在根本上有瑕疵。在拉賽福的學生蓋革和馬士登進行的 α 粒子金箔散射實驗中，一束 α 粒子被製造並射向金箔（幾百個原子薄）。以硫化鋅硫化鋅當作光屏，α 粒子撞上時會產生光線，拉賽福認為若原子的梅子布丁模型是正確的，則均勻分布的正電荷應只會讓帶正電的 α 粒子輕微偏轉。
實驗發現有些 α 粒子偏折的幅度比預期的大（在拉賽福的建議下進行確認），有些甚至幾乎反彈回來，拉賽福認為若梅子布丁模型是正確的，則被反彈的粒子就像是 15 吋的砲彈射向衛生紙，但砲彈卻被彈開一樣。只有當原子的正電荷集中在中心時，才能將 α 粒子彈開，也就是原子核。
在這之前，沒有人知道 α 粒子自己就是原子核，也不知道質子和中子。這次實驗後，湯木生的的梅子布丁模型被遺棄，之後出現了波耳模型以及現代的波動力學模型。
在 1917 年，拉賽福意外地用 α 粒子進行了核轉換。1901 年之後一個元素轉換成另一個元素被認為是自然的放射性衰變，但當拉賽福將 α 衰變產生的 α 粒子射入空氣時，產生了新的輻射，後來被確認是氫離子（拉賽福命名為質子）。之後的實驗發現質子是由空氣中的氮氣產生，而這個反應被推定是氮轉變成氧的核轉換。
: 14N + α → 17O + p
這是第一個被發現的核反應。
在右邊的布拉格曲線中，可看見 α 粒子在末段失去較多能量。
## 反 α 粒子
2011 年，STAR 的成員在美國能源部的布魯克黑文國家實驗室中，利用相對論性重離子對撞機發現了氦原子核的反物質，也被稱作反 α。該實驗將金離子以接近光速對撞以產生反粒子。
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Α輻射 | * 有些煙霧探測器裡面會有少量的 α 放射物鎇 - 241。α 粒子會游離空氣讓電流通過。火焰引起的煙霧粒子會使電流降低，進而使警報響起。此同位素無論是吸入或攝入都非常危險，但只要保持密封就可以最小化傷害。許多美國直轄市已經計畫收集並回收老舊的煙霧探測器以避免它們進入一般的垃圾處理管道。
* α 衰變可安全地提供太空探測器和心律調節器中使用的放射性同位素熱電機所需的能量。α 衰變比起其他衰變更容易被阻擋。鈽 - 238，α 粒子的來源之一，只需要 2.5 毫米的鉛就能阻隔具危險性的輻射放出。
* 防靜電裝置通常會使用釙 - 210，α 粒子放射物，游離空氣讓「靜電吸附力」更快消散。
* 科學家正嘗試利用 α 放射性同位素的破壞性，以少量同位素破壞腫瘤。
### α 粒子用於治療癌症
α 放射性同位素可以用三種方式根除惡性腫瘤：針對特定組織進行不熔放射性治療（鐳 - 223），作為放射源置入腫瘤（鐳 - 224），作為腫瘤指向分子的連結物，像是抗體與抗原（腫瘤）的關係。
#### 鐳 - 223
* 鐳 - 223 是一種 α 放射物，因為它是一種擬鈣劑，它會自然地吸附在骨頭上。鐳 - 223（在 RaCl2 的形式下）可以注入病人的靜脈，它會進入那些因為癌細胞轉移而快速增生的骨骼區域。
* 一旦進入骨骼，鐳 - 223 可以放出 α 輻射並摧毀 100 微米內的實體腫瘤。從 2013 年開始，一個名為氯化鐳的藥品（商品名稱為鐳治骨 ®）被用於治療轉移到骨骼的前列腺癌。
* 注入血液循環的放射性核種可以抵達有血管的地方。然而，這也代表對於內部並未血管化（裡面幾乎沒有血流通過）的大型腫瘤來說，無法有效地以放射線來根除。
#### 鐳 - 224
* 鐳 - 224 是放射性原子，它被當成新型癌症治療方式 DaRT（擴散 α 放射性治療）的 α 輻射來源，該治療用儀器的名稱為 Alpha DaRT™。
* Alpha DaRT 由數個以鐳 - 224 浸泡的不鏽鋼圓型管線組成。每個鐳 - 224 經過衰變會產生 6 個衰變產物，此時會放出 4 個 α 粒子。α 粒子的射程最多只到 100 微米，不足以處理大部分的腫瘤，然而鐳 - 224 的衰變產物可以在組織內擴散到 2~3 毫米，藉此產生 "死區" 有足夠的輻射以清除腫瘤（當管線設置恰當時）。 | [
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Α輻射 | * 鐳 - 224 的半衰期足夠短（3.6 天），可以有快速的臨床效果而避免長期暴露在輻射下引起的傷害。同時，它的半衰期夠長，讓它可以即時從世界上的任何地方送到癌症治療中心。
#### α 標靶治療
* 針對實體腫瘤的 α 標靶治療包含連接 α 放射性同位素到腫瘤指向分子上，就像抗體一樣，可以用靜脈注射的方式治療癌症病人。
* 傳統上，抗體放射性核種共軛物已經使用 β 放射性核種。舉例來說，長久以來碘 - 131 被用於甲狀腺癌的治療。
* 近期，α 放射物錒 - 225 被用來研究以用於轉移性攝護腺癌的治療。錒 - 225 被連結在前列腺特定膜抗原（PSMA），因為它有 10 天的半衰期，在衰變成鉍 - 209 時會放出 4 個 α 粒子，因此其在臨床上相當實用。
## α 輻射與 DRAM 錯誤
1978 年，Intel 的 DRAM（動態隨機存取記憶體）晶片的 "軟性錯誤" 被發現與 α 粒子有關。之後藉由嚴格控制半導體材料中放射性元素的包裝解決了這項錯誤。
## 參見
* α 核種
* α 作用 （也被稱作 α 捕獲或 α 梯）
* β 粒子
* 宇宙射線
* 氦核，氦 - 3 而不是氦 - 4 的原子核
* α 放射性材料列表
* 核子物理
* 粒子物理學
* 放射性同位素
* 射線：
* β 射線
* γ 射線
* δ 射線
* ε 射線
* 拉塞福散射
## 註釋
## 參考資料
## 延伸閱讀
* Tipler, Paul; Llewellyn, Ralph. Modern Physics 4th. W. H. Freeman. 2002. ISBN 978-0-7167-4345-3.
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] |
EncFS | **EncFS** 是一個基於 FUSE 的加密檔案系統，為自由軟體。通過它可以將任意目錄作為儲存空間來透明地加密並儲存檔案。
掛載一個 EncFS 檔案系統涉及到兩個目錄：源目錄和掛載點。掛載點中的每個檔案都是加密儲存在源目錄中的一個與之對應的檔案。掛載點中的檔案呈現未加密的樣子，對應源目錄中已加密的檔案。源目錄中會加密檔名。
檔案通過卷金鑰加密，卷金鑰可以儲存在加密的源目錄之內或之外。有一個密碼機制用於解密該金鑰。
## 常見用途
* 在 Linux 中，作為 eCryptfs 的替代方案加密 home 資料夾。
* 用來加密儲存到雲端儲存（Dropbox、Google Drive、OneDrive 等）的檔案和資料夾。
* 為抽取式磁碟中的資料夾進行可攜式加密。
* 作為一個跨平台的資料夾加密機制。
* 可增加雙重身分驗證（2FA）來提高儲存安全性。例如，可以將 EncFS 用來加密資料的卷金鑰儲存在加密資料之外（如隨身碟、網路掛載點、光碟或雲端儲存中），同時要求提供密碼來解密這個卷金鑰。
## 優勢
與其他磁碟加密軟體相比，EncFS 因在主機上單獨儲存每個被加密的檔案而具有一些優勢。
### 跨平台
EncFS 可在多個平台上使用，而 eCryptfs 與 Linux 核心繫結
### Bitrot 檢測
EncFS 在底層檔案系統之上實現了資料腐爛（bitrot）檢測
### 可延伸儲存
EncFS 不存在占用固定大小的「卷」—— 加密的目錄占用的空間隨著掛載點中的檔案增加而增加，也隨著檔案被刪除而縮小。
### 適用於檔案伺服器
EncFS 的加密目錄可以位於普通的檔案伺服器上（通過 NFS、SSHFS 等），並且可以使用普通的檔案系統工具（例如 Rsync）高效地進行鏡像和備份。
### 跨物理裝置
掛載點中的某些目錄可能存在於其他的物理裝置上。
### 更快的備份
檔案同步、雲端儲存等備份軟體可以只備份源目錄中已變更的檔案。
### 抗損壞
資料損壞發生時，更不易波及多個檔案。損壞也更可控，可能通過檔案系統級別的維護工具來修復某些資料損壞。磁碟加密系統遭遇資料異常損壞時可能遭遇更多意外情況。
### 最佳化
因為加密的檔案仍基於底層檔案系統來儲存，某些作業系統最佳化機制仍然奏效，例如傳遞 Trim 命令來幫助提升固態硬碟的效能。不過 dm-crypt 也支援這一點。
### 隨機檔案存取 | [
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EncFS | 支援檔案的隨機存取，無需解密整個被加密的卷。
## 不足
EncFS 也存在一些缺點。
### 相容性
掛載的 EncFS 目錄與容納源目錄的檔案系統享有相同的能力與限制。
### 不支援長檔名
因加密原因，EncFS 產生的加密檔案的檔名會比原始檔名長。因此，EncFS 不能儲存長度接近檔案系統的檔名長度上限的檔名。大多數檔案系統的一般檔名限制為 255 個位元組，這種情況下 EncFS 最多支援 190 個位元組的檔名。
### 一般性安全隱患
儘管檔名和檔案資料被加密，但任何有權存取源目錄的人都可看到加密的檔案系統內有多少檔案、檔案的權限、檔案的大致大小，以及上次存取和上次修改檔案的時間。
### EncFS 1.7 安全問題
2014 年 2 月進行的付費安全審計中發現了多個潛在的漏洞。給出的結論是：
> EncFS is probably safe as long as the adversary only gets one copy of the ciphertext and nothing more. EncFS is not safe if the adversary has the opportunity to see two or more snapshots of the ciphertext at different times. EncFS attempts to protect files from malicious modification, but there are serious problems with this feature.
>
>
> 如果對手只能看到一版被加密的內容，EncFS 可能是安全的。如果對手能看到密文在不同時間的兩版或更多版本，則 EncFS 不安全。EncFS 在嘗試避免檔案免遭惡意修改，但該功能存在嚴重問題。
>
>
### EncFS 1.8 安全問題
EncFS 1.8 的公告包括數項基礎設計變更，以回應上次的審計中提出的安全問題。儘管如此，這些安全漏洞仍受到關注。
## 檔案系統選項
在新建 EncFS 卷時，有數個用來自訂檔案系統的選項，以滿足不同需求。
### 密碼演算法
EncFS 可使用若干種存在於系統中各加密庫內的加解密演算法。Blowfish 和 AES 通常可用。 | [
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] |
EncFS | 可以為支援「可變金鑰長度」的密碼演算法選擇金鑰長度（keySize）。
### 塊大小
每個檔案均以塊的形式加密，有選項控制這些塊的大小。每次讀取單個位元組時，必須解密包含它的整個塊。同樣，每次寫入都需要解密對應塊、更改內容並重新加密。
預設的塊大小 1024（位元組）足以滿足大多數用途。
### 檔名編碼
源目錄中的檔名可以儲存為明文、加密塊或流模式加密。塊模式在一定程度上掩蓋了檔名長度，而流模式的檔名會儘可能短，這在某些檔案系統中可能節省一些儲存空間。
### 檔名 IV 鏈
啟用此選項後，加密檔名所用的初始化向量取自檔案的父目錄 —— 這將使兩個同名但位於不同目錄的檔案具有不同的加密後檔名。
如果重新命名目錄，則其中包含的所有檔案和目錄都需要重新加密其檔名，這可能花費不少時間和資源。如果經常重新命名含有大量子專案的目錄，則應該禁用此選項。
### 每個檔案的初始化向量
啟用此選項後，每個檔案都使用一個隨機的 8 位元組初始化向量來加密，該向量儲存在源目錄的加密檔案中。如果禁用此選項，則每個檔案都使用相同的初始化向量來加密，可能使卷金鑰更容易被破解。
啟用此選項可加強檔案系統的安全性，但每個檔案會多占用 8 個位元組。
### 外部 IV 鏈
使檔案資料的初始化向量衍生自檔名的初始化向量鏈。相同資料但不同檔名或目錄的檔案將加密成不同的樣子。
如此，此模式啟用時重新命名檔案需要檔案的隨機初始化向量被檔名初始化向量鏈中的更改抵消，或者重新編碼資料。 EncFS 的作者選擇了前一種方法，因為快得多，尤其是對於大檔案。
### 檔名到 IV 頭部鏈
使編碼依賴於完整的路徑名。因此重新命名或移動檔案將需要重新編碼。不支援硬連結。
### 塊 MAC 頭部
儲存每個加密塊的校驗和，使 EncFS 能檢測到加密檔案的損壞或修改。校驗和 (blockMACBytes) 為 8 個位元組，並可選擇向每個塊添加最多 8 個額外位元組的亂數據 (blockMACRandBytes)，以防止相同未加密資料的兩個塊有相同的校驗和。因為必須在讀取資料（驗證完整性）與寫入資料（更新校驗和）時計算每個塊的校驗和，此選項會帶來大量 CPU 開銷。
## 參見
* 磁碟加密
* 檔案系統級加密
* 加密檔案系統列表
* 加密檔案系統列表
* 檔案系統列表
## 參考資料
## 外部連結 | [
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EncFS | * Encfs manpage （頁面存檔備份，存於網際網路檔案館）
* Safe: another per-file implementation for Windows and Mac OS X, with no kernel mode driver (slower), but fully open source （頁面存檔備份，存於網際網路檔案館）
* Boxcryptor: Proprietary software based on EncFS for Windows, Android, and iOS （頁面存檔備份，存於網際網路檔案館）
* EncFSMP: Implementation that runs on Windows and Mac OS X （頁面存檔備份，存於網際網路檔案館） | [
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] |
化学文件格式 | 本文討論了一些常見的**分子檔案格式**，以及它們之間的使用和轉換。
## 區分這些格式
化學資訊通常以檔案或流（stream）的形式提供，人們已經建立了許多格式，分別具有不同詳細程度的文件。格式通過以下三種方式表示（參見 "Chemical MIME" 部分）
* _副檔名_（通常為 3 個字母）。這不是十分可靠，因為包括非化學檔案在內的許多檔案類型共享這些常見的字尾，例如「.mol」和「.dat」。
* 檔案中包含格式資訊的_自描述_檔案。例如 CIF 和 CML。
* 可以傳輸化學資訊的伺服器傳輸的 Chemical/MIME 流。
## 化學標記式語言（CML）
化學標記式語言（CML） 是一種用於表示分子和其他化學資料的開源格式。其開源專案包含 XML Schema、用於解析和使用 CML 資料的一整套原始碼以及一個活躍的社群。許多工具都相容 CML 資料檔案，包括 JChemPaint、Jmol、XDrawChem 和 MarvinView。
## 蛋白質資料庫（PDB）
PDB 格式通常用於蛋白質，但也可用於其他類型的分子。它最初的設計為固定列寬格式，因此具有最大原子數、殘基數和鏈數的限制；這導致較大的結構（例如核糖體）需要被拆分為多個檔案。但是也有許多工具可以讀取超出這些限制的檔案。例如，E.coli 70S 核糖體 在 2009 年被分割為 4 個 PDB 檔案： 3I1M （頁面存檔備份，存於網際網路檔案館）、3I1N （頁面存檔備份，存於網際網路檔案館）、3I1O 和 3I1P，在 2014 年又被合併為一個檔案 4V6C （頁面存檔備份，存於網際網路檔案館） 。
一些 PDB 檔案擁有描述原子連接性和位置的可選功能。因為這些檔案有時用於描述大分子組裝體或在顯式溶劑中表示的分子，所以它們的體積會非常大且被壓縮。一些工具，例如 Jmol 和 KiNG， 可以讀取 .gz 格式的 PDB 檔案。 wwPDB 制定並維護 PDB 及其 XML 替代方案 PDBML 的規範。 2007 年 8 月，PDB 格式規範（版本 3.0）發生了相當大的變化，並且修復了現有資料庫中的許多檔案問題。 PDB 檔案的典型副檔名是_.pdb_ ，一些較舊的檔案也可能使用_.ent_ 或_.brk_ 。一些分子建模工具根據自己的需要調整基本格式為非標準的 PDB 樣式檔案。
## GROMACS | [
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] |
化学文件格式 | GROMACS 系列檔案格式是為與分子類比軟體套件 GROMACS 一起使用而建立的。它與 PDB 格式非常相似，但允許額外的數值精度，並可選擇保留有關粒子速度以及類比軌跡中給定點的位置的資訊。這是因為它被設計用於儲存分子動力學類比的輸出。GROMACS 不允許儲存連接資訊，這些資訊是從單獨的分子和系統拓撲檔案中獲得的。 GROMACS 檔案的典型副檔名是_.gro_ 。
## CHARMM
CHARMM 分子動力學軟體套件可以讀寫多種標準的化學和生化檔案格式；但 CARD（coordinate, 坐標）和 PSF（Protein Structure File, 蛋白質結構檔案）幾乎是 CHARMM 獨有的。 CARD 格式是固定列寬的，類似於 PDB 格式，專門用於儲存原子坐標。 PSF 檔案包含原子連接資訊（即描述原子之間的鍵），這在在開始分子動力學類比之前是必需的。使用的典型副檔名分別是_.crd_ 和_.psf_ 。
## GSD（General Simulation Data）
通用類比資料（General Simulation Data, GSD）格式用於高效地讀取和寫入主要來自（但不限於）_HOOMD-blue_ 的粒子類比資料。該軟體套件包含的一個 python 模組可以以易於使用的語法讀取和寫入 HOOMD 模式的 gsd 檔案。 [1]
## Ghemical
_Ghemical 軟體_可以使用 OpenBabel 匯入和匯出多種檔案格式，預設使用 GPR 格式。該檔案由幾個部分組成，由 `!Header` `!Info` `!Atoms` `!Bonds` `!Coord` `!PartialCharges` `!End` 分隔。
這種格式的推薦 MIME 類型為 _application/x-ghemical_ 。
## SYBYL 線性字串（SYBYL Line Notation，SLN）
SYBYL 線性字串（SYBYL Line Notation, SLN）是一種用字串表示化學結構的方法。因其基於 SMILES，所以包含用於特定相對立體化學的完整表示方式。 SLN 具有豐富的查詢語法，允許查詢 Markush 結構，還支援 ChemDraw 組合庫的規範。
_例：_
## 簡化分子線性輸入規範（SMILES） | [
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] |
化学文件格式 | **S**implified **M**olecular **I**nput **L**ine **E**ntry **S**pecification（簡化分子線性輸入規範，SMILES）用一串字元描述分子。 SMILES 字串包括連接情況，但不包括 2D 和 3D 坐標。
氫原子會被略去，原子用在方括號內的化學元素符號表示，例如 [Au] 表示「金」。有機物中的 **C**、**N**、**O**、**P**、**S**、**Br**、**Cl**、**I** 等原子可以省略方括號，其他元素必須包括在方括號之內。離子用方括號括起，用元素符號和電荷表示，"+" 表示正電荷、"-" 表示負電荷，電荷數跟在正負號之後。如果該原子與氫原子相連，氫原子寫在元素符號後面。
符號「=」表示雙鍵，「#」表示三鍵。 () 表示分支。如果結構中有環，則要打開。斷開處的兩個原子用同一個數字標記，表示原子間有鍵相連。
芳環中的 **C**、**O**、**S**、**N** 原子分別用用小寫字母 **c**,**o**,**s**,**n** 表示。如苯表示為 c**1ccccc1**
_例：_
## XYZ
XYZ 是一種簡單的格式，通常在第一行給出原子數，在第二行給出注釋，然後是幾行原子符號（或原子序數）和坐標參數。
## MDL 編號
MDL 編號包含每個反應和變化的唯一標識號。格式為 RXXXnnnnnnnn。 R 表示一個反應（reaction），XXX 表示哪個資料庫包含反應記錄。數字部分 nnnnnnnn 是一個 8 位數字。
## 其他常見格式
最广泛使用的工业标准之一是化学表格文件（{{tsl|en|Chemical\_table\_file|Chemical\_table\_file|Chemical table file}}）格式，例如_結構資料格式_(Structure Data Format，SDF) 檔案。這些文字檔案遵循著嚴格格式，用於表示多個化學結構和相關資料。該格式最初由 Molecular Design Limited (MDL) 開發和發布。 MOL 是 MDL 的另一種檔案格式。它記錄在 _CTfile 格式_的第 4 章中。
PubChem 還有 XML 和 ASN1 格式，是 PubChem 線上資料庫的匯出選項。二者均基於文字（ASN1 通常是二進位格式）。
下表列出了大量其他格式
## 格式轉換 | [
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化学文件格式 | OpenBabel 和 JOELib 是免費提供的開源工具，專門被設計用來將檔案在各種格式之間轉換。他們的化學專家系統（chemical expert systems）支援大型的檔案轉換。
OpenBabel 的基本命令是
: `babel -i _input\_format_ _input\_file_ -o _output\_format_ _output\_file_`
例如，要將 SDF 中的 _腎上腺素.sdf_ 檔案轉換為 CML 格式：
: `babel -i sdf epinephrine.sdf -o cml epinephrine.cml`
生成的檔案是 _腎上腺素.cml_
IOData （頁面存檔備份，存於網際網路檔案館）是一個免費的開源 Python 庫，用於解析、儲存和轉換量子化學、分子動力學和平面波密度泛函理論軟體程式常用的各種檔案格式。它還能為各種軟體套件生成匯入檔案的靈活框架。有關支援格式的完整列表，請存取 https://iodata.readthedocs.io/en/latest/formats.html 。
許多用於檢視和編輯分子結構的工具能夠讀取多種格式的檔案並以其他格式輸出。例如：JChemPaint（基於 Chemistry Development Kit）、XDrawChem（基於 OpenBabel）、Chime、Jmol、Mol2mol 和 Discovery Studio。
## 化學 MIME 專案
「化學 MIME」是一種將化學資訊流附加到 MIME 類型的方法。
> 該專案於 1994 年 1 月開始，並於 1994 年 5 月在 CERN 舉行的第一屆 WWW 國際會議的化學研討會上首次宣布。…… 草案的第一個版本在 1994 年 5 月至 10 月期間發布，第二個修訂版在 1995 年 4 月至 9 月期間發布。在 1996 年 8 月的 IUPAC 會議上提交給 CPEP（印刷和電子出版物委員會）的論文尚待商榷。
>
>
1998 年該檔案在 JCIM 上正式發表。
## 化學資料來源
以下是部分免費提供的分子資料來源。網際網路上的資源比此處列出的要多得多。下面的參考資料中給出了這些來源的連結。 | [
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] |
化学文件格式 | 1. 美國國家衛生研究院 PubChem 資料庫是一個巨大的化學資料來源。所有資料都是二維的。資料包括 SDF、SMILES、PubChem XML 和 PubChem ASN1 格式。
2. 全球蛋白質資料庫（wwPDB （頁面存檔備份，存於網際網路檔案館））是蛋白質和核酸分子坐標資料的極好來源。資料是三維的，以蛋白質資料庫（PDB）格式提供。
3. eMolecules 是一個用於分子資料的商業資料庫。資料包括每個化合物的二維結構圖和微笑字串。 eMolecules 支援基於部分分子結構的快速子結構搜尋。
4. ChemExper 是用於分子資料的商業資料庫。搜尋結果包括許多化合物的二維結構圖和莫耳檔案。
5. 紐約大學 3-D 分子結構圖書館 （頁面存檔備份，存於網際網路檔案館）。
6. 美國環境保護署的分散式結構可搜尋毒性（DSSTox）資料庫網路是 EPA 計算毒理學計劃的一個專案。該資料庫提供 SDF 分子檔案，重點關注致癌物質和其他有毒物質。
## 另請參見
* 檔案格式
* OpenBabel,JOELib, OELib
* 化學開發工具包（Chemistry Development Kit）
* 化學標記式語言
* 分子建模軟體（Software for molecular modeling）
* NCI/CADD 化學識別碼解析器 （頁面存檔備份，存於網際網路檔案館）
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雅克·迪克斯米爾 | **雅克・迪克斯米耶**（法語：Jacques Dixmier；1924 年 5 月 24 日 —）是一名法國數學家，同時也是尼古拉・布爾巴基協會會員。
## 生平
雅克・迪克斯米耶出生於奧弗涅 - 羅訥 - 阿爾卑斯大區羅亞爾省聖艾蒂安，1942 年就學於巴黎高等師範學院，1944 年獲得數學聚合競賽（法語：Agrégation de mathématiques）第一名，並於 1946 年加入法國國家科學研究中心，1949 年在數學家讓 - 皮埃爾・塞爾及皮埃爾・薩米埃爾（法語：Pierre Samuel）的邀約下，加入尼古拉斯・布爾巴基數學協會。
雅克・迪克斯米耶後來在土魯斯大學及勃艮第大學任教，退休後仍在法國高等科學研究所繼續工作五年，同時也是巴黎第六大學榮譽教授。
### 著名學生
雅克・迪克斯米耶是算子代數理論（法語：Algèbre d'opérateurs）領域的研究者，門下著名的學生有：
* 阿蘭・科納
* 米歇爾・布里翁（法語：Michel Brion）
* 妮科爾・貝林納（法語：Nicole Berline）
* 米歇爾・迪弗洛（法語：Michel Duflo）
* 阿蘭・吉沙爾代（法語：Alain Guichardet）
* 米謝勒・韋爾涅（法語：Michèle Vergne）
### 著作
他寫過兩部科幻小說：
* 1993 年：眾神的黎明（法語：L'Aurore des dieux）
* 1995 年：第七次存款（法語：Le Septième arrhe）
另外，他也與達尼耶・謝羅（法語：Danye Chéreau）和阿蘭・科納合著兩部小說：
* 2013 年：量子劇院（法語：Le Théâtre Quantique）
* 2018 年：阿塔卡馬幽靈（法語：Le Spectre d'Atacama）
雅克・迪克斯米耶的艾狄胥數是 1。
## 獲獎
* 法國科學院國家獎
* 法國科學院安培電力獎
* 美國數學學會勒羅伊・斯蒂爾獎
* 法國科學院埃米爾 - 皮卡德獎章
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马克·里德 | **馬克・阿瑟・里德**（Mark Arthur Reed，1955 年 1 月 4 日 —2021 年 5 月 5 日）是美國物理學家，耶魯大學教授。
馬克・里德 1983 年畢業於雪城大學， 1983 年到 1990 年在德州儀器工作，在那裡他展示了第一台量子點設備。1990 年後擔任耶魯大學工程與應用科學學院擔教授。他於 1996 年 8 月 24 日與伊莉莎白・謝弗結婚。他曾於 2009-2019 年間擔任奈米技術雜誌的主編。
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查尔斯·高尔顿·达尔文 | **查爾斯・高爾頓・達爾文** KBE MC FRS （1887 年 12 月 18 日 - 1962 年 12 月 31 日），英國物理學家，第二次世界大戰期間擔任英國國家物理實驗室 (NPL) 主任。 他是數學家喬治・達爾文的兒子和生物學家查爾斯・達爾文的孫子。
## 早期生活
達爾文出生在劍橋的紐納姆格蘭奇，英國當時正是一個處於鼎盛的科學王朝。他是數學家喬治・達爾文的兒子和生物學家查爾斯・達爾文的孫子。他的母親是賓夕法尼亞州費城的莫德・杜皮（Maud du Puy）達爾文夫人。達爾文的姐姐是藝術家格溫・拉維拉特，妹妹瑪格麗特嫁給了經濟學家約翰・梅納德・凱恩斯的弟弟傑佛瑞・凱恩斯。他的弟弟威廉・勞勃・達爾文是倫敦的一名股票經紀人。達爾文在馬爾伯勒學院（1901-1906）接受教育，然後在劍橋大學三一學院學習數學，並於 1910 年獲得碩士學位。
## 科研生涯
達爾文在曼徹斯特維多利亞大學獲得了研究員職位，於厄尼斯特・盧瑟福和尼爾斯・玻爾手下研究盧瑟福的原子理論。 1912 年，他的興趣發展到利用他的數學技能協助亨利・莫塞萊進行 X 射線衍射。1914 年，他發表的兩篇關於完美晶體 X 射線衍射動力學理論的論文創造了反射率的達爾文曲線，獲得同行認可。在 1922 年的另一篇論文中，他介紹了鑲嵌水晶模型。
第一次世界大戰爆發時，他被任命為審查員並被派往法國。一年後，威廉・勞倫斯・布拉格將他調到皇家工兵隊，參與通過聲測距定位敵方火炮的工作。當這項研究站穩腳跟時，他被調到英國皇家空軍研究飛機噪音。從 1919 年到 1922 年，他是劍橋大學基督學院的講師和研究員，在那裡他與拉爾夫・福勒一起研究統計力學以及後來被稱為達爾文 - 福勒方法的研究。再前往加州理工學院工作了一年後，達爾文於 1924 年成為了愛丁堡大學首任泰特自然哲學教授，研究量子光學和磁光效應。1928 年，他成為了第一個根據保羅・狄拉克的電子相對論理論計算出氫原子的精細結構的人。
1936 年，達爾文聯繫上了物理學家馬克斯・波恩，希望他能考慮代替他成為泰特自然哲學教授的繼任者。波恩欣然接受了這一提議。隨後，他辭去了在愛丁堡的職務，成為基督學院的院長，開始了他作為一名積極而能幹的行政人員的職業生涯。他並 1938 年成為了英國國家物理實驗室的主任，負責研究戰爭的臨近，並於戰後期間一直擔任這個角色。但在戰爭年代的大部分時間裡他都在曼哈頓計劃上為協調美國、英國和加拿大的工作努力。 | [
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查尔斯·高尔顿·达尔文 | 達爾文晚年四處遊歷，致力於推廣國際合作，傳播科學思想。他於 1962 年除夕在劍橋的紐納姆農莊（他出生的房子）去世；於 1963 年 1 月 4 日在劍橋火葬場火化。
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超决定论 | **超決定論**（英語：superdeterminism）在量子力學中是指貝爾定理的一個漏洞。超決定論指出，當所有被測量的系統都與對它們進行測量的選擇相關時，貝爾定理的假設不再被滿足。因此，超決定性的隱變量理論可以在滿足定域性原理的同時，仍然違反貝爾定理得出的不等式。這使得構建一個能夠兼容所有量子力學預測的定域隱變量理論成為可能，也已有學者提出了一些相關的玩具模型。不過需要指出的是，「超決定論」這個詞本身具有一定的誤導性。超決定性模型的決定性就是通常意義上的決定性。只不過除了決定性之外，它們還假設測量的狀態與測量行為本身之間存在相關性。
## 概述
貝爾定理假定在每個檢測器上執行的測量可以相互獨立地做出，也獨立於可以決定測量結果的隱變量。這通常被稱為測量獨立性或統計獨立性。然而在超決定性理論中這種關係是不滿足的，即隱變量與測量行為本身相關。由於測量的選擇和隱變量是預先確定的，因此一個檢測器的結果可以取決於另一個檢測器進行的測量，而無需藉助以超光速傳播的信息。統計獨立性的假設有時被稱為自由選擇或自由意志假設，因為對它的否定意味著人類實驗者不能自由地選擇執行哪種測量。
一部分超決定論模型會額外假設隱變量和測量選擇之間的相關性是在不久的過去才建立的。對這些超決定論模型而言，可以用實驗對其進行檢驗。但總體而言，超決定論從根本上是無法檢驗的，因為可以假設這種相關性自大爆炸以來就存在。
約翰・斯圖爾特・貝爾在 1980 年代的一次 BBC 採訪中討論了超決定論：
> 有一種方法可以迴避超光速和幽靈般超距作用的結論。但它涉及到宇宙中的絕對決定論，並且不允許自由意志的存在。假設世界是超決定性的，不僅無生命的大自然在幕後發條上運行，而且我們的行為，包括我們相信我們可以自由選擇做一個實驗而不是另一個實驗，包括實驗者進行一組測量而不是另一組測量的「決定」本身，都是預先確定的了，那難題就迎刃而解了。這時不再需要比光速更快的信號來告訴粒子 A 已經對粒子 B 進行了哪些測量，因為包括粒子 A 在內的宇宙已經「知道」這一測量行為及其結果將是什麼。
>
>
儘管貝爾承認存在這一漏洞，但他覺得這並不可信。他認為，即使執行的測量是由確定性隨機數生成器選擇的，我們可以假定這些選擇「對於手頭的目的而言可以看作是自由的」，因為機器的選擇會被大量微小的影響所改變。而隱變量不太可能像隨機數生成器那樣對所有微小的影響都敏感。 | [
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] |
超决定论 | 諾貝爾獎得主傑拉德・特・胡夫特在 1980 年代初曾與貝爾討論過這一漏洞。「我提出了一個問題：假設愛麗絲和鮑伯的決定不是出於自由意志，而是由理論中的一切所決定了的。約翰說，嗯，你知道，我必須排除這種可能。如果這是可能的話，那麼我所說的就不適用了。我回復道，愛麗絲和鮑伯是出於一個原因做出決定的。這一原因在於他們的過去，所以並且必須被考慮」。
物理學家安東・塞林格指出，如果超決定論是真實的，那它會通過破壞可證偽性來質疑科學本身的價值：
> 我們總是不明言地假設實驗者的自由…… 這個基本假設對於做科學是必不可少的。如果這不是真的，那麼，我想通過實驗來尋求自然的答案就變得沒有意義了，因為這樣的話自然可以決定我們的問題是什麼，並且可以引導我們的問題以使我們得出錯誤的回答。
>
>
物理學家薩賓・霍森菲爾德和蒂姆・帕爾默則認為超決定論「是一種很有前途的方法，不僅可以解決測量問題，而且可以用於理解量子物理學表面上的非定域性」。
霍華德・懷斯曼和埃里克・卡瓦爾康蒂（Eric Cavalcanti）認為，任何超決定性理論的合理性和吸引力都和相信無所不在的外星人精神控制一樣。
## 例子
第一個超決定性隱變量模型由卡爾・H・布蘭斯於 1988 年提出。麥可・霍爾（Michael Hall）於 2010 年提出了另一個模型。傑拉德・特・胡夫特將他的量子力學元胞自動機模型稱為是超決定性的，儘管尚不清楚它是否確實滿足超決定性的定義。
一些研究者認為量子力學中的逆因果關係也是超決定論的一個例子，而其他人則認為這兩種情況不能混淆。目前還沒有被廣泛接受的定義來區分兩者。
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薛定谔-牛顿方程 | **薛丁格 - 牛頓方程**（英語：Schrödinger–Newton equation），亦稱為**牛頓 - 薛丁格方程**（Newton–Schrödinger equation）、**薛丁格 - 泊松方程**（Schrödinger–Poisson equation），是引入了牛頓引力勢的非線性薛丁格方程，其中引力勢來自作為質量密度處理的波函數，使該方程與通常的薛丁格方程相比多出了表示粒子與其自身引力場相互作用的一項。這一包含自相互作用的項是對經典量子力學原則的根本改變。該方程既可以寫成單個積分 - 微分方程，也可以寫成一個薛丁格方程和一個泊松方程的耦合系統。
薛丁格 - 牛頓方程最初由魯菲尼（Ruffini）和博納佐拉（Bonazzola）在研究自引力玻色子星時引入。在經典廣義相對論的背景下，它是彎曲時空中的克萊因 - 戈爾登方程或狄拉克方程加上愛因斯坦場方程的非相對論極限。該方程可以描述模糊暗物質，並在粒子質量很大的極限下可以用來近似弗拉索夫 - 泊松方程描述的經典冷暗物質。
此後，拉喬斯・迪西（Lajos Diósi）與羅傑・彭羅斯亦提出該方程，以作為解釋量子波函數坍縮的模型，並為其命名「薛丁格 - 牛頓方程」。在此一背景下，物質具有量子屬性，而引力則在基本層面上仍是經典的。因此，薛丁格 - 牛頓方程也可作為測試量子引力必要性的一種方法。
最後，薛丁格 - 牛頓方程還表現為大量粒子系統中相互引力作用的哈特里近似（Hartree approximation）。在此種背景下，菲利普・喬夸德（Philippe Choquard）於 1976 年洛桑庫侖系統研討會上提出了一個對應的電磁庫侖相互作用的方程來描述單組分電漿體。埃利奧特・利布證明了定基態的存在性和唯一性，並將該方程稱為**喬夸德方程**（Choquard equation）。
## 概述
薛丁格 - 牛頓方程是在普通薛丁格方程中引入了自相互作用引力勢得到的一個耦合系統
: $\mathrm {i} \hbar {\frac {\partial \Psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi +m\Phi \Psi \,,$
其中 V 是普通勢，引力勢 $\Phi $ 則表示粒子與其自身引力場的相互作用，滿足泊松方程 | [
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薛定谔-牛顿方程 | : $\nabla ^{2}\Phi =4\pi Gm|\Psi |^{2}\,.$
由于波函數與引力勢的耦合中，這是一個非線性系統。
方程的積分 - 微分形式是
: $\mathrm {i} \hbar {\frac {\partial \Psi }{\partial t}}=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V-Gm^{2}\int {\frac {|\Psi (t,\mathbf {y} )|^{2}}{|\mathbf {x} -\mathbf {y} |}}\,\mathrm {d} ^{3}\mathbf {y} \right]\Psi \,.$
該形式是在假設勢必須在無窮遠處消失的情況下，通過對泊松方程的積分而得到的。
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湯川耦合 | 在粒子物理學中，**湯川耦合**（命名自日本物理學家湯川秀樹）是描述一純量場（或贋純量場） $\phi $ 和一狄拉克場 $\psi $ 在湯川勢下產生的交互作用。其具有以下形式 :
: : $V\approx g\,{\bar {\psi }}\,\phi \,\psi \quad $ （對於純量場） $\qquad $ 或是 $\qquad V\approx g\,{\bar {\psi }}\,i\,\gamma ^{5}\,\phi \,\psi \quad $ （對於贋純量場)
湯川耦合最初發展作為強子間的強作用力模型，其可用於描述核子間藉由交換 π 介子（一種贋純量介子）所產生的核力。此外，湯川耦合亦在標準模型中用來表達希格斯粒子和無質量的基礎費米子（如夸克和輕子）間的交互作用。這些費米子會透過自發對稱破缺得到質量，其會正比於希格斯粒子的真空期望值。這樣的交互作用首先在 1967 年由史蒂文・溫伯格提出。
## 古典位能
考慮兩費米子藉由交換一帶有質量 $\mu $ 的**湯川粒子**而產生交互作用，其之間的位能（也就是湯川勢）可被寫成
: : $V(r)=-{\frac {g^{2}}{\,4\pi r\,}}\,e^{-\mu r}$
除了正負號及指數部份，此位能形式和電位能相同。負號的部分代表帶有相同正負號的荷的粒子在此交互作用下會互相吸引；另一方面，在電磁作用力下電荷同號的粒子則會互相排斥。此差異源於湯川粒子不具有自旋（自旋為 0)，而在量子場論中，交換偶數自旋的玻色子（如自旋為 0 的 π 介子、自旋為 2 的重力子）所產生的交互作用在荷同正負號的情況下會形成互相吸引的位能，交換帶有奇數自旋的玻色子（如自旋為 1 的光子、膠子及 ρ 介子）則相反。指數的部分則指出此作用力僅在有限範圍內有效，長距離下因位能隨距離增加呈指數衰減將很難發生交互作用。
## 作用量
考慮描述一純量介子場 $\phi $ 和一狄拉克重子場 $\psi $ 之間交互作用的作用量，其可拆解為 | [
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] |
湯川耦合 | : : $S[\phi ,\psi ]=\int {\bigl [}\,{\mathcal {L}}_{\mathrm {meson} }(\phi )+{\mathcal {L}}_{\mathrm {baryon} }(\psi )+{\mathcal {L}}_{\mathrm {int} }(\phi ,\psi )\,{\bigr ]}\;\operatorname {d} ^{n}x$
其中積分範圍為一 $n$ 維空間，若考慮四維時空則有 $\operatorname {d} ^{4}x\equiv \operatorname {d} x_{1}\,\operatorname {d} x_{2}\,\operatorname {d} x_{3}\,\operatorname {d} x_{4}$ 。
介子的拉格朗日量可寫成
: : ${\mathcal {L}}_{\mathrm {meson} }(\phi )={\frac {1}{2}}\partial ^{\mu }\phi \;\partial _{\mu }\phi -V(\phi )~.$
其中 $V(\phi )$ 為自身交互作用的位能項。對於一帶有質量 $\mu $ 的自由介子場，其為 $V(\phi )={\frac {1}{2}}\,\mu ^{2}\,\phi ^{2}$ ；對於一可重整化並帶有耦合常數 $\lambda $ 的自交互作用場，其為 $V(\phi )={\frac {1}{2}}\,\mu ^{2}\,\phi ^{2}+\lambda \,\phi ^{4}$ 。
自由重子的拉格朗日量可寫成
: : ${\mathcal {L}}_{\mathrm {baryon} }(\psi )={\bar {\psi }}\,\left(i\,\partial \!\!\!/-m\right)\,\psi $
其中 $m$ 為一正實數，對應到重子的質量。
交互作用的拉格朗日量為
: : ${\mathcal {L}}_{\mathrm {int} }(\phi ,\psi )=-V=-g\,{\bar {\psi }}\,\phi \,\psi $
其中 $g$ 為湯川耦合的耦合常數。
綜合以上各項可得 | [
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湯川耦合 | : : $S[\phi ,\psi ]=\int {\bigl [}{\frac {1}{2}}\,\partial ^{\mu }\phi \;\partial _{\mu }\phi -V(\phi )+{\bar {\psi }}\,\left(i\,\partial \!\!\!/-m\right)\,\psi -g\,{\bar {\psi }}\,\phi \,\psi \,{\bigr ]}\operatorname {d} ^{n}x$
## 標準模型中的湯川耦合
在標準模型中，希格斯場和費米子以湯川耦合的形式連繫在一起，藉由自發對稱性破缺提供費米子的質量。
考慮位能 $V(\phi )$ 在某非零的值 $\phi =\phi _{0}$ 具有極小值的情況，由於 $\phi _{0}$ 的真空期望值不為零，對應的拉格朗日量會產生自發對稱性破缺。位能 $V(\phi )=\mu ^{2}\,\phi ^{2}+\lambda \,\phi ^{4}$ 在 $\mu $ 為虛數時即是一個例子。
儘管手徵對稱性在標準模型中禁止了費米子藉由形如 $m{\bar {\psi }}\psi $ 的項產生質量， $\phi $ 場不為零的期望值透過另一種方式為費米子提供質量。藉由將作用量以 $\phi '=\phi -\phi _{0}$ 改寫 (其中 $\phi _{0}\neq 0$ 為 $\phi $ 場的真空期望值)，湯川耦合的形式變為
: : $V=g\,\phi \,{\bar {\psi }}\,\psi =g\,\phi '\,{\bar {\psi }}\,\psi +g\,\phi _{0}\,{\bar {\psi }}\,\psi $
因為 $g$ 和 $\phi _{0}$ 皆為常數，上式中的第二項可被視為提供了質量 $g\,\phi _{0}$ 。上述機制即為自發對稱性破缺提供費米子質量的粗略描述，其中的 $\phi '$ 被稱為希格斯場。
常數 $g$ 在標準模型中是一初始的輸入，也就是說，其無法被標準模型推導出來。湯川耦合在其中存在的根本原因仍尚未知曉，有待更完整深入的理論作解釋。
## 馬約拉納費米子
湯川耦合亦可存在於一純量場和一馬約拉納場之間。事實上，在考慮包含一純量場和一狄拉克場的湯川耦合時，其亦可視為一純量場和兩個帶有相同質量的馬約拉納場之間的交互作用。其作用量有以下形式 | [
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] |
湯川耦合 | : : $S[\phi ,\chi ]=\int \left[\,{\frac {1}{2}}\,\partial ^{\mu }\phi \;\partial _{\mu }\phi -V(\phi )+\chi ^{\dagger }\,i\,{\bar {\sigma }}\,\cdot \,\partial \chi +{\frac {i}{2}}\,(m+g\,\phi )\,\chi ^{T}\,\sigma ^{2}\,\chi -{\frac {i}{2}}\,(m+g\,\phi )^{*}\,\chi ^{\dagger }\,\sigma ^{2}\,\chi ^{*}\,\right]\;\operatorname {d} ^{n}x$
其中 $g$ 為一複數耦合常數、 $m$ 為一複數、 $n$ 為考慮的時空維度。
## 參見
* 湯川勢
## 資料來源
* Itzykson, Claude; Zuber, Jean-Bernard. Quantum Field Theory. New York: McGraw-Hill. 1980. ISBN 0-07-032071-3. 含有內容需登入查看的頁面 (link)
* Bjorken, James D.; Drell, Sidney D. Relativistic Quantum Mechanics. New York: McGraw-Hill. 1964. ISBN 0-07-232002-8. 含有內容需登入查看的頁面 (link)
* Peskin, Michael E.; Schroeder, Daniel V. An Introduction to Quantum Field Theory. Addison-Wesley. 1995. ISBN 0-201-50397-2. 含有內容需登入查看的頁面 (link) | [
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Deutsch–Jozsa算法 | **多伊奇 - 喬薩算法**（英語：Deutsch–Jozsa algorithm）是戴維・多伊奇和里查德・喬薩於 1992 年提出的一種確定性量子算法。1998 年，理察・克利夫、阿圖爾・埃克特、基婭拉・馬基亞韋洛（Chiara Macchiavello）與米凱萊・莫斯卡對其進行了改進。儘管該算法目前在現實中基本沒有用途，但可以證明它比任何可能的確定性經典算法都快指數級，是最早提出的有此特性的量子算法之一。
## 問題描述
在多伊奇 - 喬薩問題中，我們有一個被稱為預言機的黑盒量子計算機，它能實現某一函數 $f\colon \{0,1\}^{n}\rightarrow \{0,1\}$ 。該函數以 n 比特值作為輸入，輸出結果則為 0 或 1。問題承諾該函數要麼是常函數（即對任何輸入都輸出恆定值 0 或 1），要麼是平衡（balanced）函數（即對一半的輸入返回 1，另一半則返回 0）。問題的任務是通過預言機確定 $f$ 是常函數還是平衡函數。
## 問題背景
多伊奇 - 喬薩問題是專門為量子計算設計的一個問題，該問題對於量子算法而言相對容易，但對任何確定性經典算法來說都是十分困難的。其提出的動機是展示可以由量子計算機有效並且正確解決的一個黑盒問題，而同時確定性經典計算機則需要進行大量計算才能解決該問題。更準確地說，該問題表明了複雜度類 EQP（即可以由量子計算機在多項式時間內精確地解決）與 P 之間的預言分離（oracle separation）。
由於該問題在概率經典計算機上也很容易解決，因此它不會產生與複雜度類 BPP（即在概率經典計算機上以多項式時間解決且誤差有界）之間的預言分離。西蒙問題則是一個證明 BQP 和 BPP 之間產生預言分離的例子。
## 經典算法
對於傳統的確定性算法而言，假設 _n_ 是比特數，則在最壞情況下需要 $2^{n-1}+1$ 次對 $f$ 的求值。為了證明 $f$ 是常函數，必須對超過一半的輸入進行計算並且發現它們有相同的輸出。最好情況是 $f$ 為平衡函數且恰好選擇的前兩個輸入值對應不同的輸出值。對於傳統的隨機算法， $k$ 次求值足以產生高概率的正確答案（對 $k\geq 1$ 錯誤概率為 $\epsilon \leq 1/2^{k}$ ）。然而，如果想保證獲得正確答案的話，仍需要 $k=2^{n-1}+1$ 次求值。多伊奇 - 喬薩量子算法則僅需一次對 $f$ 的求值就能獲得準確的答案。
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Deutsch–Jozsa算法 | 多伊奇 - 喬薩問題是戴維・多伊奇早期工作（1985 年）的推廣。當時他提出的算法針對的是 $n=1$ 的簡單情形，即有一個輸入為 1 比特的布爾函數 $f:\{0,1\}\rightarrow \{0,1\}$ ，需要確定該函數是否是常函數。
多伊奇最早提出的算法實際上並不是確定性的。1992 年，多伊奇與喬薩共同提出了一種確定性算法，並將該算法推廣到 $n$ 比特的輸入值。與多伊奇的算法不同，該算法需要進行兩次而非一次函數求值。
後來克利夫等人又對多伊奇 - 喬薩算法進行了改進，提出了一種既具有確定性又僅需一次 $f$ 求值的算法。不過為了紀念多伊奇和喬薩兩人開創性的貢獻，該算法仍被稱為多伊奇 - 喬薩算法。
## 算法
多伊奇 - 喬薩算法成立的前提之一是由 $x$ 計算 $f(x)$ 的預言機必須是一個不會將 $x$ 退相干的量子預言機。此外，它也不能在算法結束後留下任何 $x$ 的拷貝。
假設我們有 $n+1$ 比特量子態 $|0\rangle ^{\otimes n}|1\rangle $ ，即前 n 比特分別處於 $|0\rangle $ 而最後一比特則是 $|1\rangle $ 。對每個比特應用阿達馬變換可以得到
: ${\frac {1}{\sqrt {2^{n+1}}}}\sum _{x=0}^{2^{n}-1}|x\rangle (|0\rangle -|1\rangle )$ .
$f$ 是由量子預言機實現的函數。預言機將量子態 $|x\rangle |y\rangle $ 映射到 $|x\rangle |y\oplus f(x)\rangle $ ，其中 $\oplus $ 是模 2 加法（由受控非門實現，可以看作是量子版異或門）。於是該量子預言機可以將上述量子態轉換為
: ${\frac {1}{\sqrt {2^{n+1}}}}\sum _{x=0}^{2^{n}-1}|x\rangle (|f(x)\rangle -|1\oplus f(x)\rangle )$ .
對於每個 $x,f(x)$ 的結果為 0 或 1。為了檢驗這兩種可能性，我們注意到上面的量子態等價於
: ${\frac {1}{\sqrt {2^{n+1}}}}\sum _{x=0}^{2^{n}-1}(-1)^{f(x)}|x\rangle (|0\rangle -|1\rangle )$ . | [
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Deutsch–Jozsa算法 | 此時可以忽略最後一個量子比特 ${\frac {|0\rangle -|1\rangle }{\sqrt {2}}}$ ，剩餘部分為：
: ${\frac {1}{\sqrt {2^{n}}}}\sum _{x=0}^{2^{n}-1}(-1)^{f(x)}|x\rangle $ .
再對每一量子比特進行阿達馬變換，得到
: ${\frac {1}{\sqrt {2^{n}}}}\sum _{x=0}^{2^{n}-1}(-1)^{f(x)}\left[{\frac {1}{\sqrt {2^{n}}}}\sum _{y=0}^{2^{n}-1}(-1)^{x\cdot y}|y\rangle \right]={\frac {1}{2^{n}}}\sum _{y=0}^{2^{n}-1}\left[\sum _{x=0}^{2^{n}-1}(-1)^{f(x)}(-1)^{x\cdot y}\right]|y\rangle $
其中 $x\cdot y=x_{0}y_{0}\oplus x_{1}y_{1}\oplus \cdots \oplus x_{n-1}y_{n-1}$ 是每一比特相應乘積之和。
最後，我們可以檢驗測量得到 $|0\rangle ^{\otimes n}$ 的概率
: ${\bigg |}{\frac {1}{2^{n}}}\sum _{x=0}^{2^{n}-1}(-1)^{f(x)}{\bigg |}^{2}$
當 $f(x)$ 是常函數（相長干涉）時此概率為 1，當 $f(x)$ 是平衡函數（相消干涉）時此概率為 0。換句話說，如果 $f(x)$ 是常函數的話測量結果為 $|0\rangle ^{\otimes n}$ （即所有量子比特全為零），其他結果則表明 $f(x)$ 是平衡函數。
## 多伊奇算法
多伊奇算法是一般多伊奇 - 喬薩算法的特例。此時我們需要檢查 $f(0)=f(1)$ 是否成立。這相當於檢查 $f(0)\oplus f(1)$ ，當結果為零時表明 $f$ 是常函數，否則則不是常函數。
我們從兩比特量子態 $|0\rangle |1\rangle $ 開始，對每一量子比特應用阿達馬變換，結果為
: ${\frac {1}{2}}(|0\rangle +|1\rangle )(|0\rangle -|1\rangle ).$ | [
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Deutsch–Jozsa算法 | 函數 $f$ 可以將 $|x\rangle |y\rangle $ 映射為 $|x\rangle |f(x)\oplus y\rangle $ ，將此函數應用於當前的量子態，可以得到
: ${\frac {1}{2}}(|0\rangle (|f(0)\oplus 0\rangle -|f(0)\oplus 1\rangle )+|1\rangle (|f(1)\oplus 0\rangle -|f(1)\oplus 1\rangle ))$
: $={\frac {1}{2}}((-1)^{f(0)}|0\rangle (|0\rangle -|1\rangle )+(-1)^{f(1)}|1\rangle (|0\rangle -|1\rangle ))$
: $=(-1)^{f(0)}{\frac {1}{2}}\left(|0\rangle +(-1)^{f(0)\oplus f(1)}|1\rangle \right)(|0\rangle -|1\rangle ).$
忽略最後一個比特與全局相位因子，可以得到量子態
: ${\frac {1}{\sqrt {2}}}(|0\rangle +(-1)^{f(0)\oplus f(1)}|1\rangle ).$
再次應用阿達馬變換，得到
: ${\frac {1}{2}}(|0\rangle +|1\rangle +(-1)^{f(0)\oplus f(1)}|0\rangle -(-1)^{f(0)\oplus f(1)}|1\rangle )$
: $={\frac {1}{2}}((1+(-1)^{f(0)\oplus f(1)})|0\rangle +(1-(-1)^{f(0)\oplus f(1)})|1\rangle ).$
若且唯若測量結果為 $|0\rangle $ 時有 $f(0)\oplus f(1)=0$ 。反之，若且唯若測量結果為 $|1\rangle $ 時有 $f(0)\oplus f(1)=1$ 。因此我們可以肯定地知道 $f(x)$ 是否為常函數。
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洛夫·格罗弗 | **洛夫・庫瑪爾・格羅弗**（英語：Lov Kumar Grover，1961 年 —）是一名印度裔美國電腦科學家。他是用於量子計算的格羅弗資料庫搜索算法的發起人。格羅弗於 1996 年的算法贏得了聲譽，是為量子計算提出的第二個主要算法（繼 1994 年的秀爾演算法之後），2017 年終於在可擴展的物理量子系統中實現。格羅弗的算法已經成為眾多科普文章的主題。
格羅弗於 1981 年獲得德里印度理工學院的學士學位，並於 1985 年獲得史丹佛大學的電子工程博士學位。1984 年，格羅弗去了貝爾實驗室。1987 年至 1994 年，他繼續在康乃爾大學擔任助理教授。他在 2008 年退休，成為一名獨立的研究人員，並患有帕金森氏症。
## 著作
* Grover L.K.: _A fast quantum mechanical algorithm for database search （頁面存檔備份，存於網際網路檔案館）_ , Proceedings, 28th Annual ACM Symposium on the Theory of Computing, (May 1996) p. 212
* Grover L.K.: _From Schrödinger's equation to quantum search algorithm （頁面存檔備份，存於網際網路檔案館）_ , American Journal of Physics, 69(7): 769–777, 2001. Pedagogical review of the algorithm and its history.
* Grover L.K.: Quantum Computing: How the weird logic of the subatomic world could make it possible for machines to calculate millions of times faster than they do today （頁面存檔備份，存於網際網路檔案館） _The Sciences_, July/August 1999, pp. 24–30.
* What's a Quantum Phone Book?, Lov Grover, Lucent Technologies
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