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勃格留波夫變換 | 因為 Bogoliubov 變換是算符的線性重組，所以將它們寫成矩陣變換更方便簡潔。如果一對湮滅算符 $(a,b)$ 按照下面變化
: ${\begin{pmatrix}\alpha \\\beta \end{pmatrix}}=U{\begin{pmatrix}a\\b\end{pmatrix}}$
其中 $U$ 是一個 $2\times 2$ 矩陣。那麼自然
: ${\begin{pmatrix}\alpha ^{\dagger }\\\beta ^{\dagger }\end{pmatrix}}=U^{*}{\begin{pmatrix}a^{\dagger }\\b^{\dagger }\end{pmatrix}}$
對於費米子算符，對易關係的要求體現在對矩陣 $U$ 的兩個條件，即
: $U={\begin{pmatrix}u&v\\-v^{*}&u^{*}\end{pmatrix}}$
和
: $|u|^{2}+|v|^{2}=1$
對於玻色子算子，對易關係需要
: $U={\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}$
和
: $|u|^{2}-|v|^{2}=1$
這些條件可以統一寫成
: $U\Gamma _{\pm }U^{\dagger }=\Gamma _{\pm }$
其中
: $\Gamma _{\pm }={\begin{pmatrix}1&0\\0&\pm 1\end{pmatrix}}$
而 $\Gamma _{\pm }$ 分別適用於費米子和玻色子。
### 使用矩陣描述對角化二次哈密頓量
Bogoliubov 變換讓我們通過對角化矩陣 $\Gamma _{\pm }H$ 來對角化二次哈密頓量
: ${\hat {H}}={\begin{pmatrix}a^{\dagger }&b^{\dagger }\end{pmatrix}}H{\begin{pmatrix}a\\b\end{pmatrix}}.$
在上面的符號中，區分運算符 ${\hat {H}}$ 和矩陣 $H$ 很重要。這個通過重寫 ${\hat {H}}$ 看到 | [
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] |
勃格留波夫變換 | : ${\hat {H}}={\begin{pmatrix}\alpha ^{\dagger }&\beta ^{\dagger }\end{pmatrix}}\Gamma _{\pm }U(\Gamma _{\pm }H)U^{-1}{\begin{pmatrix}\alpha \\\beta \end{pmatrix}}$
而 $\Gamma _{\pm }U(\Gamma _{\pm }H)U^{-1}=D$ 若且唯若 $U$ 對角化了 $\Gamma _{\pm }H$ ，即 $U(\Gamma _{\pm }H)U^{-1}=\Gamma _{\pm }D$ 。
下面列出了 Bogoliubov 變換的一些有用的性質。
## 相關條目
* 荷斯坦 - 普里馬科夫變換
* 約旦 - 維格納變換
* 喬丹 - 施溫格變換
* 克萊因變換
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氦-3表面自旋回声 | **氦 - 3 自旋迴聲（HeSE）** 是表面科學中的一種原子散射技術，可用於測量超高真空中晶體表面的微觀動力學。氦 - 3 自旋迴聲補充和擴展了其他非彈性散射技術，例如中子自旋迴聲和傳統的氦原子散射技術 (HAS)。
## 原理
氦 - 3 自旋迴聲的實驗原理類似於中子自旋迴聲。概括地說，氦 - 3 自旋迴聲技術利用磁場和核自旋的相互作用將氦 - 3 原子束一分為二，並使這兩束氦原子在不同的時刻和樣品相互作用，並收集反射的氦原子束的自旋數據，以此測量表面或者表面吸附物在皮秒量級的時間尺度之內的變化。
### 氦 - 3 自旋迴聲觀測表面吸附粒子運動的半經典理論詮釋
氦 - 3 原子的核自旋為 $1/2$ ，在任意方向上會有自旋向上或者自旋向下兩個自旋分量。氦 - 3 自旋迴聲技術使用自旋起偏器產生在 x 方向向上的自旋
可以發現，x 方向上的自旋可以表示成 z 方向自旋向上和向下的和。這兩個自旋分量 $\psi _{z+}$ 和 $\psi _{z-}$ 會經過由螺線管產生的 z 方向上的磁場。在磁場中兩個自旋分量會有不同的速度，所以兩個自旋分量到達表面的時間會有一定的差，稱為自旋迴聲時間，或 $t_{SE}$ 。 $t_{SE}$ 與螺線管中的磁場強度成正比，可以在幾皮秒的時間尺度內精準調控。散射後的氦原子會進入另一個螺線管，這個螺線管中的磁場和第一個螺線管大小相等方向相反，所以兩束氦原子會重新「組合」成為一束。在由表面散射時，因為到達樣品表面的時間不同，兩束氦原子所「看到」的表面也有差異，這種差異會使得兩束氦原子的相位出現 $\phi $ 的偏差。重新組合後的氦原子的自旋會變為
x 方向上的自旋緊接著會由自旋偏振裝置檢測，結果為
在 $t_{SE}=0$ ，即螺線管中不通電，磁場為零時，兩束氦原子不產生相位差， $\phi =0$ ，x 方向的自旋 $P_{x}=1$ 。隨著 $t_{SE}$ 的增大， $P_{x}$ 會逐漸衰減，從衰減信號就可以獲知表面吸附的分子或原子的運動模式。如果將第 $i$ 個吸附粒子在 $t_{SE}=0$ 中於表面產生的位移為 $\mathbf {R} _{i}(t_{SE})$ ，氦 - 3 原子在散射前後平行於表面的動量變化為 $\hbar \Delta \mathbf {K} $ ，那麼由第 $i$ 個吸附粒子產生的相位差可以表示為 | [
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] |
氦-3表面自旋回声 | 如果要考慮所有 $N$ 個吸附粒子對最終的粒子數的 x 方向自旋的影響，則會得到
更一般地，如果用范霍夫關聯函數 $G(\mathbf {R} ,t)$ 來表示吸附粒子在時間 $t$ 內移動了 $\mathbf {R} $ 的概率，那麼
一般情況下，因為吸附粒子運動的對稱性， $G(\mathbf {R} ,t)=G(-\mathbf {R} ,t)$ 。所以
由此可得
所以 $P_{x}$ 即為 $G(\mathbf {R} ,t)$ 的空間二維傅立葉變換，一般稱其為中間散射函數（intermediate scattering function，或 ISF），記作 $I(\Delta \mathbf {K} ,t)$ 。
一般的吸附粒子運動模式包括二維布朗運動（隨機漫步）、在特定晶格位置間的跳躍、以及二維理想氣體（彈性運動）。
#### 二維布朗運動（隨機漫步）
#### 特定晶格位置間的跳躍
#### 二維理想氣體（彈性運動）
## 應用
氦原子散射可以大致分為彈性散射、准彈性散射和非彈性散射。彈性散射中，氦原子的動能在散射前後沒有變化，可以用於測量表面的結構信息和選擇吸附共振。准彈性散射中的氦原子在散射過程中的動能變化相對較小，適於觀測表面吸附分子或原子的運動。非彈性散射中氦原子的動能變化較大，可以此測量固體表面的准粒子，比如表面聲子。氦 - 3 自旋迴聲可以以超高的能量解析度來觀測准彈性散射和非彈性散射，所以適於微觀擴散和聲子壽命等需要精確測量能量的研究領域。
### 微觀擴散
氦 - 3 自旋迴聲已被用於研究原子和分子在固體表面的擴散速率和運動機制，如氫原子在表面擴散中的核量子效應， 對吸附物和表面相互作用能量進行測量， 吸附物與表面之間的能量交換，表面吸附物之間的相互作用。
### 選擇吸附共振
通過測量 LiF (001) 表面和氫化 Si (111) 表面上的選擇吸附共振（束縛態共振），氦 - 3 自旋迴聲可以用於測量氦和固體表面的相互作用勢能。
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Subsets and Splits