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时间复杂性 | 常見的非合次線性時間演算法都採用了諸如平行處理（就像 NC1 matrix 行列式計算那樣）、非古典處理（如同葛羅佛搜尋那樣），又或者選擇性地對有保證的輸入結構作出假設（如冪對數時間的二分搜尋）。不過，一些情況，例如在頭 log (n) 位元中每個字串有一個位元作為索引的字串組就可能依賴於輸入的每個位元，但又符合次線性時間的條件。
「次線性時間演算法」通常指那些不符合前一段的描述的演算法。它們通常執行於傳統電腦架構系列並且不容許任何對輸入的事先假設。但是它們可以是隨機化演算法，而且必須是真隨機演算法除了特殊情況。
## 線性時間
如果一個演算法的時間複雜度為 O (_n_)，則稱這個演算法具有線性時間，或 **O(_n_)** 時間。非正式地說，這意味著對於足夠大的輸入，執行時間增加的大小與輸入成線性關係。例如，一個計算列表所有元素的和的程式，需要的時間與列表的長度成正比。這個描述是稍微不準確的，因為執行時間可能顯著偏離一個精確的比例，尤其是對於較小的 n。
## 線性對數（準線性）時間
若一個演算法時間複雜度 T (n) = O (nlog n)，則稱這個演算法具有線性對數時間。因此，從其表達式我們也可以看到，線性對數時間增長得比線性時間要快，但是對於任何含有 n，且 n 的冪指數大於 1 的多項式時間來說，線性對數時間卻增長得慢。
## 多項式時間
### 強多項式時間與弱多項式時間
### 複雜度類別
從多項式時間的概念出發，在計算複雜度理論中可以得到一些複雜度類別。以下是一些重要的例子。
* **P**：包含可以使用確定型圖靈機在多項式時間內解決的決定性問題。
* **NP**：包含可以使用非確定型圖靈機在多項式時間內解決的決定性問題。
* **ZPP**：包含可以使用概率圖靈機在多項式時間內零錯誤解決的決定性問題。
* **RP**：包含可以使用概率圖靈機在多項式時間內解決的決定性問題，但它給出的兩種答案中（是或否）只有一種答案是一定正確的，另一種則有機率不正確。
* **BPP**：包含可以使用概率圖靈機在多項式時間內解決的決定性問題，它給出的答案有錯誤的概率在某個小於 0.5 的常數之內。
* **BQP**：包含可以使用量子圖靈機在多項式時間內解決的決定性問題，它給出的答案有錯誤的概率在某個小於 0.5 的常數之內。 | [
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时间复杂性 | 在機器模型可變的情況下，P 在確定性機器上是最小的時間複雜度類別。例如，將單帶圖靈機換成多帶圖靈機可以使演算法執行速度以二次階提升，但所有具有多項式時間的演算法依然會以多項式時間執行。一種特定的抽象機器會有自己特定的複雜度類別分類。
## 超越多項式時間
如果一個演算法的時間 _T_(_n_) 沒有任何多項式上界，則稱這個演算法具有**超越多項式**（superpolynomial）時間。在這種情況下，對於所有常數 _c_ 我們都有 _T_(_n_) = ω(_n_c)，其中 _n_ 是輸入參數，通常是輸入的數據量（位元數）。指數時間顯然屬於超越多項式時間，但是有些演算法僅僅是很弱的超越多項式演算法。例如，Adleman-Pomerance-Rumely 質數測試對於 _n_ 位元的輸入需要運行 _n_O(log log _n_) 時間；對於足夠大的 _n_，這時間比任何多項式都快；但是輸入要大得不切實際，時間才能真正超過低階的多項式。
## 准多項式時間
**準多項式時間**演算法是運算慢於多項式時間的演算法，但不會像指數時間那麼慢。對一些固定的 $c>0$ ，準多項式時間演算法的最壞情況運行時間是 $2^{O((\log n)^{c})}$ 。如果準多項式時間演算法定義中的常數「c」等於 1，則得到多項式時間演算法；如果小於 1，則得到一個次線性時間演算法。
## 次指數時間
術語次指數時間用於表示某些演算法的運算時間可能比任何多項式增長得快，但仍明顯小於指數。在這種狀況下，具有次指數時間演算法的問題比那些僅具有指數演算法的問題更容易處理。「次指數」的確切定義並沒有得到普遍的認同，我們列出了以下兩個最廣泛使用的。
### 第一定義
如果一個問題解決的運算時間的對數值比任何多項式增長得慢，則可以稱其為次指數時間。更準確地說，如果對於每個 ε> 0，存在一個能於時間 O (2nε) 內解決問題的演算法，則該問題為次指數時間。所有這些問題的集合是複雜性 SUBEXP，可以按照 DTIME 的方式定義如下。
: ${\text{SUBEXP}}=\bigcap _{\varepsilon >0}{\text{DTIME}}\left(2^{n^{\varepsilon }}\right)$
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时间复杂性 | 一些作者將次指數時間定義為 2o(_n_) 的運算時間。該定義允許比次指數時間的第一個定義更多的運算時間。這種次指數時間演算法的一個例子，是用於整數因式分解的最著名古典演算法 —— 普通數域篩選法，其運算時間約為 $2^{{\tilde {O}}(n^{1/3})}$ ，其中輸入的長度為 _n_。另一個例子是圖形同構問題的最著名演算法，其運算時間為 $2^{O({\sqrt {n\log n}})}$ 。
## 指數時間
若 _T_(_n_) 是以 2poly(_n_) 為上界，其中 poly (_n_) 是 _n_ 的多項式，則演算法被稱為**指數時間**。更正規的講法是：若 _T_(_n_) 對某些常數 _k_ 是由 O (2nk) 所界定，則演算法被稱為**指數時間**。在確定性圖靈機上認定為指數時間演算法的問題，形成稱為 **EXP** 的複雜性級別。
: ${\text{EXP}}=\bigcup _{c\in \mathbb {N} }{\text{DTIME}}\left(2^{n^{c}}\right)$
有時侯，指數時間用來指稱具有 _T_(_n_) = 2O(_n_) 的演算法，其中指數最多為 _n_ 的線性函式。這引起複雜性等級 **E**。
: ${\text{E}}=\bigcup _{c\in \mathbb {N} }{\text{DTIME}}\left(2^{cn}\right)$
## 雙重指數時間
若 _T_(_n_) 是以 22poly(_n_) 為上界，其中 poly (_n_) 是 _n_ 的多項式，則演算法被稱為雙重指數時間。這種演算法屬於複雜性等級 2-EXPTIME。
: ${\mbox{2-EXPTIME}}=\bigcup _{c\in \mathbb {N} }{\mbox{DTIME}}\left(2^{2^{n^{c}}}\right)$
眾所周知的雙重指數時間演算法包括：
* 預膨脹算術的決策程式
* 計算葛洛拿基底（在最差狀況）
* 實封閉體的量詞消去至少耗費雙重指數時間，而且可以在這樣的時間內完成。
## 參見
* L-notation
* 遊戲複雜度
* 計算複雜性理論
* 零一律
* 柯爾莫哥洛夫空間
* 柯氏複雜性
* 空間複雜度
## 參考資料 | [
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費米-狄拉克統計法 | **費米 - 狄拉克統計**（英語：**Fermi–Dirac statistics**），簡稱**費米統計**或 **FD 統計**，是統計力學中描述由大量滿足包立不相容原理的費米子組成的系統中粒子分處不同量子態的統計規律。該統計規律的命名源於恩里科・費米和保羅・狄拉克，他們分別獨立地發現了該統計律。不過費米在數據定義比狄拉克稍早。
費米–狄拉克統計的適用對象是熱平衡的費米子 (自旋量子數為半奇數的粒子)。此外，應用此統計規律的前提是系統中各粒子間交互作用可忽略不計。如此便可用粒子在不同定態的分布狀況來描述大量微觀粒子組成的宏觀系統。不同的粒子分處不同能態，這點對系統許多性質會產生影響。自旋量子數為 1/2 的電子是費米–狄拉克統計最普遍的應用對象。費米–狄拉克統計是統計力學的重要組成部分，它利用了量子力學的一些原理。
## 概述
根據量子力學，費米子為自旋為半奇數的粒子，其本徵波函數反對稱，在費米子的某一個能階上，最多只能容納一個粒子。因而符合費米–狄拉克統計分布的粒子，當他們處於某一分布 $\left\{n_{j}\right\}$ （「某一分布」指這樣一種狀態：即在能量為 $\left\{\epsilon _{j}\right\}$ 的能階上同時有 $n_{j}$ 個粒子存在著，不難想像，當從宏觀觀察體系能量一定的時候，從微觀角度觀察體系可能有很多種不同的分布狀態，而且在這些不同的分布狀態中，總有一些狀態出現的機率特別的大，而其中出現機率最大的分布狀態被稱為最可幾分布）時，體系總狀態數為：
: $\Omega _{j}={\frac {g_{j}!}{n_{j}!(g_{j}-n_{j})!}}$
費米–狄拉克統計的最可幾分布的數學表達式為：
: $\left\{n_{j}^{FD}\right\}={\frac {g_{j}e^{\alpha }e^{\beta \epsilon _{j}}}{1+e^{\alpha }e^{\beta \epsilon _{j}}}}$
由於費米 - 狄拉克統計在數學處理上非常困難，因此在處理實際問題時經常引入一些近似條件，使費米 - 狄拉克統計退化成為古典的馬克士威 - 波茲曼統計。此外，對於玻色子，也有對應的玻色 - 愛因斯坦統計予以處理。
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費米-狄拉克統計法 | 1926 年發現費米–狄拉克統計之前，要理解電子的某些性質尚較為困難。例如，在常溫下，未施加電流的金屬內部的熱容比施加電流的金屬少了大約 100 倍。此外，在常溫下給金屬施加一強電場，將造成場致電子發射（Field electron emission）現象，從而產生電流流經金屬。研究發現，這個電流與溫度幾乎無關。當時的理論難以解釋這個現象。
當時，由於人們主要根據的是古典靜電學理論，因此在諸如金屬電子理論等方面遇到的困難，無法得到令人滿意的解答。他們認為，金屬中所有電子都是等效的。也就是說，金屬中的每個電子都以相同的程度對金屬的熱量做出貢獻（這個量是波爾茲曼常數的一次項）。上述問題一直困擾著科學家，直到費米–狄拉克統計的發現，才得到較好地解釋。
1926 年，恩里科・費米、保羅・狄拉克各自獨立地在發表了有關這一統計規律的兩篇學術論文。另有來源顯示，P・喬丹（Pascual Jordan）在 1925 年也對這項統計規律進行了研究，他稱之為「包立統計」，不過他並未及時地發表他的研究成果。狄拉克稱此項研究是費米完成的，他稱之為「費米統計」，並將對應的粒子稱為「費米子」。
1926 年，拉爾夫・福勒在描述恆星向白矮星的轉變過程中，首次應用了費米–狄拉克統計的原理。1927 年，阿諾・索末菲將費米–狄拉克統計應用到他對於金屬電子的研究中。1928 年，福勒和 L・W・諾德漢（Lothar Wolfgang Nordheim）在場致電子發射的研究中，也採用了這一統計規律。直至今日，費米–狄拉克統計仍然是物理學的一個重要部分。
## 費米–狄拉克分布
根據費米–狄拉克分布，給定費米子組成的系統中處於量子態 $i$ 上的平均粒子數可以通過下面的式子計算：
: ${\bar {n}}_{i}={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}+1}}$
其中 $k$ 是波爾茲曼常數， $T$ 為絕對溫度（熱力學溫標）， $\epsilon _{i}\ $ 為量子態 $i$ 上單個粒子的能量， $\mu \ $ 是化學勢。當 $T=0K$ 時，化學勢就是系統的費米能。半導體中電子的費米能，也被被稱為費米能階。 | [
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費米-狄拉克統計法 | 要應用費米–狄拉克統計，系統必須滿足一定的條件：系統的費米子數量必須足夠大，以至於再加入一個費米子所引起化學勢 $\mu \ $ 的變化可以忽略不計。由於費米–狄拉克統計的推導過程中利用了包立不相容原理，即單個量子態上最多能有一個粒子，這樣的結果就是某個量子態上的平均量子數滿足 $0<{\bar {n}}_{i}<1$ 。
* 費米–狄拉克分布
* *
（點擊圖片可以獲得完整尺寸）
### 粒子的能量分布
前面的章節敘述了給定費米子系統在不同量子態上的分布，一個量子態上最多只能具有一個費米子。利用費米–狄拉克統計，還可以獲得費米子系統不同能量值上的分布情況，這與分析量子態的原理略有不同，因為可能出現多個定態具有同一能量值，即出現所謂的簡併能量態情況。
將費米–狄拉克統計中某個量子態上的平均粒子數 ${\bar {n}}_{i}\ $ 與簡併度 $g_{i}\ $ （即能量值為 $\epsilon _{i}\ $ 的量子態數）相乘，就可以得到能量為 $\epsilon _{i}\ $ 的平均費米子數。
: ${\begin{alignedat}{2}{\bar {n}}(\epsilon _{i})&=g_{i}\ {\bar {n}}_{i}\\&={\frac {g_{i}}{e^{(\epsilon _{i}-\mu )/kT}+1}}\\\end{alignedat}}$
當 $g_{i}\geq 2\ $ 時，可能出現 $\ {\bar {n}}(\epsilon _{i})>1$ 。導致這個現象的原因前面提到過，即具有同一個能量值的粒子可能處於不同的定態，也就是說完全可能出現多個粒子處於同一能量值 $\epsilon _{i}\ $ 。
當一個系統的能量是准連續（quasi-continuum）的，定義其單位體積內單位能量域的量子態數為狀態密度。，單位能量域的平均費米子數為
: ${\bar {\mathcal {N}}}(\epsilon )=g(\epsilon )\ F(\epsilon )$
這裡 $F(\epsilon )\ $ 被稱為費米函數，它與前面用來表達量子態 ${\bar {n}}_{i}$ 上粒子數分布的函數具有相同的形式。
: $F(\epsilon )={\frac {1}{e^{(\epsilon -\mu )/kT}+1}}$
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] |
費米-狄拉克統計法 | : ${\bar {\mathcal {N}}}(\epsilon )={\frac {g(\epsilon )}{e^{(\epsilon -\mu )/kT}+1}}$
## 量子範疇和古典範疇
如果古典範疇中涉及的位移、動量之間的關係還遠未達到不確定性原理所設定的極限，通常可以採用馬克士威 - 波茲曼統計來代替費米–狄拉克統計，這樣做可以簡化數學計算的難度。如果粒子平均間距 ${\bar {R}}$ 遠大於粒子的平均物質波波長 ${\bar {\lambda }}$ ，就可以採用上述古典範疇的處理方式。
: ${\bar {R}}\ \gg \ {\bar {\lambda }}\ \approx \ {\frac {h}{\sqrt {3mkT}}}$
這裡， $h$ 為普朗克常數， $m$ 為粒子的質量。
對於常溫（約 300 克耳文）下金屬中的電子，由於 ${\bar {R}}\approx {\bar {\lambda }}/25$ ，因此該系統遠離古典範疇。這是因為電子質量較小，並且在金屬中聚集程度較高。這樣，為了分析金屬中的傳導電子，必須採用費米–狄拉克統計。
由恆星演變而來的白矮星，是另一個不屬於古典範疇、必須採用費米–狄拉克統計的例子。儘管白矮星的溫度很高（其表面溫度通常能達到 10,000 克耳文），但是它內部高度聚集的電子和每個電子的低質量，使得處理這問題必須採用費米–狄拉克統計，而不能用古典的波爾茲曼統計近似處理。
## 參考文獻
## 相關條目
* 量子統計
* 盒中氣體
* 費米氣體
* 白矮星
* 中子星
* 全同粒子
* 馬克士威 - 玻茲曼統計 | [
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玻色－爱因斯坦统计 | **玻色 - 愛因斯坦統計**是玻色子所依從的統計規律。
根據量子力學，玻色子是自旋為整數的粒子，其本徵波函數對稱，在玻色子的某一個能階上，可以容納無限個粒子。因而符合玻色 - 愛因斯坦統計分布的粒子，當他們處於某一分布 $\left\{n_{j}\right\}$ （「某一分布」指這樣一種狀態：即在能量為 $\left\{\epsilon _{j}\right\}$ 的能階上同時有 $n_{j}$ 個粒子存在著，不難想像，當宏觀觀察體系能量一定的時候，從微觀角度觀察體系可能有很多種不同的分布狀態，而且在這些不同的分布狀態中，總有一些狀態出現的機率特別的大，而其中出現機率最大的分布狀態被稱為最可幾分布）時，體系總狀態數為：
: $\Omega _{j}={\frac {(g_{j}+n_{j}-1)!}{n_{j}!(g_{j}-1)!}}$
對這一公式的理解是這樣的：把 $g_{j}$ 個簡併能階看作一個擁有 $g_{j}$ 個隔室的大盒子，把 $n_{j}$ 個粒子看作準備放入盒子中的 $n_{j}$ 個不可區分的小球，則可以把這個向盒子裡面放小球的過程看作 $n_{j}$ 個小球和盒子中 $(g_{j}-1)$ 個隔室壁的隨機排列過程，則這樣的排列一共有 $(g_{j}+n_{j}-1)!$ 種可能出現的狀態；另一方面，小球和小球是不可區分的，隔室壁和隔室壁也是不可區分的，因此對小球和隔室壁的計數都有重複，需要除以這種重複計數 $(g_{j}-1)!$ 和 $(n_{j})!$ ，最終得到的結果就是上述結果。
: $\Omega _{j}={\frac {(g_{j}+n_{j}-1)!}{n_{j}!(g_{j}-1)!}}g_{j}=3;n_{j}=2;\Omega _{j}=6$
玻色 - 愛因斯坦統計的最可幾分布的數學表達式為：
: $\left\{n_{j}^{BE}\right\}={\frac {g_{j}e^{\alpha }e^{\beta \epsilon _{j}}}{1-e^{\alpha }e^{\beta \epsilon _{j}}}}$
由於量子統計在數學處理上非常困難（對於非物理系所的人員而言的確如此），因此在處理實際問題時經常引入一些近似條件，使費米 - 狄拉克統計和玻色 - 愛因斯坦統計退化成為古典的馬克士威 - 波茲曼統計。
## 參考文獻
## 參見
* 量子統計
* 盒中氣體 | [
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玻色－爱因斯坦统计 | * 玻色 - 愛因斯坦凝聚
* 玻色氣體
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波茲-愛因斯坦統計 | **玻色 - 愛因斯坦統計**是玻色子所依從的統計規律。
根據量子力學，玻色子是自旋為整數的粒子，其本徵波函數對稱，在玻色子的某一個能階上，可以容納無限個粒子。因而符合玻色 - 愛因斯坦統計分布的粒子，當他們處於某一分布 $\left\{n_{j}\right\}$ （「某一分布」指這樣一種狀態：即在能量為 $\left\{\epsilon _{j}\right\}$ 的能階上同時有 $n_{j}$ 個粒子存在著，不難想像，當宏觀觀察體系能量一定的時候，從微觀角度觀察體系可能有很多種不同的分布狀態，而且在這些不同的分布狀態中，總有一些狀態出現的機率特別的大，而其中出現機率最大的分布狀態被稱為最可幾分布）時，體系總狀態數為：
: $\Omega _{j}={\frac {(g_{j}+n_{j}-1)!}{n_{j}!(g_{j}-1)!}}$
對這一公式的理解是這樣的：把 $g_{j}$ 個簡併能階看作一個擁有 $g_{j}$ 個隔室的大盒子，把 $n_{j}$ 個粒子看作準備放入盒子中的 $n_{j}$ 個不可區分的小球，則可以把這個向盒子裡面放小球的過程看作 $n_{j}$ 個小球和盒子中 $(g_{j}-1)$ 個隔室壁的隨機排列過程，則這樣的排列一共有 $(g_{j}+n_{j}-1)!$ 種可能出現的狀態；另一方面，小球和小球是不可區分的，隔室壁和隔室壁也是不可區分的，因此對小球和隔室壁的計數都有重複，需要除以這種重複計數 $(g_{j}-1)!$ 和 $(n_{j})!$ ，最終得到的結果就是上述結果。
: $\Omega _{j}={\frac {(g_{j}+n_{j}-1)!}{n_{j}!(g_{j}-1)!}}g_{j}=3;n_{j}=2;\Omega _{j}=6$
玻色 - 愛因斯坦統計的最可幾分布的數學表達式為：
: $\left\{n_{j}^{BE}\right\}={\frac {g_{j}e^{\alpha }e^{\beta \epsilon _{j}}}{1-e^{\alpha }e^{\beta \epsilon _{j}}}}$
由於量子統計在數學處理上非常困難（對於非物理系所的人員而言的確如此），因此在處理實際問題時經常引入一些近似條件，使費米 - 狄拉克統計和玻色 - 愛因斯坦統計退化成為古典的馬克士威 - 波茲曼統計。
## 參考文獻
## 參見
* 量子統計
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波茲-愛因斯坦統計 | * 玻色 - 愛因斯坦凝聚
* 玻色氣體
* 全同粒子
* | [
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] |
顯微鏡 | **顯微鏡**泛指將微小不可見或難見物品之影像放大，而能被肉眼或其他成像儀器觀察之工具。日常用語中之顯微鏡多指光學顯微鏡，放大倍率和清析度（聚焦）為顯微鏡重要因素。
顯微鏡是在 1590 年由荷蘭的查哈里亞斯・楊森及其子所首創。顯微鏡的類型有許多。最常見的（和第一個被發明的）是光學顯微鏡，其他主要的顯微鏡類型包括電子顯微鏡、掃描探針顯微鏡等。
## 發明
最早的顯微鏡是 16 世紀在荷蘭製造出來，發明者是亞斯・詹森，荷蘭眼鏡商。同時另一位荷蘭科學家漢斯・利珀希也製造了顯微鏡。後來有兩個人開始在科學上使用顯微鏡，第一個是義大利科學家伽利略。他在 1611 年通過顯微鏡觀察到一種昆蟲後，第一次對它的複眼進行了描述。第二個是荷蘭亞麻織品商人雷文霍克（1635 年 - 1703 年），他自己學會了磨製透鏡。他第一次描述了許多肉眼所看不見的微小植物和動物。
1953 年，弗里茨・塞爾尼克因為對相襯法的證實，發明相襯顯微鏡獲得諾貝爾物理學獎。
1986 年，恩斯特・魯斯卡因研製第一台透視電子顯微鏡獲得諾貝爾物理學獎。格爾德・賓寧、海因里希・羅雷爾因研製掃描隧道顯微鏡獲得諾貝爾物理學獎。
2014 年 10 月 8 日，諾貝爾化學獎頒給了艾力克・貝齊格 (Eric Betzig)，W・E・莫爾納爾 (William Moerner) 和斯特凡・W・赫爾 (Stefan Hell)，獎勵其發展超分辨熒光顯微鏡 (Super-Resolved Fluorescence Microscopy)，帶領光學顯微鏡進入奈米級尺度中。
2017 年，雅克・杜博歇、約阿希姆・弗蘭克、理察・亨德森因研製用於溶液內生物分子的高解析度結構測定的低溫電子顯微鏡獲得諾貝爾化學獎。
## 用途
* 物質成分分析。
* 礦物質分析。
* 分子、中子、原子等分析。
* 細胞、基因等分析。
* 細菌、病毒分析。
* 金相分析
* 集成電路生產中各種檢測。
* 電子器件檢測，如晶振、連接器、液晶屏扽。
## 種類
下文並未把所有種類顯微鏡列表，只是簡介較知名的類型。其他尚有像紫外線顯微鏡、X 光顯微鏡、**場離子顯微鏡**等，僅用於較專門需要而開發的，少量生產的特種用途顯微鏡。
### 光學顯微鏡 | [
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顯微鏡 | 利用透鏡放大物像送到眼睛或成像儀器，解析度大約為一微米，可以看到細胞大小的物品。一般來說顯微鏡大都是指光學顯微鏡，光學顯微鏡依設計的不同，又可分為正立顯微鏡、倒立顯微鏡（又稱**倒置顯微鏡**）和解剖顯微鏡（又稱**實體顯微鏡**或**立體顯微鏡**）；又有偏光顯微鏡：又稱為岩石顯微鏡、礦物顯微鏡或金屬顯微鏡，用以觀察岩石、礦物及金屬表面，是利用光的不同性質（偏光）而做成的；相襯顯微鏡：觀察變形蟲、草履蟲等透明生物時，所使用的顯微鏡。它的特殊裝置可以將光透過生物體所產生的偏差，改變為明暗不同；又結合光學顯微鏡並利用雷射光作為光源，以達到特殊觀察需求的有共聚焦顯微鏡（又譯作共軛焦顯微鏡）。
### 電子顯微鏡
在 20 世紀初的一種光學顯微鏡顯著替代被開發，利用電子而不利用光線來產生圖像。於 1931 年，恩斯特・魯斯卡（Ernst Ruska）開始開發第一個電子顯微鏡 - 透射電子顯微鏡（TEM）。透射電子顯微鏡的工作原理和光學顯微鏡有相同的原理，但在使用光的地方用電子代替，在使用玻璃透鏡的地方用電磁鐵代替。使用電子而不是光線允許更高的解析度。
緊接著透射電子顯微鏡的開發，是馬克斯・諾爾在 1935 年開發的掃描電子顯微鏡（SEM）。
不使用光線而利用電子流來照射標本來觀察的顯微鏡。由於電子用肉眼看不出，因此就使電子透過觀察材料，而映在塗有螢光劑的板子上，這種方法稱為穿透式電子顯微鏡。另一種方法是以電流在觀察材料的表面移動，然後使觀察材料所放出的二次電子流映在真空管上，以這種方式觀察的稱為掃描式電子顯微鏡。穿透式電子顯微鏡可放大 80 萬倍，可以看出分子的形象；掃描式電子顯微鏡可用以觀察立體的表面，放大倍率約 20 萬倍。電子顯微鏡分為透射電子顯微鏡、能量過濾透過式電子顯微鏡、掃描電子顯微鏡、場發射掃描電子顯微鏡、掃描透射電子顯微鏡等類型。某些電子顯微鏡甚至能看到單一原子。原理：物質波理論告訴我們，電子也具有波動性質，所以可以用類似光學顯微鏡的原理，做成顯微鏡。不一樣的是，這裡將凸透鏡改成磁鐵，由於電子的波長比可見光短，所以他可以比光學顯微鏡「看」到更小的東西，如：病毒。
### 掃描探針顯微鏡
是機械式地用探針在樣本上掃描移動以探測樣本影像的顯微鏡。
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] |
顯微鏡 | STM 用來看金屬表面，它是利用量子物理的穿隧效應。古典物理認為，物質不能穿過位壘，但量子物理告訴我們：物質有機會穿過位壘，而他穿過位壘的機率和位壘的寬度有關。利用一通電的針狀物體，靠近金屬表面，則電場使電子附近的位能出現位壘形式，此時就有機會觀測到跑出金屬表面的電子，再利用穿過位壘的機率和位壘的寬度有關的特性，就可以推出針到金屬表面的距離，因此可 "看" 到金屬表面。
#### 原子力顯微鏡
原子力顯微鏡（Atomic Force Microscope，簡稱 AFM）用來探測樣本表面與探針交互作用力，推出探針到樣本表面的距離，因此可「看」到非金屬或金屬表面。
## 顯微鏡展示框
* * * * * * * * * * * *
## 顯微鏡的機械部件
*
## 參看
* 放大鏡
* 望遠鏡
* 數碼顯微鏡
* 相機
* 監視器
* 攝影機
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单位阵 | 在線性代數中， $n$ 階**單位矩陣**，是一個 $n\times n$ 的方形矩陣，其主對角線元素為 1，其餘元素為 0。單位矩陣以 $I_{n}$ 表示；如果階數可忽略，或可由前後文確定的話，也可簡記為 $I$ （或者 $E$ ）。（在部分領域中，如量子力學，單位矩陣是以粗體字的 **1** 表示，否則無法與 $I$ 作區別。）
: $I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},\ \cdots ,\ I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}$
一些數學書籍使用 $U$ 和 $E$ （分別意為「單位矩陣」和「基本矩陣」），不過 $I$ 更加普遍。
特別是單位矩陣作為所有 $n$ 階矩陣的環的單位，以及作為由所有 $n$ 階可逆矩陣構成的一般線性群 $GL(n)$ 的單位元素（單位矩陣明顯可逆，單位矩陣乘自己，仍是單位矩陣）。
這些 $n$ 階矩陣經常表示來自 $n$ 維向量空間自己的線性變換， $I_{n}$ 表示恆等函數，而不理會基。
有時使用這個記法簡潔的描述對角線矩陣，寫作：
: $I_{n}=\operatorname {diag} (1,1,...,1)$
也可以克羅內克爾 δ 記法寫作：
: $(I_{n})_{ij}=\delta _{ij}$
## 性質
根據矩陣乘法的定義，單位矩陣 $I_{n}$ 的重要性質為：
: $AI_{n}=A$ 且 $I_{n}B=B$
單位矩陣的特徵值皆為 1，任何向量都是單位矩陣的特徵向量。具有重數 $n$ 。因為特徵值之積等於行列式，所以單位矩陣的行列式為 1。因為特徵值之等於跡數，單位矩陣的跡為 $n$ 。
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群的表示 | 在群論中，**群表示論**（英語：Group representation theory）是一個非常重要的理論。它包含了（局部）緊緻群、李群、李代數及群概形的表示等種種分支，近來無限維表示理論也漸露頭角。表示理論在量子物理與數學的各領域中均有重要應用。
## 基本定義
表示理論早期是藉矩陣的語言描述的，具體定義如下：
* 如果任何非零方陣的集合的乘法關係和給定群的乘法關係相同，則這個矩陣集合形成群的一個**表示**，這套矩陣的階稱為表示的**維數**。
* 如果兩個同維表示的矩陣以同一相似變換相關聯，則稱這兩個表示是**等價**的。
* 如果任何維數大於一的表示的所有矩陣都可以用相同的相似變換轉換為相同的塊對角矩陣結構，則稱此表示為**可約表示**，反之稱為**不可約表示**。
形式地說，一個群 $G$ 的表示乃一同態 $\rho :G\rightarrow \mathrm {GL} (V)$ ，其中 $V$ 為給定的有限維向量空間，係數佈於一個域 $F$ ，通常取 $F=\mathbb {C} $ ，但在一般域（如局部體或有限體）上的表示也有重要應用。 $\mathrm {GL} (V)$ 表從 $V$ 上的自同構，或對一給定的基底來說，是 $n=\dim V$ 階可逆方陣的集合。若 $\mathrm {Ker} (\rho )$ 是平凡的，則稱此表現是**忠實**的。
若所考慮的群 $G$ 帶有額外的結構（如拓撲群、李群或群概形），我們通常要求 $\rho $ 滿足相應的條件（如連續性、可微性或者要求它是概形間的態射）；在有限群及緊緻群以外的情況，通常也須考慮無窮維表示。
一個群 $G$ 的所有有限維表示構成一個張量範疇，記為 $\mathrm {Rep} _{G}$ ；其態射定義如下：
$$
\mathrm {Hom} _{G}((\rho ,V),(\sigma ,W)):=\{f\in \mathrm {Hom} _{F}(V,W):f(\rho (g)v)=\sigma (g)f(v)\}
$$ | [
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群的表示 | 它等價於有限維 $F[G]$ - 模所構成的範疇。不難驗證表示間的同構確由矩陣的相似變換給出。一個表示被稱作**不可約**的，若且唯若它沒有在 $G$ 的作用下不變的非平凡子空間。若一個表示能表成不可約表示的直和，則稱之為**完全可約**的。若取 $F=\mathbb {C} $ ，則緊緻群的表示均為完全可約的，對於一般的李群及群概形則複雜得多，完全可約與否通常與半單性有關。
## 特徵標
給定 $G$ 的一個表示，可以得到一個**特徵標** $\chi :G\rightarrow F$ ，它是個類函數。特徵標理論在有限群分類中佔關鍵地位；在緊緻群上，特徵標滿足舒爾正交關係，又根據**彼得 - 外爾定理**，不可約表現的特徵標相對於 $L^{\infty }$ 範數在類函數中稠密。請參見特徵標理論。
## 誘導與限制
設 $H$ 為 $G$ 之子群， $(G:H)<\infty $ 。以下將定義兩個函子 $\mathrm {Res} _{H}^{G}:\mathrm {Rep} _{G}\rightarrow \mathrm {Rep} _{H}$ （**限制**）與 $\mathrm {Ind} _{H}^{G}:\mathrm {Rep} _{H}\rightarrow \mathrm {Rep} _{G}$ （**誘導**）。
* 若 $\rho :G\rightarrow \mathrm {GL} (V)$ 為 G 的表示，則 ρ 限制於 H 給出 H 的表示，記為 $\mathrm {Res} _{H}^{G}(V)$ 。
* 若 $\rho :H\rightarrow \mathrm {GL} (V)$ 為 H 的表示，我們定義 $V^{G}:=\{f:G\rightarrow V:\forall h\in H\;f(hg)=\rho (h)f(g)\}$ 。 $G$ 以右乘法作用在 $V^{G}$ 上。 $V^{G}$ 仍是有限維，記此表示為 $\mathrm {Ind} _{H}^{G}(V)$ 。
誘導表示亦可用矩陣直接計算，或定義為某個主齊性空間的截面；後者可推廣至李群與群概形的表示，此時誘導表示的性狀與 $G/H$ 的幾何構造密切相關。 | [
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群的表示 | 弗羅貝尼烏斯互反定理言明：若 $V,W$ 分別為 $G,H$ 的表示，則有自然的同構 $\mathrm {Hom} _{H}(W,\mathrm {Res} _{H}^{G}(V))=\mathrm {Hom} _{G}(\mathrm {Ind} _{H}^{G}(W),V)$ 。換言之： $(\mathrm {Ind} _{H}^{G},\mathrm {Res} _{H}^{G})$ 為一對伴隨函子。
若以特徵標表之，上述同構化為一個較弱但較具體的等式： $(\chi _{\mathrm {Ind} _{H}^{G}(W)},\chi _{V})=(\chi _{W},\chi _{\mathrm {Res} _{H}^{G}(V)})$ 。
## 例子
* 任意一個群 $G$ 都自然地作用在其群代數 $\mathbb {C} [G]$ 上，稱為**正則表現**。
* 對稱群 $S_{n}$ 以 $\sigma \cdot e_{i}=e_{\sigma (i)}$ 作用在 $\mathbb {C} ^{n}$ 上。
* $\mathrm {SO} _{n}(\mathbb {R} )$ 以 $g\cdot f(x)=f(g(x))$ 作用於 m 次調和多項式上。
## 與物理學的關係
迄今已知的物理定律通常在某個李群的作用下保持不變，如空間的旋轉群 $\mathrm {SO} (3)$ 或其覆蓋 $\mathrm {Spin} (3)$ ，其不可約表示關係到角動量的量子化。進一步的例子是：任何與狹義相對論相容的量子力學系統都帶有 $G:=AH$ （半直積）的酉表示，其中 $A$ 是時空的平移而 $H$ 是 勞侖茲變換群，藉著研究 $G$ 的不可約酉表示，可分類粒子的質量和自旋。
## 參見
* 舒爾正交關係
* 特徵標理論
## 文獻
* J.L. Alperin, Rowen B. Bell, _Groups and Representations_ (1995), Graduate Texts in Mathematics 162 ,Springer. ISBN 0387945261
* J.C. Jantzen, _Representations of Algebraic Groups_ (2003), American Mathematical Society. ISBN 0821835270 | [
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] |
群的表示 | * V.S. Varadarajan, _An Introduction to Harmonic Analysis on Semisimple Groups_ (1989), Cambridge University Press. ISBN 0-521-34156-6 | [
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全像 | **全像攝影**（英語：Holography），又稱**全像投影、全像 3D**，是一種記錄被攝物體反射（或透射）光波中全部訊息（振幅、相位）的照相技術，而物體反射或者透射的光線可以透過記錄膠片完全重建，彷彿物體就在那裡一樣。透過不同的方位和角度觀察相片，可以看到被拍攝的物體的不同的角度，因此記錄得到的像可以使人產生立體視覺。
## 全像術歷史與概述
1947 年，英國匈牙利裔物理學家丹尼斯・蓋伯發明了全像術，他因此項工作獲得了 1971 年的諾貝爾物理學獎。其它的一些科學家在此之前也曾做過一些研究工作，解決了一些技術上的問題。全像術的發明是蓋伯在英國 BTH 公司研究增強電子顯微鏡效能手段時的偶然發現，而這項技術由該公司在 1947 年 12 月申請了專利（專利號 GB685286）。這項技術從發明開始就一直應用於電子顯微技術中，在這個領域中被稱為電子全像術技術，但是全像術一直到 1960 年雷射的發明才取得了實質性的發展。
第一張實際記錄了三維物體的光學全像術相片是在 1962 年由蘇聯科學家尤里・丹尼蘇克拍攝的。 與此同時，美國密西根大學雷達實驗室的工作人員艾米特・利思和尤里斯・烏帕特尼克斯也發明了同樣的技術。 尼古拉斯・菲利普斯改進了光化學加工技術，以生產高品質的全像術圖片。
全像術可以分為如下若干類。透射全像術，如利思和烏帕特尼克斯所發明的技術，這種技術通過向全像術膠片照射雷射，然後從另一個方向來觀察重建的圖像。後來經過改進，彩虹全像術可以使用白色光來照明，以觀察重建的圖像。彩虹全像術現在廣泛的應用於諸如信用卡安全防偽和產品包裝等領域。這些種類的彩虹全像術通常在一個塑料膠片形成了表面浮雕圖案，然後通過在背面鍍上鋁膜使光線透過膠片以重建圖像。另一種常見的全像術稱為反射全像術，或稱為丹尼蘇克全像術。這種技術可以通過使用白色光源從和觀察者相同的方向來照射膠片，通過反射來重建彩色的圖像，以重建圖像。鏡面全像術是一種通過控制鏡面在二維表面上的運動來製造三維圖像的相關技術。它通過控制反射光線或者折射光線來構造全像圖像，而蓋伯的全像術是通過繞射光來重建波前的。 | [
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全像 | 促使全像術在短短的一段時間內就蓬勃發展的關鍵原因是低成本的固體雷射器的大規模生產，如 DVD 播放機和其他的一些常用裝置中所使用的雷射器。這些雷射器對全像術的發展也產生了極大的促進作用。這些廉價的體積又很小的固體雷射器可以在某些條件下與最初用於全像術的那些大型的昂貴的氣體雷射器相媲美，因此使得預算較低的研究者、藝術家甚至業餘愛好者都可以參與到全像術的研究中來。
## 全像術理論
儘管全像術經常被稱為三維攝影，這是一個不正確的說法。一個更好地類比是在錄音的過程中通過將聲場編碼，使得隨後可以將其重現。在全像術中，一個物體或者一組物體散射的光線會照射到記錄媒介上，此時，第二束被稱為參考光的光線也照射在記錄媒介上，這樣，兩束光發生了干涉。產生的光場產生了看起來隨機的圖案，而變化的密度被記錄媒介記錄了下來。可以證明，如果使用與參考光相同的光線，參考光可以在相片上產生繞射，而繞射的光場和物體散射的光場相同。這樣，觀察全像術的相片就會看到那個物體，儘管物體其實並不在那裡。包括攝影膠捲在內的多種記錄媒介都可以用於全像術。
全像術的發明人丹尼斯・蓋伯解決的問題是怎樣為所有穿過一個大窗口的光線拍照，而不僅僅是為穿過一個很小的針孔的光線拍照。在透過這個窗口進行觀察的時候，由於每隻眼睛觀察到不同的場景，觀察者會產生立體的感覺。而且，如果觀察者能夠將他的頭圍繞著窗口外部移動，他可以看到物體的不同的角度（1960 年代早期的的一個全像術實驗拍攝了一個物體，物體前面幾厘米的位置擺放了一個放大鏡，觀察者可以通過將頭上下擺動，看到物體透過透鏡成的像和物體本身）。
丹尼斯・蓋伯為了進行全像術，需要使用一個高速的快門，快門的速度非常快，使得它可以將光波穿過窗口時的相位固定住，也就是說，這個快門需要以光速工作。如果光線閃爍的時間和物體運動的週期一致，每次看到的都是物體同樣的部位，這樣物體看起來就是固定的，頻閃燈就是用這個原理來「固定」快速移動的物體如發動機，蓋伯採用了類似的辦法來實現。在全像術中，和頻閃燈類似的功能由**參考光**來完成。在上邊的示意圖中，光線的一部分**照明光**被物體散射，直接照射在膠片上（這裡沒有使用針孔或者透鏡來成像），而另一部分**參考光**沒有照在物體上，而是從原始雷射束中分離後直接照射在膠片上。 | [
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全像 | 在全像術重構過程中，為了回放膠片中拍攝的內容，需要重新提供參考光線，將它照射在沖洗出來的膠片上，也就是我們所說的窗口中。這使得在攝影過程中捕捉的穿過窗口的光線相位和他們當初離開物體照在膠片上的時候的相位完全一致的重現。實際上，如示意圖所示，現在可以通過窗口觀察到其後的物體了。
丹尼斯・蓋伯的發明的原理並不像頻閃燈那樣簡單。為了進一步的理解全像術的理論，我們需要理解光波的干涉和繞射。對於那些不熟悉這些概念的人，在閱讀本節以下部分之前可以通過閱讀那些專門介紹的文章來學習這些概念。
### 干涉與繞射
在一個或者多個波前疊加的時候會出現干涉現象。而在一個波前接觸到一個物體的時候就會產生繞射。全像術重建的過程在下面完全是用干涉和折射進行了解釋。這個解釋有所簡化，但是足以理解全像術過程的工作原理了。
#### 平面波前的情況
繞射光柵是一種具有週期性結構的器件。一個刻有均勻的刻線的金屬板就是一個簡單的光柵。光線穿過它的時候會產生彎曲，彎曲的角度 θ 由光線的波長 λ 和刻線的間距 d 決定， 關係為 sinθ = λ/d。
非常簡單的全像術可以通過將從同一光源射出的兩束平面波疊加來示意。參考光垂直照射在攝影膠片上，而另一束光以一定的角度 θ 照射在膠片上。兩束光之間的相對相位差在膠片上不停地變化，變化的關係為 2π y sinθ/λ，其中 y 是沿著膠片的距離。這兩束光的干涉會產生干涉圖樣。由於光線的相位差每經過 d = λ/sinθ 的距離變化 2π，干涉條紋的間距也是 d。這樣參考光和物體發出的光之間的相對相位就由干涉條紋的最亮處和最暗處記錄下來了。
在攝影膠片沖洗出來以後，干涉條紋可以用作繞射光柵。這樣，當參考光線照射在膠片上的時候，部分光線會以相同的角度 θ 發生繞射。這樣，物體發出的光線就被重建出來了。使用兩個波的干涉建立出的繞射光柵可以重建出物體發射的光線，這樣它可以看作是上面定義的全像術。
#### 點光源的情況
更複雜的全像術可以使用點光源作為物體，同時使用一束平面波來作為參考光來照射在攝影膠片上。這時，產生的干涉圖樣是曲線，曲線的形狀為一圈一圈的圓環，離中心越遠，圓環的間距越小，這個形狀也被稱為波帶板。 | [
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全像 | 照相底片沖洗以後會顯示出複雜的干涉圖樣，這個圖樣可以被認為是有不同間隔的繞射圖案的疊加。當沖洗後的膠片單單被參考光照明的時候，膠片上的圖案可以看作是一個光柵，這個光柵會根據光柵刻線的間距將光線繞射至不同的角度。可以證明，這個效果的淨效應是重建了物體（點光源）發出的光線。由於從膠片發出的光線與點光源所發射出來的光線一模一樣，觀察者可以看到光線是從膠片後面的一個點發射出來的，儘管這個物體其實並不存在。
這張全像術的相片可以產生凹透鏡的效果，因為它使平面波前轉變為發散的波前。它也可以增加照射在其上的任何波前的分散性，就和普通的透鏡一樣。它的焦距就是點光源與膠片的間距。
#### 複雜物體的情況
為了記錄複雜物體的全像術，首先要將一束雷射用分光器分成兩束：一束光用於照亮物體，物體會將它散射，反射光會照在記錄媒介上；另一束光直接照射在記錄媒介上面。
根據繞射理論，物體上面的每一個點都可以看作是一個點光源。每個點光源所發射出來的光都會和參考光發生干涉，產生干涉圖樣。結果產生的干涉圖樣是所有的點光源和參考光源產生的干涉圖樣的疊加。
當移除了物體並沖洗膠片以後，將參考光重新照射記錄了全像術的膠片，每一個點光源繞射光柵都可以繞射部分的參考光線，重建他們對應的點光源的波前。這些單獨的波前疊加起來以後就可以重建整個物體散射的光線的波前。由於觀察者感知到的波前和物體散射出的光線的波前完全一致，因此觀察者仍然可以看到在那裡的物體。這個圖像可以看作是虛像，因為那裡並沒有實際物體發出光線。而且可以看到照亮物體的光源的方向和原始的照明光線方向是一致的。
### 數學模型
光波可以使用一個複數變數 **U** 來表示其光波中的電場和磁場。光波的幅度和相位可以使用複數的模和輻角來表示。在全像術的系統中，每一點處的物體發出的光和參考光可以用變數 **U**O 和 **U**R 來表示，這樣聯合起來的光波可以表示為 **U**O + **U**R。這個光波的能量和電場幅度的平方成正比：
$|U_{O}+U_{R}|^{2}=U_{O}U_{R}^{*}+|U_{R}|^{2}+|U_{O}|^{2}+U_{O}^{*}U_{R}$ 。
如果一張攝影膠片暴露在這兩束光中，然後沖洗出來，它的透射函數將於照射在其上的光線能量成正比，可以表示為：
$T=k[U_{O}U_{R}^{*}+|U_{R}|^{2}+|U_{O}|^{2}+U_{O}^{*}U_{R}]$ ， | [
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全像 | 其中 $k$ 是常數。沖印出來的膠片在使用參考光照射的時候，透過膠片的光波可以用 **U**H 表示：
$U_{H}=TU_{R}=k[U_{O}U_{R}^{*}+|U_{R}|^{2}+|U_{O}|^{2}+U_{O}^{*}U_{R}]U_{R}=k[U_{O}|U_{R}|^{2}+|U_{R}|^{2}U_{R}+|U_{O}|^{2}U_{R}+U_{O}^{*}U_{R}^{2}]$ 。
可以看出 **U**H 中包含四項，第一項正比於 **U**O，可以用來重建物體發射出來的光。第二項代表了參考光，其幅度變成了 **U**R2。第三項同樣代表了參考光，其幅度為 **U**O2，這個修改可以引起參考光線在其中心方向周圍發生繞射。第四項被稱為**共軛物體光線**。它的凹凸性和物體正好相反，而且在全像膠片的前方形成了一個實像。由於全像術在拍攝時都要讓物體和參考光垂直的照射在膠片上，這意味著全像術被參考光照射後產生的四束光會疊加在一起。由利思和烏帕特尼克斯發明的離軸全像術可以解決這個問題。物體光和參考光以一個角度照射在全像記錄媒介上，因此虛像、實像和參考光波前以不同的角度射出，使得可以清晰的觀察到重建的像的光線。
## 全像術膠片
右圖所示為一張在漫射光背景下拍攝的全像相片，相片上已經記錄了全像的光資訊。圖中所示的區域的大小為大約 8x8 毫米。全像攝影中記錄下來的是光線強度的隨機變化，這個變化被稱為客觀散斑。膠片中規則的線條是由光線在裝載膠片的玻璃板中多次反射產生的干涉條紋。直接觀察全像術的膠片時是無法根據記錄的結構辨識出所拍攝的物體的，就像無法根據留聲機在唱片上留下的刻痕直接讀出音樂一樣。當使用雷射束照亮全像攝影膠片的時候，觀察者可以看到拍攝的物體（右邊這張相片拍攝的一輛玩具汽車），這是因為雷射可以被膠片繞射，從而重建物體散射出的光線。 | [
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全像 | 當一個人觀察的時候，每隻眼睛都捕捉到物體散射的光線的一部分，而人眼中的晶狀體可以作為透鏡將物體在視網膜上成像，而從不同角度上射出的光線都在相平面的不同角位上成像。由於全像術系統可以將射在膠片上的光場完整的重構出來，觀察者看到的光場和物體散射出來的光場完全一致，換句話說，觀察者無法區分看到的是真實的物體散射的光線，還是僅僅是一個虛像。如果觀察者移動，它看到的物體看起來也在移動，使用者仍然無法區分他看到的是究竟是原始的光場還是重構出來的光場。如果場景中有若干個物體，使用者者還可以觀察到視差現象。如果觀察者用兩隻眼睛同時觀察，就可以產生立體視覺，也就是在觀察全像術相片的時候得到深度的資訊，這和他在觀察真實的場景的時候感受到立體視覺的原理是完全一致的。然而，全像術並不是三維相片。相片可以從一個觀察點將場景的像記錄下來，這個觀察點是由照相機的透鏡位置決定的。而全像術記錄下來的並不是像，而是將需要重建的散射光光場編碼記錄下來。在任何位置使用照相機或者眼睛都可以記錄下重建的散射光線。在早期的全像術研究中，通常使用棋盤作為拍攝物體，然後可以通過不同角度對重建的光線拍照來展示棋子相對位置的變化。
由於全像術中每個點都包含了原始場景的光線的資訊，從原理上說，整個場景可以通過任意小的一部分全像術膠片上還原出來。為了展示這個概念，可以將全像相片分成若干部分，通過每個部分都可以觀察到整個的物體。如果一個人將全像相片看作是觀察物體的窗口，每一小片全像相片僅僅是窗口的一部分，但是通過這個窗口仍然可以觀察到物體，儘管其他的窗口已經被關閉了。然而，在全像相片的尺寸減小了以後，解析度會隨之降低，因此物體會變得模糊。這是繞射的結果。在普通的光學成像系統中，也可以觀察到類似的現象，當透鏡或者透鏡的光圈直徑降低的時候，成像的解析度會受繞射光斑的影響而降低。
## 觀察和創作全像攝影
在記錄全像影像的過程中，物體散射光和參考光必須能夠產生穩定的干涉圖樣。為了達到這個效果，這些光線必須具有相同的頻率，在曝光時也保持相同的相對相位，這也就是說，這些光線必須相干。很多雷射光束符合這個條件，因此自從全像攝影發明開始就使用雷射來進行全像攝影了，儘管最早蓋伯提出的全像攝影使用的是准單色光。從原理上說，如果兩個不同的光源可以產生相干光，那麼就可以使用這兩個分離的光源來進行拍攝，但是實際上總是使用單一的雷射光源。 | [
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全像 | 另外，用於記錄全像攝影的干涉條紋的媒介必須擁有足夠的解析度，以使干涉條紋可以分辨出來。下面列出了一些通常使用的記錄媒介。干涉條紋的距離和物體與參考光之間的角度有關。例如，如果這個角度是 45°，光波的波長為 0.5 微米，那麼條紋的間距大約是 0.7 微米，也就是 1400 線 / 毫米。當然即使無法解析出所有的條紋仍然可以看到全像攝影的拍攝結果，只是圖像的解析度會隨著記錄媒介解析度的下降而下降。
在拍攝全像相片的時候，機械穩定性也是非常重要的。物體和參考光的震動甚至是空氣的運動都會產生相對的相位變化，這會使得記錄媒介上的條紋發生移動。如果相位變化超過 π，就會導致干涉圖樣平均起來消失了，也就無法得到全像記錄的結果。通常的記錄時間需要若干秒，而相對的相位差小於 π 相當於要求位移小於 λ/2，這是一個相當嚴格的穩定性要求了。
一般來說，光線的相干長度決定了全像攝影能夠記錄的場景深度。通常擁有良好效能的雷射的相干長度可達數米，足夠用於拍攝很深的全像相片了。某些雷射筆也被用來製造較小的全像相片，這些全像相片的深度並不是由雷射筆產生的雷射的相干長度的限制的，而是受限於雷射筆的功率（低於 5 毫瓦）。
場景中待拍攝的物體必須有光學上粗糙的表面，因此可以在很廣的角度上散射光線。鏡面反射的表面會照射其上每一點的光線反射至一個特定的角度，因此，大多數的光線不會接觸到記錄的媒介。而從物體粗糙表面散射的光線會產生具有隨機幅度和相位的客觀散斑。
參考光線一般並不是一個平面波前，而常用的是一束分散的波前。這種波前可以通過在雷射的光路中插入一個凹透鏡來實現。
為了通過透射全像相片準確的重建物體的像，照在其上的參考光必須和拍照時的參考光有相同的波長和曲率，也必須以和拍照相同的角度照在相片上。唯一可以有不同之處的地方只能是參考光的相位。違反任意一條條件都會導致重建過程失真。幾乎所有的全像相片都是使用雷射拍攝的，但是窄頻的燈甚至日光都可以辨別出重構的像。
如果用於重構全像圖像的光線的波長變長，那麼重構出來的圖像會放大。最開始人們希望能夠使用 X 射線來拍攝全像圖像，然後使用可見光來觀察像，然而直到現在仍然沒能成功的使用 X 射線來拍攝全像圖像。 但是這個效應可以通過使用能夠發出不同頻率光線的光源來觀察到
。
全像干涉度量中，會將重建的全像圖像的波前與真正的波前進行干涉，以找出任何物體的位移。如果物體沒有移動，就不會產生干涉條紋，而這裡需要精確的重建全像圖像。 | [
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全像 | ## 全像攝影記錄媒介
根據上面的討論，記錄全像影像的媒介需要有能力解析出干涉條紋。媒介也必須足夠敏感，以能夠在儘量短的時間內完成拍攝，使得系統儘可能的保持其光學穩定性。光學穩定性是指兩束光之間的相對位移需要遠小於 λ/2。使用大功率脈衝雷射器可以在幾奈秒的時間內在特定的材料上記錄下全像圖樣。
記錄媒介需要將干涉圖樣轉換成能夠改射在其上的光線的幅度或者相位的光學元素，這被稱為幅度全像和相位全像。在幅度全像相片中，相片上不同位置對光線的吸收率不同，這是由於在沖洗出來的相片中，膠片上的感光乳劑根據照射其上的光強度不同剩餘的數量也不相同。在相位全像相片中，材料的光學距離（折射率或者厚度）隨著光強的變化而發生變化。
大多數用於相位全像相片的感光材料可以達到理論上的繞射效率 ，對於厚全像相片來說，效率達到了 100%（布拉格繞射區域），而對於薄全像相片，效率達到 33.9%（拉曼 - 奈斯繞射區域，全像相片通常只有幾微米厚）。幅度全像相片的效率要比相位全像相片的效率低，因此較少使用。
下面的列表顯示了用於全像攝影的主要的感光材料。注意這些材料並不包括用於大規模複製已有全像相片的那些材料。表中解析度的極限表示曝光後形成的光柵每毫米最多的線條數。曝光需要很長的曝光時間，而短曝光時間（少於 1 毫秒，如使用脈衝雷射）需要大曝光量。
### 全像印表機
全像印表機是一種全像圖像的列印裝置，它可以根據一個三維模型或影片序列輸出全彩色的數字全像圖像。一台這樣的機器價值可能達到 50 萬美元，體積大概能占據一個小房間。它使用紅色、綠色和藍色的雷射在全像膠片上刻印上一系列全像像素。全像像素包含有從它的位置可以觀察到的整個圖像的資訊。每個全像像素的資訊是根據產生的電腦圖像計算得出的。全像膠片的媒介是一曝光後可能還需要衝洗。隨後，這層薄膜將被壓在一個硬塑背板上。由於每個全像像素都需要使用三種顏色單獨印刷，列印一張數字全像圖像可能需要若干個小時。每個全像像素的大小大約是 1 平方毫米。
目前全世界僅有少數幾家數字全像印表機製造商。
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全像 | 全像相片製作成功以後可以對它進行複製，複製可以採用於全像攝影類似的光學方法，或者通過壓印來製作表面浮雕全像相片。表面浮雕全像相片使用光阻材料或者光致熱敏材料進行記錄，而且可以以較低的成本進行大規模的複製生產。現在，這種壓印的全像圖像已經得到了廣泛的應用，如信用卡或合格產品上印刷的安全資訊。加拿大皇家造幣廠甚至還通過複雜的沖印工藝在金幣和銀幣上製造全像圖像。1984 年，國家地理雜誌出版了第一本封面印有全像圖案的雜誌。
壓印工藝的第一步是通過電子沉積法在記錄了全像圖像的光阻材料或者光致熱敏材料上鍍上鎳，以製造壓模。當鎳層達到了要求的厚度，就將它與全像相片分離，隨後裝在金屬背板上。用於複製全像相片的材料包括聚酯薄膜、樹脂分離層、以及用以構成全像圖像層的熱塑薄膜。
壓印的過程可以通過簡單的熱壓來實現。底層用來複製全像圖像的，為熱塑層，這層首先需要加熱，當超過軟化點溫度後，使用壓模壓制。這個形狀在薄膜冷卻以後依然保持，然後從壓模剝離。為了允許通過反射來觀察到壓印的全像圖像，還需要在記錄了全像圖像的薄膜後添加一層反射層。這層薄膜通常使用鋁來製造。
目前研究表明，可以通過表面炸藥爆炸來建立需要的表面浮雕，以在鋼鐵上直接印刷全像圖像。
## 應用
### 資料儲存
除了記錄圖像，全像攝影技術還有很多其他應用。全像儲存就是一種能夠以很高的密度在晶體內部和光聚材料上儲存資訊的技術。由於許多電子產品都需要包含儲存裝置，這種能夠在某些媒介上儲存大量資訊的技術非常重要。目前的儲存技術如藍光光碟已經達到了繞射所限制的最大的資料儲存密度，因此全像儲存可能成為下一代主要的儲存技術。這種資料儲存技術的優點是資料不僅僅是記錄在表面上，而且也記錄在材料的內部。目前可用的空間光調製器可以在 1 秒鐘內產生 1000 幅不同的圖像，圖像的解析度是 1024×1024 位元。使用合適的記錄材料（可能是聚合材料或者鈮酸鋰）可以達到每秒 1Gb 的寫入速度。讀取速度比寫入速度快許多，一般認為可以達到每秒 1Tb 的讀取速度。
2005 年，一些公司如 Optware 和 Maxell 生產了 120mm 的全像光碟，這個全像光碟使用全像記錄層，最多可以儲存 3.9TB 的資訊。他們計劃以全像通用光碟來將這項技術推向市場。其他的公司如 InPhase 科技也在研究類似的技術格式。 | [
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全像 | 大多數的全像資料儲存模型都採用了基於頁的儲存方式，每一記錄的全像圖像都包含有大量地資訊。最近的研究計劃使用亞微米的微型全像圖像來實現可能的三維光儲存解決方案。儘管這種資料儲存方式無法達到基於頁的資料儲存的高資料率，這種方案的生產成本更低，技術障礙也更小。
### 安全領域
由於複製全像圖像需要價格昂貴的專門先進裝置，安全全像圖像非常難以偽造。許多貨幣都使用了全像防偽圖像，如巴西 20 雷亞爾鈔票、英國的 5/10/20 英鎊鈔票、愛沙尼亞 25/50/100/500 克朗鈔票、加拿大的 5/10/20/50/100 元的鈔票、5/10/20/50/100/200/500 歐元鈔票、韓國 5000/10000/50000 韓元鈔票、日本 5000/10000 日元鈔票等等。他們也經常用於銀行儲蓄卡、信用卡以及護照、身分證明、書籍、DVD 以及體育器材等等。
另外，全像技術還在全像武器照準器上應用。以 EOTech 公司產品為代表的各種全像瞄準器被廣泛使用在各種槍械上。
### 藝術作品
藝術家很早就意識到了全像攝影的作為一種藝術媒介的潛能，因此他們來到科學實驗室來創造他們的藝術品。全像攝影藝術經常是科學家和藝術家的合作結果，儘管某些全像攝影家認為他們自己既是科學家又是藝術家。薩爾瓦多・達利聲稱他是第一個在藝術中應用全像攝影的人。可以肯定，他是第一位著名的應用全像攝影的超現實主義藝術家，但是在 1972 年紐約達利全像攝影展之前，就已經先後有 1968 年的密西根克蘭布魯克藝術學院全像藝術展覽和 1970 年的芬奇學院畫廊全像藝術展了。這些展覽得到了全國媒體的關注。
在二十世紀七十年代，一些藝術工作室和學校成立，每一家都以其獨特的方式來研究全像攝影。比較著名的有舊金山全像攝影學校、紐約全像攝影博物館、倫敦皇家藝術學院和湖林學院研討會等等。目前，這些工作室都不再存在了。然而，紐約全像藝術中心和首爾 HOLO 中心仍然為藝術家提供創作和展覽全像藝術的場所。
在八十年代，許多使用全像攝影的藝術家在藝術世界中推廣了這種所謂的新媒體。每個藝術家都找到了一種合適的表達方式來展現他們的三維藝術作品，避免了簡單的使用全像攝影再現是一個雕像或物體。例如，在巴西，許多仿形體詩人發現全像攝影可以用來表達自己的想法，更新了仿形體詩的創作。 | [
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全像 | 目前還有一群人數雖然較少但仍很活躍的藝術家仍然在使用全像攝影作為他們的主要載體，更多的藝術家將全像攝影的元素整合到了他們的作品中。有一些人採用了創新的全像攝影技術，例如藝術家馬特・布蘭德使用電腦鏡面設計來消除鏡面全像攝影的像失真。
麻省理工學院博物館和喬納森・羅斯都收藏了大量的全像攝影作品，同時也提供了線上作品目錄。
### 業餘愛好
自從全像攝影發明以來，許多人都探索了全像攝影的可能應用。1971 年，勞埃德・克羅斯開辦了舊金山全像攝影學校，進行對入門者使用便宜的裝置進行全像攝影的方法。這種方法需要使用一大桌子的深沙來固定光學器件，減弱可能毀壞圖像的振動。
許多全像攝影家都在製作全像攝影藝術作品。1983 年，弗雷德・安特爾捨出版了全像攝影手冊，用非常淺顯的文字描述了在家中拍攝全像相片的方法。這本書帶來了新一波的全像攝影家，他們採用非常簡單的方法使用鹵化銀來記錄全像影像。
2000 年，法蘭克・德弗萊伊塔斯出版了全像攝影書，向無數全像攝影愛好者介紹了使用雷射筆進行全像攝影的方法。這個方法對於初學者非常重要，因為一支 5 毫瓦的雷射筆價格僅 5 美元，以前使用的雷射器價格高達 1200 美元。目前，全世界有成百上千的業餘全像攝影愛好者。
2006 年，用於全像攝影的綠色雷射器開始大量出現，而業餘全像攝影家們也可以使用重鉻酸鹽明膠來進行全像攝影。全像攝影界對重鉻酸鹽明膠對綠色光的高感光性感到非常驚異，因為以前人們認為這種敏感性應該是不存在的。傑夫・布萊斯認為採用 G307 配方的重鉻酸鹽明膠可以增加拍攝的速度和敏感性。
許多膠片提供商在鹵化銀市場進進出出。儘管越來越多的膠片製造商開始出現填補出現的空白，許多業餘愛好者開始自己製造膠片。比較流行的配方是重鉻酸鹽明膠、感光亞甲藍、以及擴散方法製備鹵化銀。傑夫・布萊斯發表了非常準確的製備膠片的配方，人們可以在小型實驗室甚至車庫中製備膠片。
目前甚至還有一小批業餘愛好者自製脈衝雷射器來拍攝運動物體的全像相片。
### 全像干涉
全像干涉是一種能夠靜態和動態的檢查有粗糙表面的物體位移的技術，測量的精度可以達到光學干涉的精度（小於光線的波長）。這種技術也可以用來檢測透明媒介中的光路長度的變化，因此可以顯示並分析液體的流動。它也可以用於產生物體表面的等高線。
目前這種技術被廣泛的用於測量機械結構的應力、張力和震動情況。
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全像 | 全像圖像儲存了光場的幅度和相位資訊，有一些全像圖像儲存的資訊可以接近向各個方向輻射的光分布所包含的全部資訊。對這些全像圖像的數值分析可以仿真非常大的數值孔徑，因此能夠提高光學顯微鏡的解析度。相應的技術稱為干涉顯微技術。目前的干涉顯微技術可以達到 1/4 波長的解析度極限。
### 全像感測器
使用某種特定材料製成的全像攝影的膠片可以在與特定的分子發生反應的時候引起條紋週期性或者折射率的變化，因此，全像圖像反射光的顏色也將為此發生變化。
### 動態全像攝影術
在穩定的全像攝影中，記錄、沖洗和重建等步驟依次執行，而最終會產生永久的全像相片。
還有一種不需要進行沖洗的全像感光材料，它可以在很短的時間內記錄一張全像圖像。這樣，人們就可以使用全像攝影來完成一些簡單的全光操作。這種即時的全像圖像的應用包括相位共軛鏡，光快取、圖像處理（對時變圖像的圖型識別）以及光計算。
由於計算過程是對整個圖像的並列處理，需要進行計算的資訊量可能非常巨大（Tb/s）。這樣大的計算量可以補償通常幾微秒的記錄時間。幾微秒的時間對於通常使用的電腦來說已經是很長的時間了。對動態全像圖像的光處理也不如電腦的處理方法靈活。從一方面來說，人們總是需要對整個圖像進行處理，但是從另一方面來說，對全像圖像的處理又是非常基本的，通常是乘法或者相位共軛等等。但是在光學處理中，加法和傅立葉變換線上形材料中都是非常實現的（傅立葉變換可以簡單的通過透鏡來實現）。這樣就使得在某些應用中，裝置可以使用光學方法對圖像進行比較。
目前的研究中，人們正在積極尋找一些新型的非線性光學材料。最常見的材料就是光折變晶體，還有半導體、半導體異質（如量子阱）、原子蒸汽和氣體、電漿，甚至是能夠產生全像圖像的液體。
一個非常可能得到重要應用的研究是光學相位共軛。它可以通過讓光線再次穿過具有共軛相位的媒介來移除光線在穿過致像差媒介時產生的波前失真。在自由空間光通信中，這個技術可以用於補償大氣干擾（這是造成星光閃爍的原因）。
### 非光學應用
從原理上說，可以對任何波進行全像記錄。
電子全像攝影是將全像攝影技術應用於電子波中。電子全像攝影技術由丹尼斯・蓋伯發明，可以用於改進解析度並防止透射電子顯微鏡的吸收。現在，這種技術仍然在用於研究薄膜的電場和磁場，這是因為電場和磁場可以改變穿過樣品的干擾波的相位。電子全像攝影的原理也可以用於干涉光刻。 | [
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全像 | 全像聲學是一種能夠通過測量遠離聲源的一組壓力 / 粒子速度感測器來估計聲源附近的聲場的方法。全像聲學等測量技術在許多領域都越來越重要，特別是在運輸、運載工具和飛船的設計等領域。全像聲學的基本思想已經導致了不同種類的全像聲學，如近場全像聲學、統計最佳近場全像聲學。在聲學再現領域，波場合成是最相關的過程了。
**全像原子學**的進展已經超越了原子光學領域的基本要素的發展。在菲涅耳繞射透鏡和原子鏡的幫助下，全像原子學和原子束物理學的一起發展。最近的關於原子反射鏡，特別是脊反射鏡的進展為拍攝原子全像攝影圖像提供了重要工具。然而，到目前為止，原子全像攝影還沒有被商業化。
### 其它應用
在郵局、大型貨運公司以及自動化傳輸系統中使用了全像圖像掃描器來確定包裹的三維尺寸。這個技術經常與重量選別秤一起使用，以在給定的體積內自動的打包，可以更好地用於卡車等大型貨物運輸裝置。
## 易混淆概念
光柵印刷與投影、佩珀爾幻象都是容易與全像圖像相混淆的概念。
## 另見
* 鏡面全像攝影
* 體積全像圖像
* 數字全像攝影
* 數字平面全像攝影
* 全像感測器
* 全像腦理論
* 積分成像
* 全像原理
* 斷層攝影術
* 相位相干全像
* 全像儲存
* 電腦生成全像攝影
## 參考
## 擴充閱讀
* 國際全像製造商協會（頁面存檔備份，存於網際網路檔案館）
* 美國專利第 3,506,327 號 — "_Wavefront reconstruction using a coherent reference beam_" — E. N. Leith et al.
* 丹尼斯・蓋伯的諾貝爾獲獎演講（頁面存檔備份，存於網際網路檔案館）
* 麻省理工學院的空間成像研究小組及其關於全像理論與全像影片的論文（頁面存檔備份，存於網際網路檔案館）
* 全像影像的醫學應用 Medical Applications of Holograms（頁面存檔備份，存於網際網路檔案館）
* 全像影像的工作原理（頁面存檔備份，存於網際網路檔案館）
* 紐約全像攝影藝術中心，推廣全像攝影藝術的非營利組織（頁面存檔備份，存於網際網路檔案館）
* 鏡面全像藝術網站（頁面存檔備份，存於網際網路檔案館）
* 更快的方法來生產全像圖片（頁面存檔備份，存於網際網路檔案館）
* 手工繪製的全像圖（頁面存檔備份，存於網際網路檔案館） | [
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分子动力学 | **分子動力學**是一套分子模擬方法，該方法主要是依靠計算機來模擬分子、原子體系的運動，是一種多體模擬方法。通過對分子、原子在一定時間內運動狀態的模擬，從而以動態觀點考察系統隨時間演化的行為。通常，分子、原子的軌跡是通過數值求解牛頓運動方程式得到，位能（或其對笛卡爾坐標的一階偏導數，即力）通常可以由分子間交互作用位能函數、分子力學力場、全始計算給出。對於考慮分子本身的量子效應的體系，往往採用波包近似處理或採用量子力學的費恩曼路徑積分表述方式處理。
分子動力學也常常被採用作為研究複雜體系熱力學性質的採樣方法。在分子體系的不同狀態構成的系綜中抽取樣本，從而計算體系的構型積分，並以構型積分的結果為基礎進一步計算體系的熱力學量和其他宏觀性質。
分子動力學最早在 20 世紀 50 年代由物理學家提出，如今廣泛應用於物理、化學、生物體系的理論研究中。
## 分子動力學簡史
* 1957 年：基於剛球勢的分子動力學法（Alder and Wainwright）
* 1964 年：利用 Lennard-Jones 勢函數法對液態氬性質的模擬（Rahman）
* 1971 年：模擬具有分子團簇行為的水的性質（Rahman and Stillinger）
* 1977 年：約束動力學方法（Rychaert, Ciccotti & Berendsen; van Gunsteren）
* 1980 年：恆壓條件下的動力學方法（Andersen 法、Parrinello-Rahman 法）
* 1983 年：非平衡態動力學方法（Gillan and Dixon）
* 1984 年：恆溫條件下的動力學方法 (Berendsen et al.)
* 1984 年：恆溫條件下的動力學方法（Nosé-Hoover 法）
* 1985 年：第一原理分子動力學法（→Car-Parrinello 法）
* 1991 年：巨正則系綜的分子動力學方法（Cagin and Pettit）
* 1993 年：路徑積分分子動力學
## 基本步驟
* 確定起始構型
: 進行分子動力學模擬的第一步是確定起始構型， 一個能量較低的起始構型是進行分子模擬的基礎 ，一般分子的起始構型主要來自實驗數據或量子化學計算。 | [
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分子动力学 | : 在確定起始構型之後要賦予構成分子的各個原子速度，這一速度是根據波茲曼分布隨機生成的，由於速度的分布符合波茲曼統計，因此在這個階段，體系的溫度是恆定的。另外，在隨機生成各個原子的運動速度之後須進行調整，使得體系總體在各個方向上的動量之和為零，即保證體系沒有平動位移。
* 進入平衡相
: 由上一步確定的分子組建平衡相，在構建平衡相的時候會對構型、溫度等參數加以監控。
* 進入生產相
: 進入生產相之後體系中的分子和分子中的原子開始根據初始速度運動，可以想像其間會發生吸引、排斥乃至碰撞，這時就根據牛頓力學和預先給定的粒子間交互作用勢來對各個粒子的運動軌跡進行計算，在這個過程中，體系總能量不變，但分子內部位能和動能不斷相互轉化，從而體系的溫度也不斷變化，在整個過程中，體系會遍歷位能面上的各個點（理論上，如果模擬時間無限）。計算分析所用樣本正是從這個過程中抽取的。
* 計算結果
: 用抽樣所得體系的各個狀態計算當時體系的位能，進而計算構型積分。
## 作用勢與動力學計算
作用勢的選擇與動力學計算的關係極為密切，選擇不同的作用勢，體系的位能面會有不同的形狀，動力學計算所得的分子運動 和 分子內部運動的軌跡也會不同，進而影響到抽樣的結果和抽樣結果的位能計算，最初的分子動力學計算採用比較簡單的剛球勢，現在更多地採用蘭納 - 瓊斯勢，後者能夠更好的與粒子間交互作用擬合。
## 時間步長與約束動力學
分子動力學計算的基本思想是賦予分子體系初始運動狀態之後利用分子的自然運動在相空間中抽取樣本進行統計計算，時間步長就是抽樣的間隔，因而時間步長的選取對動力學模擬非常重要。太長的時間步長會造成分子間的激烈碰撞，體系數據溢出；太短的時間步長會降低模擬過程搜索相空間的能力，因此一般選取的時間步長為體系各個自由度中最短運動週期的十分之一。
但是通常情況下，體系各自由度中運動週期最短的是各個化學鍵的振動，而這種運動對計算某些 宏觀性質 並不產生影響，因此就產生了屏蔽分子內部振動或其他無關運動的**約束動力學**，約束動力學可以有效地增長分子動力學模擬的時間步長，提高搜索相空間的能力。
## 應用
分子動力學的計算過程給定了體系的總能量，因此適用於對微正則系綜的模擬計算，另外由於分子動力學計算過程始終是時間的函數，因此一些與時間有關的巨觀量如擴散係數的模擬必須應用分子動力學。
另外，在實際應用中，經常把分子動力學方法和蒙特・卡羅方法聯合使用。 | [
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] |
分子动力学 | 在近年，多尺度模擬計算已經得到了很多學者的關注。在多尺度模擬計算中，分子動力學方法研究奈米級現象，蒙特・卡羅方法研究微觀形態，有限元方法應用於宏觀領域。通過多種尺度的多種模擬計算方法的聯合應用，令納觀與宏觀聯結起來。
## 參見
* 計算化學
* 分子模擬
* 量子化學
## 參考資料
## 外部連結
* （英文）The GPUGRID.net Project （頁面存檔備份，存於網際網路檔案館） (GPUGRID.net)
* （英文）The Blue Gene Project (IBM)JawBreakers.org
* （英文）Materials modelling and computer simulation codes （頁面存檔備份，存於網際網路檔案館）
* （英文）A few tips on molecular dynamics （頁面存檔備份，存於網際網路檔案館）
* （英文）Movie of MD simulation of water (Youtube) （頁面存檔備份，存於網際網路檔案館）
* （英文）Live molecular dynamics simulation rendered at 1 frame per second （頁面存檔備份，存於網際網路檔案館）
* （英文）Molecular Dynamics Simulation; 7 Essential Concepts You Need to Learn （頁面存檔備份，存於網際網路檔案館） } | [
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分子力场 | **分子力場**根據量子力學的波恩 - 奧本海默近似，一個分子的能量可以近似看作構成分子的各個原子的空間坐標的函數，簡單地講就是分子的能量隨分子構型的變化而變化，而描述這種分子能量和分子結構之間關係的就是分子力場函數。分子力場函數為來自實驗結果的經驗公式，可以講對分子能量的模擬比較粗糙，但是相比於精確的量子力學從頭計算方法，分子力場方法的計算量要小數十倍，而且在適當的範圍內，分子力場方法的計算精度與量子化學計算相差無幾，因此對大分子複雜體系而言，分子力場方法是一套行之有效的方法。以分子力場為基礎的分子力學計算方法在分子動力學、蒙特卡羅方法、分子對接等分子模擬方法中有著廣泛的應用。
## 構成
一般而言，分子力場函數由以下幾個部分構成：
* 鍵伸縮能：構成分子的各個化學鍵在鍵軸方向上的伸縮運動所引起的能量變化
* 鍵角彎曲能：鍵角變化引起的分子能量變化
* 二面角扭曲能：單鍵旋轉引起分子骨架扭曲所產生的能量變化
* 非鍵相互作用：包括范德華力、靜電相互作用等與能量有關的非鍵相互作用
* 交叉能量項：上述作用之間耦合引起的能量變化
構成一套力場函數體系需要有一套聯繫分子能量和構型的函數，還需要給出各種不同原子在不同成鍵狀況下的物理參數，比如正常的鍵長、鍵角、二面角等，這些力場參數多來自實驗或者量子化學計算。
## 常用力場函數和分類
不同的分子力場會選取不同的函數形式來描述上述能量與體系構型之間的關係。到目前，不同的科研團隊設計了很多適用於不同體系的力場函數，根據他們選擇的函數和力場參數，可以分為以下幾類
* 傳統力場
* AMBER 力場：由 Kollman 課題組開發的力場，是目前使用比較廣泛的一種力場，適合處理生物大分子。
* CHARMM 力場：由 Karplus 課題組開發，對小分子體系到溶劑化的大分子體系都有很好的擬合。
* CVFF 力場：CVFF 力場是一個可以用於無機體系計算的力場
* MMX 力場：MMX 力場包括 MM2 和 MM3，是目前應用最為廣泛的一種力場，主要針對有機小分子
* 第二代力場
: 第二代的勢能函數形式比傳統力場要更加複雜，涉及的力場參數更多，計算量也更大，當然也相應地更加準確。
* CFF 力場 CFF 力場是一個力場家族，包括了 CFF91、PCFF、CFF95 等很多力場，可以進行從有機小分子、生物大分子到分子篩等諸多體系的計算 | [
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分子力场 | * COMPASS 力場由 MSI 公司開發的力場，擅長進行高分子體系的計算
* MMFF94 力場 Hagler 開發的力場，是目前最準確的力場之一
* 通用力場
: 通用力場也叫基於規則的力場，它所應用的力場參數是基於原子性質計算所得，用戶可以通過自主設定一系列分子作為訓練集來生成合用的力場參數
* ESFF 力場 MSI 公司開發的力場，可以進行有機、無機分子的計算
* UFF 力場可以計算周期表上所有元素的參數
* Dreiding 力場適用於有機小分子、大分子、主族元素的計算
* 反應力場
* 是一種基於鍵級的分子力場，常用於分子動力學模擬
* 傳統力場因不能滿足斷裂和形成鍵的要求而不能模擬化學反應，因而 ReaxFF 避開了顯式的鍵並基於鍵級，從而允許連續的鍵的形成或斷裂
## 參見
* 計算化學
* 量子化學
* 分子模擬 | [
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統計類比方法 | **蒙地卡羅方法**（英語：Monte Carlo method），也稱**統計類比方法**，是 1940 年代中期由於科學技術的發展和電腦的發明，而提出的一種以機率統計理論為指導的數值計算方法。是指使用亂數（或更常見的偽亂數）來解決很多計算問題的方法。
20 世紀 40 年代，在科學家馮・紐曼、斯塔尼斯拉夫・烏拉姆和尼古拉斯・梅特羅波利斯於洛斯阿拉莫斯國家實驗室為核武器計劃工作時，發明了蒙地卡羅方法。因為烏拉姆的叔叔經常在摩納哥的蒙地卡羅賭場輸錢得名，而蒙地卡羅方法正是以機率為基礎的方法。
與它對應的是確定性演算法。
蒙地卡羅方法在金融工程學、總體經濟學、生物醫學、計算物理學（如粒子輸運計算、量子熱力學計算、空氣動力學計算）、機器學習等領域應用廣泛。
## 基本概念
通常蒙地卡羅方法可以粗略地分成兩類：一類是所求解的問題本身具有內在的隨機性，藉助電腦的運算能力可以直接類比這種隨機的過程。例如在核物理研究中，分析中子在反應爐中的傳輸過程。中子與原子核作用受到量子力學規律的制約，人們只能知道它們相互作用發生的機率，卻無法準確獲得中子與原子核作用時的位置以及裂變產生的新中子的行進速率和方向。科學家依據其機率進行隨機抽樣得到裂變位置、速度和方向，這樣類比大量中子的行為後，經過統計就能獲得中子傳輸的範圍，作為反應爐設計的依據。
另一種類型是所求解問題可以轉化為某種隨機分布的特徵數，比如隨機事件出現的機率，或者隨機變數的期望值。通過隨機抽樣的方法，以隨機事件出現的頻率估計其機率，或者以抽樣的數字特徵估算隨機變數的數字特徵，並將其作為問題的解。這種方法多用於求解複雜的多維積分問題。
假設我們要計算一個不規則圖形的面積，那麼圖形的不規則程度和分析性計算（比如，積分）的複雜程度是成正比的。蒙地卡羅方法基於這樣的想法：假設你有一袋豆子，把豆子均勻地朝這個圖形上撒，然後數這個圖形之中有多少顆豆子，這個豆子的數目就是圖形的面積。當你的豆子越小，撒的越多的時候，結果就越精確。藉助電腦程式可以生成大量均勻分布坐標點，然後統計出圖形內的點數，透過它們占總點數的比例和坐標點生成範圍的面積就可以求出圖形面積。
## 工作過程
在解決實際問題的時候應用蒙地卡羅方法主要有兩部分工作：
1. 用蒙地卡羅方法類比某一過程時，需要產生各種機率分布的隨機變數。
2. 用統計方法把模型的數字特徵估計出來，從而得到實際問題的數值解。
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統計類比方法 | 使用蒙地卡羅方法進行分子類比計算是按照以下步驟進行的：
1. 使用亂數生成器產生一個隨機的分子構型。
2. 對此分子構型的其中粒子坐標做無規則的改變，產生一個新的分子構型。
3. 計算新的分子構型的能量。
4. 比較新的分子構型與改變前的分子構型的能量變化，判斷是否接受該構型。
* 若新的分子構型能量低於原分子構型的能量，則接受新的構型，使用這個構型重複再做下一次迭代。
* 若新的分子構型能量高於原分子構型的能量，則計算玻爾茲曼因子，並產生一個亂數。
* 若這個亂數大於所計算出的玻爾茲曼因子，則放棄這個構型，重新計算。
* 若這個亂數小於所計算出的玻爾茲曼因子，則接受這個構型，使用這個構型重複再做下一次迭代。
5. 如此進行迭代計算，直至最後搜尋出低於所給能量條件的分子構型結束。
## 在數學中的應用
通常蒙地卡羅方法透過構造符合一定規則的亂數來解決數學上的各種問題。對於那些由於計算過於複雜而難以得到解析解或者根本沒有解析解的問題，蒙地卡羅方法是一種有效的求出數值解的方法。一般蒙地卡羅方法在數學中最常見的應用就是蒙特卡羅積分。下面是蒙地卡羅方法的兩個簡單應用：
### 積分
非權重蒙特卡羅積分，也稱確定性抽樣，是對被積函數變數區間進行隨機均勻抽樣，然後對抽樣點的函數值求平均，從而可以得到函數積分的近似值。此種方法的正確性是基於機率論的中央極限定理。當抽樣點數為 m 時，使用此種方法所得近似解的統計誤差只與 m 有關（與 ${\begin{smallmatrix}{\frac {1}{\sqrt[{}]{m}}}\end{smallmatrix}}$ 正相關），不隨積分維數的改變而改變。因此當積分維度較高時，蒙地卡羅方法相對於其他數值解法更優。
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統計類比方法 | 蒙地卡羅方法可用於近似計算圓周率：讓電腦每次隨機生成兩個 0 到 1 之間的數，看以這兩個實數為橫縱坐標的點是否在單位圓內。生成一系列隨機點，統計單位圓內的點數與總點數，（圓面積和正方形面積之比為 PI:4，PI 為圓周率），當隨機點取得越多時，其結果越接近於圓周率（然而準確度仍有爭議：即使取 10 的 9 次方個隨機點時，其結果也僅在前 4 位元與圓周率吻合）。用蒙地卡羅方法近似計算圓周率的先天不足是：第一，電腦產生的亂數是受到儲存格式的限制的，是離散的，並不能產生連續的任意實數；上述做法將平面分割成一個個網格，在空間也不是連續的，由此計算出來的面積當然與圓或多或少有差距。
## 在機器學習中的應用
蒙地卡羅演算法也常用於機器學習，特別是強化學習的演算法中。一般情況下，針對得到的樣本資料集建立相對模糊的模型，透過蒙地卡羅方法對於模型中的參數進行選取，使之於原始資料的殘差儘可能的小。從而達到建立模型擬合樣本的目的。
## 參見
* 遺傳演算法
* 粒子濾波器
* 擬蒙地卡羅方法
## 注釋
## 參考
* Anderson, Herbert L. Metropolis, Monte Carlo and the MANIAC (PDF). Los Alamos Science. 1986, **14**: 96–108 [2018-04-29]. （原始內容 (PDF) 存檔於 2017-07-02）.
* Benov, Dobriyan M. The Manhattan Project, the first electronic computer and the Monte Carlo method. Monte Carlo Methods and Applications. 2016, **22** (1): 73–79 [2018-04-29]. doi:10.1515/mcma-2016-0102. （原始內容存檔於 2019-06-03）. | [
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統計類比方法 | * Int Panis, L; Rabl, A; De Nocker, L; Torfs, R. P. Sturm , 編. Diesel or Petrol ? An environmental comparison hampered by uncertainty. Mitteilungen Institut für Verbrennungskraftmaschinen und Thermodynamik (Technische Universität Graz Austria). 2002,. Heft 81 Vol 1: 48–54.
* Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. Numerical Recipes in Fortran 77: The Art of Scientific Computing. Fortran Numerical Recipes **1** Second. 劍橋大學出版社. 1996 [1986]. ISBN 0-521-43064-X.
* Ripley, B. D. Stochastic Simulation. 約翰威立. 1987.
* Robert, C. P.; Casella, G. Monte Carlo Statistical Methods 2nd. New York: Springer. 2004. ISBN 0-387-21239-6.
* Rubinstein, R. Y.; Kroese, D. P. Simulation and the Monte Carlo Method 2nd. New York: John Wiley & Sons. 2007. ISBN 978-0-470-17793-8.
* Savvides, Savvakis C. Risk Analysis in Investment Appraisal. Project Appraisal Journal. 1994, **9** (1). doi:10.2139/ssrn.265905. | [
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統計類比方法 | * Sawilowsky, Shlomo S.; Fahoome, Gail C. Statistics via Monte Carlo Simulation with Fortran. Rochester Hills, MI: JMASM. 2003. ISBN 0-9740236-0-4.
* Sawilowsky, Shlomo S. You think you've got trivials? (PDF). Journal of Modern Applied Statistical Methods. 2003, **2** (1): 218–225.
* Silver, David; Veness, Joel. Monte-Carlo Planning in Large POMDPs (PDF). Lafferty, J.; Williams, C. K. I.; Shawe-Taylor, J.; Zemel, R. S.; Culotta, A. (編). Advances in Neural Information Processing Systems 23. Neural Information Processing Systems Foundation. 2010 [2018-04-29]. （原始內容 (PDF) 存檔於 2012-05-25）.
* Szirmay-Kalos, László. Monte Carlo Methods in Global Illumination - Photo-realistic Rendering with Randomization. VDM Verlag Dr. Mueller e.K. 2008. ISBN 978-3-8364-7919-6.
* Tarantola, Albert. Inverse Problem Theory. Philadelphia: Society for Industrial and Applied Mathematics. 2005 [2018-04-29]. ISBN0-89871-572-5. （原始內容存檔於 2021-02-25）.
* Vose, David. Risk Analysis, A Quantitative Guide Third. John Wiley & Sons. 2008. | [
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波函數 | : 在這篇文章內，向量與純量分別用粗體與斜體顯示。例如，位置向量通常用 $\mathbf {r} \,\!$ 表示；而其大小則用 $r\,\!$ 來表示。
在量子力學裏，量子系統的量子態可以用**波函數**（英語：Wave function）來描述。薛丁格方程式設定波函數如何隨著時間流逝而演化。
波函數 $\Psi (\mathbf {r} ,t)$ 是一種複值函數，表示粒子在位置 $\mathbf {r} $ 、時間 $t$ 的機率幅，它的絕對值平方 $|\Psi (\mathbf {r} ,t)|^{2}$ 是在位置 $\mathbf {r} $ 、時間 $t$ 找到粒子的機率密度。以另一種角度詮釋，波函數 $\Psi (\mathbf {r} ,t)$ 是「在某時間、某位置發生交互作用的機率幅」。
## 歷史
在 1920 年代與 1930 年代，理論量子物理學者大致分為兩個陣營。第一個陣營的成員主要為路易・德布羅意和埃爾溫・薛丁格等等，他們使用的數學工具是微積分，他們共同創建了波動力學。第二個陣營的成員主要為維爾納・海森堡和馬克斯・玻恩等等，使用線性代數，他們建立了矩陣力學。後來，薛丁格證明這兩種方法完全等價。
德布羅意於 1924 年提出的德布羅意假說表明，每一種微觀粒子都具有波粒二象性。電子也不例外，具有這種性質。電子是一種波動，是電子波。電子的能量與動量分別決定了它的物質波頻率與波數。既然粒子具有波粒二象性，應該會有一種能夠正確描述這種量子特性的波動方程式，這點子給予埃爾溫・薛丁格極大的啟示，他因此開始尋找這波動方程式。薛丁格參考威廉・哈密頓先前關於牛頓力學與光學之間的類比這方面的研究，在其中隱藏了一個奧妙的發現，即在零波長極限，物理光學趨向於幾何光學；也就是說，光波的軌道趨向於明確的路徑，而這路徑遵守最小作用量原理。哈密頓認為，在零波長極限，波傳播趨向於明確的運動，但他並沒有給出一個具體方程式來描述這波動行為，而薛丁格給出了這方程式。他從哈密頓 - 雅可比方程式成功地推導出薛丁格方程式。他又用自己設計的方程式來計算氫原子的譜線，得到的答案與用波耳模型計算出的答案相同。他將這波動方程式與氫原子光譜分析結果，寫為一篇論文，1926 年，正式發表於物理學界。從此，量子力學有了一個嶄新的理論平台。 | [
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波函數 | 薛丁格給出的薛丁格方程式能夠正確地描述波函數的量子行為。那時，物理學者尚未能解釋波函數的涵義，薛丁格嘗試用波函數來代表電荷的密度，但遭到失敗。1926 年，玻恩提出機率幅的概念，成功地解釋了波函數的物理意義。可是，薛丁格本人不贊同這種統計或機率方法，和它所伴隨的非連續性波函數塌縮，如同愛因斯坦認為量子力學只是個決定性理論的統計近似，薛丁格永遠無法接受哥本哈根詮釋。在他有生最後一年，他寫給玻恩的一封信內，薛丁格清楚地表明了這意見。
1927 年，道格拉斯・哈特里與弗拉基米爾・福克在對於多體波函數的研究踏出了第一步，他們發展出哈特里－福克方程式來近似方程式的解。這計算方法最先由哈特里提出，後來福克將之加以改善，能夠符合包立不相容原理的要求。
薛丁格方程式不具有勞侖茲不變性 ，無法準確給出符合相對論的結果。薛丁格試著用相對論的能量動量關係式，來尋找一個相對論性方程式，並且描述電子的相對論性量子行為。但是這方程式給出的精細結構不符合阿諾・索末菲的結果，又會給出違背量子力學的負機率和怪異的負能量現象，他只好將這相對論性部分暫時擱置一旁，先行發表前面提到的非相對論性部分。
1926 年，奧斯卡・克萊因和沃爾特・戈爾登將電磁相對作用納入考量，獨立地給出薛丁格先前推導出的相對論性部分，並且證明其具有勞侖茲不變性。這方程式後來稱為克萊因 - 戈爾登方程式。
1928 年，保羅・狄拉克最先成功地統一了狹義相對論與量子力學，他推導出狄拉克方程式，適用於電子等等自旋為 1/2 的粒子。這方程式的波函數是一個旋量，擁有自旋性質。
## 概述
### 位置空間波函數
假設一個自旋為零的粒子移動於一維空間。這粒子的量子態以波函數表示為 $\Psi (x,t)$ ；其中， $x$ 是位置， $t$ 是時間。波函數是複值函數。測量粒子位置所得到的結果不是決定性的，而是機率性的。粒子的位置 $x$ 在區間 $[a,b]$ （即 $a\leq x\leq b$ ）的機率 $P_{a\leq x\leq b}$ 為
: $P_{a\leq x\leq b}=\int _{a}^{b}\,|\Psi (x,t)|^{2}\mathrm {d} x$ ；
其中， $t$ 是對於粒子位置做測量的時間。
換句話說， $|\Psi (x,t)|^{2}$ 是粒子在位置 $x$ 、時間 $t$ 的機率密度。
這導致歸一化條件：在位置空間的任意位置找到粒子的機率為 100%： | [
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波函數 | : $\int _{-\infty }^{\infty }\,|\Psi (x,t)|^{2}\mathrm {d} x=1$ 。
### 動量空間波函數
在動量空間，粒子的波函數表示為 $\Phi (p,t)$ ；其中， $p$ 是一維動量，值域從 $-\infty $ 至 $+\infty $ 。測量粒子動量所得到的結果不是決定性的，而是機率性的。粒子的動量 $p$ 在區間 $[a,b]$ （即 $a\leq p\leq b$ ）的機率為
: $P_{a\leq p\leq b}=\int _{a}^{b}\,|\Phi (p,t)|^{2}\mathrm {d} p$ 。
動量空間波函數的歸一化條件也類似：
: $\int _{-\infty }^{\infty }\,\left|\Phi (p,t)\right|^{2}\mathrm {d} p=1$ 。
### 兩種波函數之間的關係
位置空間波函數與動量空間波函數彼此是對方的傅立葉變換。他們各自擁有的資訊相同，任何一種波函數都可以用來計算粒子的相關性質。兩種波函數之間的關係為
: $\Phi (p,t)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\,e^{-ipx/\hbar }\Psi (x,t)\mathrm {d} x$ 、
: $\Psi (x,t)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\,e^{ipx/\hbar }\Phi (p,t)\mathrm {d} p$ 。
## 薛丁格方程式
在一維空間裏，運動於位勢 $V(x)$ 的單獨粒子，其波函數滿足含時薛丁格方程式
: $-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi (x,t)+V(x)\Psi (x,t)=i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)$ ；
其中， $m$ 是質量， $\hbar $ 是約化普朗克常數。
不含時薛丁格方程式與時間無關，可以用來計算粒子的本徵能量與其它相關的量子性質。應用分離變數法，猜想 $\Psi (x,\,t)$ 的函數形式為 | [
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波函數 | : $\Psi (x,\,t)=\psi _{E}(x)e^{-iEt/\hbar }$ ；
其中， $E$ 是分離常數，稍加推導可以論定 $E$ 就是能量， $\psi _{E}(x)$ 是對應於 $E$ 的本徵函數。
代入這猜想解，經過一番運算，可以推導出一維不含時薛丁格方程式：
: $-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi _{E}(x)+V(x)\psi _{E}(x)=E\psi _{E}(x)$ 。
## 波函數的機率詮釋
波函數 $\Psi (\mathbf {r} ,t)$ 是機率波。其模的平方 $\vert \Psi (\mathbf {r} ,t)\vert ^{2}\,$ 代表粒子在該處出現的機率密度，並且具有歸一性，全空間的積分
: $\int \vert \Psi (\mathbf {r} ,t)\vert ^{2}\,d^{3}\,x=1$ 。
波函數的另一個重要特性是相干性。兩個波函數疊加，機率的大小取決於兩個波函數的相位差，類似光學中的楊氏雙縫實驗。
## 波函數的本徵值和本徵態
在量子力學中，可觀察量 $A$ 以算符 ${\hat {A}}$ 的形式出現。 ${\hat {A}}$ 代表對於波函數的一種運算。例如，在位置空間裏，動量算符 ${\hat {\mathbf {p} }}$ 的形式為
: ${\hat {\mathbf {p} }}=-i\hbar \nabla $ 。
可觀察量 $A$ 的本徵方程式為
: ${\hat {A}}\psi =a\psi $ 。
對應的 $a$ 稱為算符 ${\hat {A}}$ 的本徵值， $\psi $ 稱為算符 ${\hat {A}}$ 的本徵態。假設對於 ${\hat {A}}$ 的本徵態 $\psi $ 再測量可觀察量 $A$ ，則得到的結果是本徵值 $a$ 。
## 態疊加原理
假設對於某量子系統測量可觀察量 $A$ ，而可觀察量 $A$ 的本徵態 $|a_{1}\rangle $ 、 $|a_{2}\rangle $ 分別擁有本徵值 $a_{1}$ 、 $a_{2}$ ，則根據薛丁格方程式的線性關係，疊加態 $|\psi \rangle $ 也可以是這量子系統的量子態： | [
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波函數 | : $|\psi \rangle =c_{1}|a_{1}\rangle +c_{2}|a_{2}\rangle $ ；
其中， $c_{1}$ 、 $c_{2}$ 分別為疊加態處於本徵態 $|a_{1}\rangle $ 、 $|a_{2}\rangle $ 的機率幅。
假設對這疊加態系統測量可觀察量 $A$ ，則測量獲得數值是 $a_{1}$ 或 $a_{2}$ 的機率分別為 $|c_{1}|^{2}$ 、 $|c_{2}|^{2}$ ，期望值為
: $\langle \psi |A|\psi \rangle =|c_{1}|^{2}a_{1}+|c_{2}|^{2}a_{2}$ 。
## 定態
在量子力學中，一類基本的問題是哈密頓算符 ${\hat {H}}$ 不含時間的情況。對於這問題，應用分離變數法，可以將波函數 $\Psi (\mathbf {r} ,t)$ 分離成一個只與位置有關的函數 $\psi (\mathbf {r} )$ 和一個只與時間有關的函數 $f(t)$ ：
: $\Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )f(t)$ 。
將這公式代入薛丁格方程式，就會得到
: $f(t)=\exp {(-iEt/\hbar )}$ 。
而 $\psi (\mathbf {r} )$ 則滿足本徵能量薛丁格方程式：
: ${\hat {H}}\psi (\mathbf {r} )=E\psi (\mathbf {r} )$ 。
## 例子
### 自由粒子
3D 空間中的自由粒子，其波向量 為 **k** ， 角頻率 為 _ω_，其波函數為：
: $\Psi (\mathbf {r} ,t)=Ae^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}\,.$
### 無限深方形阱
粒子被限制在 _x_ = 0 和 _x_ = _L_ 之間的 1D 空間中，其波函數為:
: ${\begin{aligned}\Psi (x,t)&={\sqrt {\frac {2}{L}}}\sin \left({\frac {n\pi x}{L}}\right)e^{-i\omega _{n}t},&\quad 0\leq x\leq L\\\Psi (x,t)&=0,&x<0,x>L\\\end{aligned}}$ | [
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波函數 | 其中， $\hbar \omega _{n}={\frac {n^{2}h^{2}}{8mL^{2}}}$ 是能量本徵值， $n$ 是正整數， $m$ 是質量。
### 有限位勢壘
在 1D 情況下，粒子處於如下勢壘中：
: $V(x)={\begin{cases}V_{0}&|x|<a\\0&{\text{otherwise,}}\end{cases}}$
其波函數的定態解為 ( $k,\kappa $ 為常數)
: $\psi (x)={\begin{cases}A_{\mathrm {r} }\exp(ikx)+A_{\mathrm {l} }\exp(-ikx)&x<-a,\\B_{\mathrm {r} }\exp(\kappa x)+B_{\mathrm {l} }\exp(-\kappa x)&|x|\leq a,\\C_{\mathrm {r} }\exp(ikx)+C_{\mathrm {l} }\exp(-ikx)&x>a.\end{cases}}$
### 量子點
**量子點**是在把激子在三個空間方向上束縛住的半導體奈米結構。粒子在三個方向上都處在勢阱中。勢阱可以由於靜電位（由外部的電極，摻雜，應變，雜質產生），兩種不同半導體材料的界面（例如：在自組量子點中），半導體的表面（例如：半導體奈米晶體），或者以上三者的結合。量子點具有分離的量子化的能譜。所對應的波函數在空間上位於量子點中，但延伸於數個晶格週期中。其中的能階可以用類似無限深方形阱的模型來描述，能階位置取決於勢阱寬度。
## 參閱
* 波包
## 參考文獻
## 注釋 | [
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] |
博學者 | **博學家**（**polymath**）或**通才**，是指精通多個不同範疇而且表現超群的人。
## 文藝復興人
**文藝復興人**（**Renaissance man**，或拉丁語、義大利語中的 **Homo universalis**、**Uomo Universale**），是「通才」的一個別稱，特別是在藝術、科學方面。這個詞語的產生是因為在文藝復興時出現了不少這類人，其中巴迪斯蒂・阿伯拉蒂（Leone Battista Alberti，1404－1472）、李奧納多・達・文西就是當中的佼佼者。不過，「文藝復興人」一詞並非專指文藝復興時期所產生的通才。在現代英語辭典，通才也指「對多方範疇感興趣，或對多方範疇有基本、片面知識」。
## 著名博學者
* 巴迪斯蒂・阿伯拉蒂 - 義大利文藝復興時期建築師、建築理論家、作家、詩人、哲學家、密碼學家。
* 李奧納多・達文西 - 義大利文藝復興時期藝術家、博學家，在數十個領域都有極高成就。
* 西蒙・斯蒂文 - 弗蘭德數學家、物理學家、建築師、軍事工程師、樂理學家。
* 布萊茲・帕斯卡 - 法國神學家、哲學家、數學家、物理學家、化學家、音樂家、教育家、氣象學家。
* 阿塔納斯・珂雪 - 17 世紀德國學者，於細菌學、醫學、聲學、天文學、力學、埃及學、地質學、數學、語言學、音樂理論皆有成就。
* 哥特佛萊德・萊布尼茲 - 德國哲學家、數學家，也從事神學理論研究。
* 米哈伊爾・瓦西里耶維奇・羅蒙諾索夫 - 俄國化學家、哲學家、詩人，並通曉礦物學、冶金學、語言學。
* 伊曼努爾・康德 - 德國哲學家、數學家，並通曉地理學、教育學、人類學。
* 皮埃爾・博馬舍 - 法國劇作家、軍火商、金融鉅子、間諜、革命者，也是一名發明家、音樂家、外交官、出版商、園藝家、諷刺作家。
* 約翰・沃爾夫岡・馮・歌德 - 德國戲劇家、詩人、自然科學家、文藝理論家、政治家。
* 湯瑪士・楊格 - 英國物理學家、生理學家、醫師，也通曉樂理、語言學、埃及學。
* 艾薩克・牛頓 - 英國物理學家、數學家、天文學家、自然哲學家、煉金術士。
* 亞歷山大・波菲里耶維奇・鮑羅定 - 俄國作曲家、化學家、醫師。
* 約翰・馮・諾伊曼 - 在泛函分析、遍歷理論、幾何學、拓撲學、數值分析等眾多數學領域、電腦學、量子力學、經濟學中有重大貢獻。
* 班傑明・富蘭克林 - 美國開國元勛、政治家、外交家、科學家、發明家。 | [
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] |
博學者 | * 顧毓琇 - 中華民國教育家、科學家、詩人、戲劇家、音樂家、禪學家。
* 梁啟超 - 中國清末民初時期政治家、思想家、史學家、文學家、法學家。
* 亞里斯多德 - 古希臘哲學家、政治家、經濟學家、文學家、天文學家、地理學家、氣象學、物理學等等，他研究了當時幾乎所有的學科。
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* Edward Carr. Last Days of the Polymath. Intelligent Life. The Economist Group. 1 October 2009 [12 January 2017]. （原始內容存檔於 2019-12-20）.
* Edmonds, David (August 2017). Does the world need polymaths? （頁面存檔備份，存於網際網路檔案館）, BBC.
* Frost, Martin, "Polymath: A Renaissance Man".
* Grafton, A, "The World of the Polyhistors: Humanism and Encyclopedism", Central European History, 18: 31–47. (1985).
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] |
博學者 | * Sher, Barbara. Refuse to Choose!: A Revolutionary Program for Doing Everything that You Love. [Emmaus, Pa.]: Rodale. 2007. ISBN 978-1594866265. 含有內容需登入查看的頁面 (link)
* Twigger, Robert, "Anyone can be a Polymath" We live in a one-track world, but anyone can become a polymath – Robert Twigger | Aeon Essays （頁面存檔備份，存於網際網路檔案館）.
* Waqas, Ahmed. The Polymath: Unlocking the Power of Human Versatility. West Sussex, UK: John Wiley & Sons. 2019 [6 August 2019]. ISBN9781119508489. （原始內容存檔於 2021-04-17）.
* Waquet, F, (ed.) "Mapping the World of Learning: The 'Polyhistor' of Daniel Georg Morhof" (2000).
* Wiens, Kyle, "In defense of polymaths" （頁面存檔備份，存於網際網路檔案館）.
* Brown, Vincent Polymath-Info Portal （頁面存檔備份，存於網際網路檔案館）.
## 注釋
## 外部連結
## 參見
* 天才
* 神童 | [
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波尔兹曼 | **路德維希・愛德華・波茲曼**（德語：Ludwig Eduard Boltzmann；1844 年 2 月 20 日 —1906 年 9 月 5 日）是奧地利物理學家、哲學家。作為物理學家，他最偉大的功績是發展了通過原子性質（如原子量、電荷量、結構等）來解釋預測物質物理性質（如黏性、熱傳導、擴散等）的統計力學，並且從統計概念出發，完美闡釋了熱力學第二定律。
## 生平簡介
### 幼年及受教育經歷
波茲曼生於奧地利首都維也納。其父路德維希・格奧爾格・波茲曼是一名稅吏，而他的祖父是一個自柏林移居至維也納的鐘表製造商。他的母親卡特琳那・玻恩芬德（Katharina Pauernfeind）來自薩爾茨堡。他從他的家教那裡接受了基礎教育，而後在上奧地利的林茨就讀高中。當他 15 歲時，他的父親去世。
波茲曼自 1863 年開始在維也納大學攻讀物理學。指導過他的老師有約瑟夫・洛施密特、約瑟夫・斯特凡、安德烈亞斯・馮・厄廷格豪森和約瑟夫・佩茲伐。波茲曼在斯特凡的指導下在 1866 年獲得理學博士學位，他的學位論文主題是分子運動論。1867 年，他成為無俸講師。在獲得博士學位後，波茲曼又當了兩年斯特凡的助手，而斯特凡讓波茲曼了解了馬克士威的工作。
### 學術生涯
1869 年，波茲曼 25 歲時，藉助斯特凡的推薦信， 他受聘為格拉茨大學數學物理學教授。1869 年，他與羅伯特・本生和利奧・格尼斯伯格在海德爾堡共事數月，而後他 1871 年與古斯塔夫・克希荷夫、赫爾曼・馮・亥姆霍茲在柏林合作過。1873 年，波茲曼成為維也納大學的數學教授，擔任至 1876 年為止。
1872 年，波茲曼與格拉茨的一位有抱負的數學和物理老師亨麗埃特・艾根特拉相遇。當時奧地利的大學不錄取女性，她在試圖旁聽當地大學講授的課程時被拒。她在波茲曼建議下進行了申訴，並獲得了成功。1876 年 7 月 17 日，他們結為伉儷。他們育有三個女兒和兩個兒子。之後，波茲曼回到格拉茨成為實驗物理學教授。斯凡特・奧古斯特・阿倫尼烏斯和瓦爾特・能斯特都在格拉茨受過他的教導。他在格拉茨度過了十四年快樂的時光。而正是在那裡，他發展起對自然界的統計概念。1885 年，他成為奧地利皇家科學院院士，而後，1887 年，他成為格拉茨大學的校長。1888 年，他被推選為瑞典皇家科學院院士。 | [
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波尔兹曼 | 1890 年，波茲曼受聘為慕尼黑大學的理論物理學教授。1893 年，他繼承他的導師約瑟夫・斯特凡成為維也納大學的理論物理學教授。
### 晚年
波茲曼晚年投入了大量精力來捍衛他的理論。他與他在維也納的同事交惡，特別是 1895 年成為哲學及科學史教授的恩斯特・馬赫。同是在 1895 年，格奧爾格・亥姆和威廉・奧斯特瓦爾德在呂貝克的一次學術會議上提出了「活力說」。他們認為能量，而非物質，是宇宙的主要組成。波茲曼及支持他原子理論的物理學家與之進行了曠日持久的爭論。1900 年，波茲曼受威廉・奧斯特瓦爾德之邀來到萊比錫大學在馬赫因為健康狀況欠佳而退休之後，1902 年，波茲曼回到了維也納。1903 年，他與古斯塔夫・馮・埃舍里希及埃米爾・穆勒一起創建了奧地利數學學會。他在維也納的學生包括卡爾・普里布拉姆，保羅・埃倫費斯特和莉澤・邁特納。
在維也納，波茲曼教授物理學，同時也講授哲學。他關於自然哲學的演講非常受歡迎，在當時引起相當大的關注，首次演講即大告成功。演講在當地最大的報告廳舉行，可報告廳內依舊人滿為患，以至樓梯上都站滿了人。由於波茲曼的哲學演講大獲成功，當時的奧匈帝國皇帝也在皇宮接見了他。
波茲曼晚年精神狀況欠佳，情緒經常起伏不定，與躁鬱症症狀類似。他自嘲式地將他變化不定的情緒歸咎於他在懺悔星期二和聖灰星期三間出生的緣故。邁特納指出與波茲曼親近的人都能看出他有嚴重的抑鬱和自殺傾向。
1906 年 9 月 5 日，在的里雅斯特附近的杜伊諾度假時，波茲曼在情緒失控中自縊身亡。他被葬於維也納中央公墓，墓碑上鐫刻著波茲曼熵公式：
: $S=k\cdot \log W\,$
其中 $\log $ 表示自然對數。
## 哲學觀點
波茲曼的分子運動論是在預設原子和分子確實存在前提下建立的。但當時幾乎所有的德國哲學家和許多科學家，像恩斯特・馬赫及物理化學家威廉・奧斯特瓦爾德，都不認為它們實際存在。十九世紀九十年代，他試圖通過建立一種繞過討論原子是否存在來折衷原子論和反原子論的立場。他的解決方法是引用赫茲的理論，將原子僅僅歸為一種物理模型 —— 原子論者可以認為這個模型就是實實在在的原子，而反原子論者可以認為原子是一個有用但並非實際存在的模型。但這並沒有使雙方滿意。而由於波茲曼對於原子和分子存在的假定及對熱力學第二定律統計意義上的解釋，奧斯特瓦爾德及眾多「純粹熱力學」的擁護者更進一步試圖去否定分子運動論和統計力學的合理性。 | [
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波尔兹曼 | 而在世紀之交，波茲曼的科學工作受到一個新興思潮的威脅。一些物理學家，包括馬赫的學生，古斯塔夫・焦曼，認為赫茲的理論中的所有的電磁現象都是連續變化的，而所有的物理現象最終都可以還原為電磁現象。由於原子和分子的存在會破壞這種連續性，因而它們並不存在。這一思潮深深打擊了波茲曼，因為它可能意味著他的分子運動論及對熱力學第二定律統計意義解釋的終結。
而 1901 年，馬赫離職後， 波茲曼重返維也納，並決定自己從哲學意義上來解釋自己的物理學理論來應對對其的質疑，但不久又再次受挫。1904 年，在聖路易斯舉辦的一個物理學會議上，與會的大多數物理學家否定原子的存在。他沒有受邀參加物理學部分的討論，更在應用數學部分討論受阻。他認為對於科學家來說，去克服從過去繼承來的哲學理念是非常困難的。
1905 年，波茲曼試圖通過與弗朗茲・布倫塔諾的廣泛交流來進一步理解哲學的本質以使科學擺脫它的影響，但他本人也對這個想法沒有什麼信心。1906 年，他的精神狀態已經糟糕到他不得不離職。當年 9 月他在與他妻子及女兒在義大利的的里亞斯特度假時自縊身亡。
## 物理學成就
波茲曼最重要的科學方面的貢獻是分子運動論，其中包括研究氣體分子運動速度的馬克士威 - 波茲曼分布，基於古典力學的研究能量的馬克士威 - 波茲曼統計和波茲曼分布。它們能在非必須量子統計時解釋許多現象，並且更深入的揭示了溫度等熱力學系統的狀態函數的物理意義。
當時多數物理學家並不像他一樣深信原子和分子的切實存在。而蘇格蘭的詹姆斯・克拉克・馬克士威和美國的約西亞・吉布斯，以及自約翰・道爾頓 1808 提出原子論來的大多數化學家卻深信原子和分子的存在。波茲曼和當時德國首級的物理學刊物的編輯進行了曠日持久的爭論。這些編輯只是將原子和分子當作方便的理論模型而並不願將它們與現實聯繫起來。在波茲曼去世後數年，讓・佩蘭在阿爾伯特・愛因斯坦 1905 年的研究基礎上對於膠體懸浮物的研究（1908–1909），測定了亞佛加厥常數和波茲曼常數，並向世界證明了原子和分子確實存在。
波茲曼還在分子運動論中發現了熵和微觀狀態的機率分布的對數關係，並提出著名的波茲曼熵公式：
: $S=k_{B}\ln W\,$ | [
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波尔兹曼 | 其中 _k_ = 1.3806505(24) × 10−23 J |K−1，稱作波茲曼常數。 $W$ 是德語中機率（_Wahrscheinlichkeit_）的縮寫，，這裡，更準確地說，是系統的微觀狀態數。波茲曼的範式是有 _N_ 個分子的理想氣體，_N_i 是分子位置和動量的第 _i_ 個微觀狀態。 $W$ 可以用下面這個排列式計算：
: $W={\frac {N!}{\prod _{i}N_{i}!}}$
其中 _i_ 的範圍是分子所有可能的微觀狀態。 由於系統中相同狀態的相同粒子是不可區別的，因而對分母進行了「修正」。
波茲曼由於他在 1877 年暗示一個物理學系統的能階離散成為量子力學的先驅。
波茲曼熵公式被鐫刻在他維也納中央墓地的墓碑上。
## 波茲曼方程式
波茲曼方程式敘述了理想氣體系統內部粒子的運動情況。
: ${\frac {\partial f}{\partial t}}+v{\frac {\partial f}{\partial x}}+{\frac {F}{m}}{\frac {\partial f}{\partial v}}={\frac {\partial f}{\partial t}}\left.{\!\!{\frac {}{}}}\right|_{\mathrm {collision} }$
其中_ƒ_為分布函數（參見馬克士威 - 波茲曼分布），代表在某一時刻在一位置和具有該動量的粒子數目、密度或發現粒子的機率， _F_ 是力 (包含相對於所討論系統的外力與粒子之間的交互作用)，_m_ 是單個粒子的質量，_t_ 是時間，_v_ 是具有該動量粒子所擁有的速度。
這個方程式描述了粒子位置和動量機率分布在相空間中的密度分布雲圖隨時間和空間的演化（參見哈密爾頓力學）。等式左邊第一項代表分布函數隨時間的變化，第二項給出隨空間的變化，然後第三項描述了某個力對粒子的影響效果。等式右邊代表碰撞（collision）的所造成的分布函數的變化 (如交換動量或能量)。從原理上，在適當的邊界條件下，這個方程式可以描述氣體粒子集合體的動態。 這個一階偏微分方程式看起來非常易解， 因為_ƒ_可以表示任意的單粒子分布函數，並且作用在粒子上的力直接取決於速度分布函數，但卻以難以積分著稱。大衛・希爾伯特多年努力去解它但卻沒有獲得實質成功。 | [
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波尔兹曼 | 由波茲曼給出的表示碰撞的項是近似的。但波茲曼方程式對於理想氣體的恰普曼一恩斯科格定理解卻是高度嚴格的。只有在激波條件下，才有可能得到對於理想氣體的錯誤的解。波茲曼使用許多年試圖利用這個方程式證明 H 定理，進而驗證熱力學第二定律。這之中他做了一個假設 -- 碰撞的項是針對分子混沌的。然而這個假設破壞了時間反演對稱，暗示了第二定律的必然成立，所以最終他在與洛施密特以及其他基於洛施密特悖論的觀點的長期爭論中落敗。
到了 20 世紀 70 年代，E.G.D. 柯漢和 J.R. 多夫曼證明了波茲曼方程式對於高密度物質的系統擴展，在數學上是不可能的，因此最終對於稠密氣體及液體的非平衡態的統計力學研究焦點，已經轉移到格林 - 久保關係（Green-Kubo relations）、漲落定理及其他方法。
## 對熱力學第二定律的闡釋
現在認為熱力學第二定律是描述系統無序性的定律的觀點，源於波茲曼對其的理解，他亦試圖將其還原為描述機械粒子間隨機碰撞機率的定律。波茲曼沿用馬克士威的方法，也將氣體分子模型化為箱中相互碰撞的撞球，隨著分子間的碰撞，速度分布會變得更為無序，最終導致了系統的宏觀性質均一，而微觀處於最為無序的狀態 -- 或者說系統的熵趨於最大值。由此，他提出熱力學第二定律是微觀世界的最概然狀態，為無序在宏觀的表現。微觀世界的最概然狀態（即最可能處於的狀態）之所以是無序態，是由於無序的微觀狀態數遠多於有序的微觀狀態數。波茲曼最後總結道，分子以同樣的速率在同一方向運動的有序狀態，「可以想像是最不可能處於的狀態，亦是不可能的能量組態」。
波茲曼將熱力學第二定律歸結為統計現象的這一功績，使熱力學中的熵增加原理得到統計上的解釋，從而更易理解。
## 另見
* 能量均分定理
* 波茲曼大腦
* 波茲曼模型
* 分子理論史
* 計算流體力學的格子波茲曼法 (LBM)
* 熱統計力學哲學
* 波茲曼獎
* 波茲曼（隕石坑）
## 參考文獻及注釋
## 延伸閱讀
## 外部連結
* Uffink, Jos. Boltzmann's Work in Statistical Physics. Stanford Encyclopedia of Philosophy. 2004 [2007-06-11]. （原始內容存檔於 2007-06-09）.
* 約翰・J・奧康納； 埃德蒙・F・羅伯遜, Boltzmann, MacTutor 數學史檔案 （英語） | [
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波尔兹曼 | * "Ludwig Boltzmann, （頁面存檔備份，存於網際網路檔案館）" Universität Wien (German).
* Ruth Lewin Sime, _Lise Meitner: A Life in Physics_ Chapter One: Girlhood in Vienna （頁面存檔備份，存於網際網路檔案館） gives Lise Meitner's account of Boltzmann's teaching and career.
* E.G.D. Cohen, 1996, "Boltzmann and Statistical Mechanics."
* Eftekhari, Ali, "Ludwig Boltzmann (1844–1906). （頁面存檔備份，存於網際網路檔案館）" Discusses Boltzmann's philosophical opinions, with numerous quotes.
* Rajasekar, S.; Athavan, N. Ludwig Edward Boltzmann. 2006-09-07. arXiv:physics/0609047 .
* 路德維希・波茲曼在數學譜系計畫的資料。
* Template:ScienceWorldBiography
* 在 Find a Grave 上的路德維希・波茲曼
* Jacob Bronowski from "The Ascent Of Man （頁面存檔備份，存於網際網路檔案館）" | [
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不含時薛丁格方程式 | 在量子力學中，**薛丁格方程式**（Schrödinger equation）是描述物理系統的量子態隨時間演化的偏微分方程式，為量子力學的基礎方程式之一，其以發表者奧地利物理學家埃爾溫・薛丁格而命名。關於量子態與薛丁格方程式的概念涵蓋於基礎量子力學假說裏，無法從其它任何原理推導而出。
在古典力學裏，人們使用牛頓第二定律描述物體運動。而在量子力學裏，類似的運動方程式為薛丁格方程式。薛丁格方程式的解完備地描述物理系統裏，微觀尺寸粒子的量子行為；這包括分子系統、原子系統、次原子系統；另外，薛丁格方程式的解還可完備地描述宏觀系統，可能乃至整個宇宙。
薛丁格方程式可以分為「含時薛丁格方程式」與「不含時薛丁格方程式」兩種。含時薛丁格方程式與時間有關，描述量子系統的波函數怎樣隨著時間而演化。不含時薛丁格方程式則與時間無關，描述了定態量子系統的物理性質；該方程式的解就是定態量子系統的波函數。量子事件發生的機率可以用波函數來計算，其機率幅的絕對值平方就是量子事件發生的機率密度。
薛丁格方程式所屬的波動力學可以數學變換為維爾納・海森堡的矩陣力學，或理察・費曼的路徑積分表述。薛丁格方程式是個非相對論性方程式，不適用於相對論性理論；對於相對論性微觀系統，必須改使用狄拉克方程式或克萊因 - 戈爾登方程式等。
## 方程式的數學形式
### 含時薛丁格方程式
含時薛丁格方程式描述物理系統隨時間演化，其最廣義形式為：
: ${\hat {H}}\Psi =i\hbar {\frac {\partial }{\partial t}}\Psi $
其中， ${\hat {H}}$ 是表徵波函數總能量的哈密頓算符， $\Psi $ 是物理系統的波函數， $i$ 是虛數單位， $\hbar $ 是約化普朗克常數， $\partial /\partial t$ 是對於時間 $t$ 的偏微分。
在三維空間裏，移動於位勢 $V(\mathbf {r} ,t)$ 的單獨粒子，其含時薛丁格方程式可以更具體地表示為
: $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\Psi (\mathbf {r} ,t)=i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)$ | [
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] |
不含時薛丁格方程式 | 其中， $m$ 是質量， $\Psi (\mathbf {r} ,t)$ 是參數為位置 $\mathbf {r} $ 、時間 $t$ 的波函數， $\nabla ^{2}$ 是拉普拉斯算符。
術語「薛丁格方程式」可以指廣義形式的薛丁格方程式，也可指具體形式的薛丁格方程式。廣義形式的薛丁格方程式名如其實，可以應用於廣泛量子力學領域，表達從狄拉克方程式到量子場論的各種方程式，只要將哈密頓算符的各種複雜表達式代入即可。通常，具體形式的薛丁格方程式所描述的系統是實際系統的簡化近似模型，這是為了要避開不必要的複雜數學運算。對於大多數案例，所得到的結果相當準確；但是對於相對論性案例，結果則並不令人滿意。對於更詳盡的細節，請參閱 相對論性量子力學。
應用薛丁格方程式時，必須先給出哈密頓算符的表達式，因此會涉及到計算系統的動能與位能；將算符表達式代入薛丁格方程式，再將所得偏微分方程式加以解析，即可找到波函數。關於系統的量子態的資訊，全部都會包含在波函數中。
### 由含時薛丁格方程式到不含時薛丁格方程式
含時薛丁格方程式 $\Psi (\mathbf {r} ,t)$ 為偏微分方程式，假定位勢與時間無關：
: $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t)+V(\mathbf {r} )\Psi (\mathbf {r} ,t)=i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)$
使用分離變量法，令 $\Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )\varphi (t)$ ，方程式變為
: $i\hbar {\frac {1}{\varphi (t)}}{\frac {d\varphi (t)}{dt}}=-{\frac {\hbar ^{2}}{2m}}{\frac {1}{\psi (\mathbf {r} )}}\nabla ^{2}\psi (\mathbf {r} )+V(\mathbf {r} )$
注意到等號左手邊是時間的函數，而右手邊則是位置的函數，所以兩邊都等於常數 $E$ ： | [
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不含時薛丁格方程式 | : $i\hbar {\frac {1}{\varphi }}{\frac {d\varphi }{dt}}=-{\frac {\hbar ^{2}}{2m}}{\frac {1}{\psi }}\nabla ^{2}\psi +V=E$
左手邊的方程式 $i\hbar {\frac {1}{\varphi (t)}}{\frac {d\varphi (t)}{dt}}=E$ 的解為
: $\varphi (t)=e^{\frac {-iEt}{\hbar }}$
右手邊的方程式可轉化為不含時薛丁格方程式：
: $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} )+V(\mathbf {r} )\psi (\mathbf {r} )=E\psi (\mathbf {r} )$
不含時薛丁格方程式也可寫為
: ${\hat {H}}\psi =E\psi $
其中， ${\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )$ 是哈密頓算符。
### 不含時薛丁格方程式
不含時薛丁格方程式與時間無關，它預言波函數可以形成駐波，稱為定態（在原子物理學裏，又稱為軌道，例如，原子軌道或分子軌道），假若能夠計算出這些定態，分析出其量子行為，則解析含時薛丁格方程式會變得更為簡易。不含時薛丁格方程式為描述定態的方程式。只有當哈密頓量不與時間顯性相關，才會使用這方程式。廣義形式的不含時薛丁格方程式為
: ${\hat {H}}\psi =E\psi $
其中， $\psi $ 是不含時波函數， $E$ 是能量。
這方程式的詮釋為，假若將哈密頓算符作用於波函數 $\psi $ 時，得到的結果與同樣波函數 $\psi $ 成正比，則波函數 $\psi $ 處於定態，比例常數 $E$ 是量子態 $\psi $ 的能量。在這裏， $\psi $ 標記設定的波函數和其對應的量子態。這方程式為又稱為「定態薛丁格方程式」，引用線性代數術語，這方程式為「能量本徵薛丁格方程式」， $E$ 是「能量本徵值」，或「本徵能量」。
在三維空間裏，處於位勢 $V(\mathbf {r} )$ 的單獨粒子，其不含時薛丁格方程式可以更具體地表示為 | [
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不含時薛丁格方程式 | : $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} )+V(\mathbf {r} )\psi (\mathbf {r} )=E\psi (\mathbf {r} )$
## 歷史背景與發展
1900 年，馬克斯・普朗克在研究黑體輻射中作出將電磁輻射能量量子化的假設，因此發現將能量 $E$ 與頻率 $\nu $ 關聯在一起的普朗克關係式 $E=h\nu $ 。1905 年，阿爾伯特・愛因斯坦從對於光電效應的研究又給予這關係式嶄新的詮釋：頻率為 $\nu $ 的光子擁有的能量為 $h\nu $ ；其中， $h$ 因子是普朗克常數。 這一點子成為後來波粒二象性概念的早期路標之一。由於在狹義相對論裏，能量與動量的關聯方式類似頻率與波數的關聯方式，因此可以揣測，光子的動量 $p$ 與波長 $\lambda $ 成反比，與波數 $k$ 成正比，以方程式來表示這關係式，
: $p=h/\lambda =\hbar k$
路易・德布羅意認為，不單光子遵守這關係式，所有粒子都遵守這關係式。他於 1924 年進一步提出的德布羅意假說表明，每一種微觀粒子都具有波動性與粒子性，這性質稱為波粒二象性。電子也不例外的具有這種性質。電子是一種物質波，稱為「電子波」。電子的能量與動量分別決定了伴隨它的物質波所具有的頻率與波數。在原子裏，束縛電子形成駐波；這意味著他的旋轉頻率只能呈某些離散數值。這些量子化軌道對應於離散能級。從這些點子，德布羅意複製出波耳模型的能級。
在 1925 年，瑞士蘇黎世每兩周會舉辦一場物理學術研討會。有一次，主辦者彼得・德拜邀請薛丁格講述關於德布羅意的波粒二象性博士論文。那段時期，薛丁格正在研究氣體理論，他從閱讀愛因斯坦關於玻色 - 愛因斯坦統計的論述中，接觸德布羅意的博士論文，在這方面有很精深的理解。在研討會裡，他將波粒二象性闡述的淋漓盡致，大家都聽的津津有味。德拜指出，既然粒子具有波動性，應該有一種能夠正確描述這種量子性質的波動方程式。他的意見給予薛丁格極大的啟發與鼓舞，他開始尋找這波動方程式。檢試此方程式最簡單與基本的方法就是，用此方程式來描述氫原子內部束縛電子的物理行為，而必能複製出波耳模型的理論結果，另外，這方程式還必須能解釋索末菲模型給出的精細結構。 | [
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不含時薛丁格方程式 | 很快，薛丁格就通過德布羅意論文的相對論性理論，推導出一個相對論性波動方程式，他將這方程式應用於氫原子，計算出束縛電子的波函數。但很可惜。因為薛丁格沒有將電子的自旋納入考量，所以從這方程式推導出的精細結構公式不符合索末菲模型。他只好將這方程式加以修改，除去相對論性部分，並用剩下的非相對論性方程式來計算氫原子的譜線。解析這微分方程式的工作相當困難，在其好朋友數學家赫爾曼・外爾鼎力相助下，他複製出了與波耳模型完全相同的答案。因此，他決定暫且不發表相對論性部分，只把非相對論性波動方程式與氫原子光譜分析結果，寫為一篇論文。1926 年，他正式發表了這論文。
這篇論文迅速在量子學術界引起震撼。普朗克表示「他已閱讀完畢整篇論文，就像被一個迷語困惑多時，渴慕知道答案的孩童，現在終於聽到了解答」。愛因斯坦稱讚，這著作的靈感如同泉水般源自一位真正的天才。愛因斯坦覺得，薛丁格已做出決定性貢獻。由於薛丁格所創建的波動力學涉及到眾所熟悉的波動概念與數學，而不是矩陣力學中既抽象又陌生的矩陣代數，量子學者都很樂意地開始學習與應用波動力學。自旋的發現者喬治・烏倫貝克驚嘆，「薛丁格方程式給我們帶來極大的解救！」沃爾夫岡・包立認為，這論文應可算是近期最重要的著作。
薛丁格給出的薛丁格方程式能夠正確地描述波函數的量子行為。在那時，物理學者尚不清楚如何詮釋波函數，薛丁格試圖以電荷密度來詮釋波函數的絕對值平方，但並不成功。1926 年，玻恩提出機率幅的概念，成功地詮釋了波函數的物理意義。但是薛丁格與愛因斯坦觀點相同，都不贊同這種統計或機率方法，以及它所伴隨的非連續性波函數塌縮。愛因斯坦主張，量子力學是個決定性理論的統計近似。在薛丁格有生的最後一年，寫給玻恩的一封信中，他清楚地表示他不接受哥本哈根詮釋。
## 含時薛丁格方程式導引
雖然含時薛丁格方程式能夠啟發式地由幾個假設推導出來，但為便於論述，在作理論量子力學研究時，經常會直接將這方程式當作一個基本假定。
### 啟發式導引 1
含時薛丁格方程式的啟發式導引建立於幾個前提：
* 粒子的總能量 $E$ 可以古典地表示為動能 $T$ 與位能 $V$ 的總和：
: : $E=T+V={\frac {p^{2}}{2m}}+V$
: 其中， $p$ 是動量， $m$ 是質量。
: 特別注意，能量 $E$ 與動量 $p$ 也出現於下述兩個關係式。 | [
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不含時薛丁格方程式 | * 愛因斯坦於提出光電效應時，指出光子的能量 $E$ 與對應的電磁波的頻率 $f$ 成正比：
: : $E=hf=\hbar \omega $
: 其中， $h$ 是普朗克常數， $\omega =2\pi f$ 是角頻率。
* 德布羅意提出的德布羅意假說表明，每一種微觀粒子都具有波粒二象性，都是一種波動。微觀粒子的動量 $p$ 與伴隨的物質波波長 $\lambda $ 有關：
: : $p=h/\lambda =\hbar k$
: 其中， $k=2\pi /\lambda $ 是波數。
: 延伸至向量，
: $\mathbf {p} =\hbar \mathbf {k} $
假設波函數是個複值平面波：
: $\Psi (x,t)=Ae^{i(kx-\omega t)}$
則其對於時間的偏導數為
: ${\frac {\partial }{\partial t}}\Psi =-i\omega \Psi $
這偏導數與能量有關：
: $E\Psi =\hbar \omega \Psi =i\hbar {\frac {\partial }{\partial t}}\Psi $
類似地，波函數對於位置的二次偏導數為
: ${\frac {\partial ^{2}}{\partial x^{2}}}\Psi =-k^{2}\Psi $
這偏導數與動量有關：
: $p^{2}\Psi =\hbar ^{2}k^{2}\Psi =-\hbar ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi $
引用古典力學的能量守恆定律，單獨粒子的總能量 $E$ 為
: $E={\frac {p^{2}}{2m}}+V$
因此，單獨粒子移動於一維位勢 $V(x)$ 的薛丁格方程式為
: $-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi +V(x)\Psi =i\hbar {\frac {\partial }{\partial t}}\Psi $
設定哈密頓函數 ${\hat {H}}$ 為 | [
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不含時薛丁格方程式 | : ${\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x)$
就可以得到廣義形式的薛丁格方程式：
: ${\hat {H}}\Psi =i\hbar {\frac {\partial }{\partial t}}\Psi $
### 啟發式導引 2
「哈密頓類比」是威廉・哈密頓在研究古典力學時給出的理論，又稱為「光學 - 力學類比」；哈密頓指出，在古典力學裏粒子的運動軌道，就如同在幾何光學裏光線的傳播路徑；垂直於這軌道的等作用量曲面，就如同垂直於路徑的等傳播時間曲面；描述粒子運動的最小作用量原理，就如同描述光線傳播的費馬原理。哈密頓發現，使用哈密頓 - 雅可比方程式，可以推導出最小作用量原理與費馬原理；同樣的形式論，可以描述光的物理行為，不論光是由遵守費馬原理的光線組成，還是由遵守最小作用量原理的粒子組成。
很多光的性質，例如，繞射、干涉等等，無法用幾何光學的理論來作解釋，必須要用到波動光學的理論來證實。這意味著幾何光學不等價於波動光學，幾何光學是波動光學的波長超短於粒子軌道曲率半徑的極限案例。哈密頓又研究發現，使用哈密頓 - 雅可比方程式也可以描述波動光學裏遵守惠金斯原理的光波，只要將光線的等傳播時間曲面改為光波的波前。薛丁格尋思，古典力學與量子力學之間的關係，就如同幾何光學與波動光學之間的關係；哈密頓 - 雅可比方程式應該對應於量子力學的波動方程式在某種極限的案例，而這極限應該也是物質波波長超短於粒子軌道曲率半徑的極限（或按照對應原理，普朗克常數趨於 0 的極限）；按照先前哈密頓類比的模式，依樣畫葫蘆，應該可以找到正確形式的波動方程式。這想法很正確，經過一番努力，他成功地推導出薛丁格方程式。
假設一個粒子移動於顯不含時位勢 $V(\mathbf {r} )$ ，它的哈密頓 - 雅可比方程式為
: ${\frac {1}{2m}}\left({\boldsymbol {\nabla }}S\right)^{2}+V+{\frac {\partial S}{\partial t}}=0$
其中， $S(\mathbf {r} ,{\boldsymbol {a}};t)$ 是哈密頓主函數， ${\boldsymbol {a}}$ 是運動常數向量。
由於位勢顯性不含時，哈密頓主函數可以分離成兩部分： | [
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不含時薛丁格方程式 | : $S=W(\mathbf {r} ,{\boldsymbol {a}})-Et$
其中，顯性不含時的函數 $W(\mathbf {r} ,{\boldsymbol {a}})$ 是哈密頓特徵函數， $E$ 是能量。
將哈密頓主函數公式代入粒子的哈密頓 - 雅可比方程式，稍加運算，可以得到
: $|{\boldsymbol {\nabla }}S|={\sqrt {2m(E-V)}}$
哈密頓主函數對於時間的全導數是
: ${\frac {dS}{dt}}={\frac {\partial S}{\partial t}}+\nabla S\cdot {\frac {d\mathbf {r} }{dt}}$
哈密頓主函數 $S$ 的常數等值曲面 $\sigma _{0}$ 在空間移動的方程式為
: $0={\frac {\partial S}{\partial t}}+\nabla S\cdot {\frac {d\mathbf {r} }{dt}}=-E+\nabla S\cdot {\frac {d\mathbf {r} }{dt}}$
所以，在設定等值曲面的正負面之後， $\sigma _{0}$ 朝著法線方向移動的速度 $u$ 是
: $u={\frac {dr}{dt}}={\frac {E}{|\nabla S|}}={\frac {E}{\sqrt {2m(E-V)}}}$
這速度 $u$ 是相速度，而不是粒子的移動速度 $v$ ：
: $v={\frac {|{\boldsymbol {\nabla }}S|}{m}}={\sqrt {\frac {2(E-V)}{m}}}$
試想 $\sigma _{0}$ 為一個相位曲面。既然粒子具有波粒二象性，假設粒子的波函數所擁有的相位與 $S$ 成正比：
: $\Psi (\mathbf {r} ,t)=A(\mathbf {r} )e^{iS/\kappa }$
其中， $\kappa $ 是常數， $A(\mathbf {r} )$ 是參數為位置的係數函數。
將哈密頓主函數的公式代入 $\Psi (\mathbf {r} ,t)$ 波函數，
: $\Psi (\mathbf {r} ,t)=A(\mathbf {r} )e^{i(W-Et)/\kappa }$ | [
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不含時薛丁格方程式 | 注意到 $E/\kappa $ 的因次必須是頻率，薛丁格靈機一動，想到愛因斯坦的光電效應理論 $E=\hbar \omega $ ；其中， $\hbar $ 是約化普朗克常數， $\omega $ 是角頻率。他嘗試設定 $\kappa =\hbar $ ，粒子的波函數 $\Psi $ 變為
: $\Psi (\mathbf {r} ,t)=A(\mathbf {r} )e^{i(W-Et)/\hbar }=\psi (\mathbf {r} )e^{-iEt/\hbar }$
其中， $\psi (\mathbf {r} )=A(\mathbf {r} )e^{iW(\mathbf {r} )/\hbar }$
$\Psi (\mathbf {r} ,t)$ 的波動方程式為
: $\nabla ^{2}\Psi -{\frac {1}{u^{2}}}{\frac {\partial ^{2}\Psi }{\partial t^{2}}}=0$
將 $\Psi (\mathbf {r} ,t)$ 波函數代入波動方程式，經過一番運算，可以得到
: $\nabla ^{2}\Psi +{\frac {E^{2}}{\hbar ^{2}u^{2}}}\Psi =\nabla ^{2}\Psi +{\frac {2m(E-V)}{\hbar ^{2}}}\Psi =0$
注意到 $E\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}$ 。稍加編排，即可推導出含時薛丁格方程式：
: $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t)+V\Psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}$
### 重要性質
#### 歸一性
在量子力學裏，所有事件發生的機率，其總和等於 1，這特性稱為歸一性，以方程式表示為
: $\int _{-\infty }^{\infty }\ \Psi ^{*}(x,t)\Psi (x,t)\ \mathrm {d} x=1$ | [
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不含時薛丁格方程式 | 為了滿足這特性，必須將波函數歸一化。薛丁格方程式能夠自動地維持波函數的歸一性。假若，某波函數 $\Phi (x,t)$ 尚未被歸一化。由於薛丁格方程式為線性方程式， $\Phi (x,t)$ 與任何常數的乘積還是這個薛丁格方程式的波函數。設定 $\phi (x)=A\Phi (x,0)$ ；其中， $A$ 是歸一常數，使得
: $\int _{-\infty }^{\infty }\ \phi ^{*}(x)\phi (x)\ \mathrm {d} x=1$
這樣，新波函數 $\Phi _{A}(x,t)=A\Phi (x,t)$ 還是這個薛丁格方程式的解答，而且， $\Phi _{A}(x,0)$ 已經被歸一化了。在這裏，特別注意到歸一性方程式的波函數 $\Psi (x,t)$ 含時間，而對於位置的積分仍舊可能含時間。在某個時間的歸一化，並不保證隨著時間的流易，波函數仍舊保持歸一化。薛丁格方程式有一個優良性質：它可以自動地保持波函數的歸一化。這樣，量子系統永遠地滿足歸一性。所以，薛丁格方程式能夠自動地維持波函數的歸一性。
##### 證明
總機率對於時間的導數為
: ${\frac {\mathrm {d} }{\mathrm {d} t}}\int _{-\infty }^{\infty }\ \Psi ^{*}(x,t)\Psi (x,t)\ \mathrm {d} x=\int _{-\infty }^{\infty }\ ({\frac {\partial \Psi ^{*}}{\partial t}}\Psi +\Psi ^{*}{\frac {\partial \Psi }{\partial t}}\ )\mathrm {d} x$
思考含時薛丁格方程式，
: $-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V(x)\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}$
其複共軛是
: $-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi ^{*}}{\partial x^{2}}}+V(x)\Psi ^{*}=-i\hbar {\frac {\partial \Psi ^{*}}{\partial t}}$ | [
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] |
不含時薛丁格方程式 | 將這兩個方程式相減，可以得到
: ${\begin{aligned}{\frac {\partial \Psi ^{*}}{\partial t}}\Psi +\Psi ^{*}{\frac {\partial \Psi }{\partial t}}&=-{\frac {i\hbar }{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}\Psi ^{*}\right)\Psi +{\frac {i}{\hbar }}V\Psi ^{*}\Psi -\Psi ^{*}{\frac {i}{\hbar }}V\Psi +\Psi ^{*}{\frac {i\hbar }{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}\Psi \right)\\&=-{\frac {i\hbar }{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}\Psi ^{*}\right)\Psi +\Psi ^{*}{\frac {i\hbar }{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}\Psi \right)\\&={\frac {i\hbar }{2m}}{\frac {\partial }{\partial x}}\left(\Psi ^{*}{\frac {\partial }{\partial x}}\Psi -\Psi {\frac {\partial }{\partial x}}\Psi ^{*}\right)\\\end{aligned}}$
所以，總機率對於時間的導數為 | [
0.1049005538225174,
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不含時薛丁格方程式 | : ${\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{-\infty }^{\infty }\ \Psi ^{*}(x,t)\Psi (x,t)\ \mathrm {d} x&=\int _{-\infty }^{\infty }\ {\frac {i\hbar }{2m}}{\frac {\partial }{\partial x}}\left(\Psi ^{*}{\frac {\partial }{\partial x}}\Psi -\Psi {\frac {\partial }{\partial x}}\Psi ^{*}\right)\ \mathrm {d} x\\&={\frac {i\hbar }{2m}}\left.\left(\Psi ^{*}{\frac {\partial }{\partial x}}\Psi -\Psi {\frac {\partial }{\partial x}}\Psi ^{*}\right)\right|_{-\infty }^{\infty }\\\end{aligned}}$
在無窮遠的極限，符合實際物理的波函數必須等於零：
: ${\frac {\mathrm {d} }{\mathrm {d} t}}\int _{-\infty }^{\infty }\ \Psi ^{*}(x,t)\Psi (x,t)\ \mathrm {d} x=0$
因此，薛丁格方程式會維持波函數的歸一化性質，這性質不會隨著時間的流易而改變。
#### 線性方程式
薛丁格方程式是一個線性方程式。滿足薛丁格方程式的波函數擁有線性關係。假設波函數 $\Psi _{A}$ 與 $\Psi _{B}$ 是薛丁格方程式的解，則任意線性組合 $\Psi $ 也是薛丁格方程式的解：
: $\Psi =a\Psi _{A}+b\Psi _{B}$
其中， $a$ 與 $b$ 是常數。
這線性組合可以延伸至任意多個波函數。因此，波函數的疊加也是同樣薛丁格方程式的解。這種疊加性質是量子力學最為奧妙的性質之一。量子系統可以同時處於兩個以上的古典狀態；一個粒子可以同時出現在幾個不同位置，可以同時擁有不同的能量。
##### 證明
根據含時薛丁格方程式， | [
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不含時薛丁格方程式 | : $-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi _{A}+V\Psi _{A}=i\hbar {\partial \over \partial t}\Psi _{A}$
: $-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi _{B}+V\Psi _{B}=i\hbar {\partial \over \partial t}\Psi _{B}$
因此，這兩個解的線性組合 $\Psi =a\Psi _{A}+b\Psi _{B}$ 為
: ${\begin{aligned}i\hbar {\frac {\partial }{\partial t}}\Psi &=i\hbar {\frac {\partial }{\partial t}}(a\Psi _{A}+b\Psi _{B})\\&=i\hbar {\frac {\partial }{\partial t}}(a\Psi _{A})+i\hbar {\frac {\partial }{\partial t}}(b\Psi _{B})\\&=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}(a\Psi _{A})+V(a\Psi _{A})\right]+\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}(b\Psi _{B})+V(b\Psi _{B})\right]\\&=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}(a\Psi _{A}+b\Psi _{B})+V(a\Psi _{A}+b\Psi _{B})\\&=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi +V\Psi \\\end{aligned}}$ | [
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不含時薛丁格方程式 | 所以， $\Psi $ 也是這含時薛丁格方程式的解，這證明了含時薛丁格方程式是一個線性方程式。類似地，也可以證明不含時薛丁格方程式是一個線性方程式。
## 不含時薛丁格方程式導引
不含時薛丁格方程式與時間無關，又稱為「能量本徵薛丁格方程式」或「定態薛丁格方程式」，可以用來計算粒子的本徵能量和其它相關的量子性質。應用分離變數法，猜想 $\Psi (x,t)$ 的形式為
: $\Psi (x,t)=\psi _{E}(x)e^{-iEt/\hbar }$ ；
其中， $E$ 是分離常數，稍後，會推論出 $E$ 就是能量， $\psi _{E}(x)$ 是對應於 $E$ 的函數。
將這猜想解代入含時薛丁格方程式，經過一番運算，可以推導出一維不含時薛丁格方程式
: $-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi _{E}(x)+V(x)\psi _{E}(x)=E\psi _{E}(x)$
類似地，可以推導出三維不含時薛丁格方程式
: $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{E}(\mathbf {r} )+V(\mathbf {r} )\psi _{E}(\mathbf {r} )=E\psi _{E}(\mathbf {r} )$
### 重要性質
#### 定態
波函數 $\Psi (x,t)=\psi _{E}(x)e^{-iEt/\hbar }$ 所代表的量子態稱為定態，雖然波函數本身與時間有關，機率密度 $P(x)=\Psi ^{*}(x,t)\Psi (x,t)=|\psi _{E}(x)|^{2}$ 只與位置有關。由於能量 $E$ 是個常數，定態所有與時間無關的可觀察量 $O$ 的期望值都是常數：
: $\langle O\rangle =\int _{-\infty }^{\infty }\ \Psi ^{*}(x,t){\hat {O}}\Psi (x,t)\ \mathrm {d} x=\int _{-\infty }^{\infty }\ \psi _{E}^{*}(x){\hat {O}}\psi _{E}(x)\ \mathrm {d} x$ | [
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不含時薛丁格方程式 | 波函數 $\Psi (x,t)$ 的相位因子 $e^{-iEt/\hbar }$ 在計算過程中會自動刪除，因此可以忽略此相位因子，而改使用不含時波函數 $\psi _{E}(x)$ 來指稱定態。處於定態的系統永遠是固定不變的。
#### 明確能量
在古典力學裏，哈密頓量 $H$ 是系統的總能量：
: $H={\frac {p^{2}}{2m}}+V(\mathbf {r} )$
在量子力學裏，對應的哈密頓算符 ${\hat {H}}$ 的形式為
: ${\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )$
其本徵函數為 $\psi _{E}(x)$ ，本徵值為 $E$ ，是系統的總能量：
: ${\hat {H}}\psi =E\psi $
${\hat {H}}$ 、 ${\hat {H}}^{2}$ 的期望值為
: $\langle {\hat {H}}\rangle =E$
: $\langle {\hat {H}}^{2}\rangle =E^{2}$
因此，對於定態系統多次重複測量哈密頓量，所得到數據的標準差為 0，換句話說，每次測量都會得到同樣的答案 $E$ 。
#### 線性組合
不含時薛丁格方程式有無窮多個本徵函數解 $\psi _{n}(x)$ ，每一個解對應一個能量本徵值 $E_{n}$ ：
: ${\hat {H}}\psi _{n}=E_{n}\psi _{n}$
含時薛丁格方程式的一般解是這些解的線性組合：
: $\Psi =\sum _{n}c_{n}\psi _{n}e^{-iE_{n}t/\hbar }$
其中， $c_{n}$ 是權重係數。
為了滿足歸一性，
: $\sum _{n}|c_{n}|^{2}=1$
這線性組合與時間有關，對應的機率密度與各種期望值都與時間有關。
## 物理意義
薛丁格方程式與其解在物理學領域造成思維方面的突破性發展。薛丁格方程式是一種嶄新的方程式，關於它的解析引導出很多不尋常、出乎意料之中的結果。
### 統計詮釋
在古典力學裏，運動於空間的粒子在任何時刻，都具有確定的位置與動量。這些物理量按照牛頓運動定律進行決定性的演化。在量子力學裏，粒子並不具有確定的位置與動量，對於這些物理量進行測量，會得到遵守粒子運動的機率分佈的隨機結果。 | [
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不含時薛丁格方程式 | 從含時薛丁格方程式可以計算出粒子的波函數。按照廣義統計詮釋，由波函數 $\Psi (x,t)$ ，可以計算出粒子運動的機率分佈 $P(x,t)$ ：
: $P(x,t)=\Psi ^{*}(x,t)\Psi (x,t)$
因此，可以預測在某時刻，粒子處於某區域的機率。薛丁格方程式描述粒子的波函數怎樣隨著時間流易而產生決定性演化。儘管可以計算出波函數的完整形式，也可以計算出粒子運動的機率分佈，但薛丁格方程式無法準確地預測粒子在哪個時刻會處於哪個區域。
從波動觀分析，薛丁格方程式乃是一個波動方程式，它完美地描述一個與時間、位置有關的量子波所發生的運動行為與所具有的量子性質，而解答這波動方程式的波函數可以詮釋為「在某時間、某位置發生交互作用的機率輻」。這寬鬆的詮釋方式可以適用於波動觀或粒子觀。
### 不確定性原理
描述粒子物理行為的薛丁格方程式是一種波動方程式，它的波函數解答是一種延伸於空間的物質波，具有波動性。在波動力學裏，做傅立葉分析可以得到一個重要結果，即假設波的波長越為明確，則波的位置越為不明確；反之亦然。物質波也遵守這結果，在量子力學裏，這結果蛻化為不確定性原理，即粒子的位置與動量不可同時被確定，位置的不確定性 $\Delta {x}$ 與動量的不確定性 $\Delta {p}$ 遵守不等式
: $\Delta {x}\Delta {p}\geq \hbar /2$
不確定性原理表明了量子測量的不確定性，這是量子系統內秉的性質。由此性質還可以推導出粒子的波動性。
### 量子測量
根據哥本哈根詮釋，粒子的運動遵守薛丁格方程式，直到因被測量而發生波函數塌縮為止。假設對於某系統的某可觀察量做測量，而描述這系統的波函數是由這可觀察量的幾個本徵函數量子疊加而成，每次對於這可觀察量做測量只能得到本徵函數的本徵值，不能得到任何其它數值。當波函數塌縮現象發生時，由於粒子與測量儀器彼此交互作用，系統的波函數會按照機率分佈隨機的約化為原本幾個本徵函數中的單獨一個本徵函數。這是量子測量的關鍵要素，將波函數與可觀察量，如位置或動量，關聯在一起。 | [
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不含時薛丁格方程式 | 量子系統隨著時間流易而演化的兩個過程為薛丁格方程式預測的演化、波函數塌縮。有些教科書會將這兩種過程分別當作量子力學的假設，然後從假設推導出量子力學的其他理論結果。很多物理學者認為，從薛丁格方程式無法推導出波函數塌縮。這兩種過程具有迥然不同的性質。薛丁格方程式預測的演化具有決定性，能夠從最初波函數預測未來的最終波函數；它還具有逆反性，能夠將時間逆反地從最終態演化回最初態。波函數塌縮具有非決定性，從最初態按照機率分佈隨機地約化至最終態，無法預測這最終態到底是甚麼；它還具有非逆反性，測量動作將量子態的資訊發掘出來，這是一種無法時間逆反的程序，獲得的額外資訊無法再還原。
### 量子穿隧效應
在古典力學裏，當一個圓球慢慢地滾上一座高山，假若它沒有足夠能量翻過山頂到另一邊，它會停止滾動，往反方向滾回。但是，薛丁格方程式預測，這圓球跑到另一邊的機率大於零，儘管它的能量不足以爬到山頂，這種波動性行為稱為量子穿隧效應，無法用微粒說來解釋這種效應。特別是對於微觀粒子與適當形狀的勢壘，做實驗很容易就可觀察到這種效應。阿爾法衰變 就是因為阿爾法粒子擺脫了本來不可能擺脫的強作用力束縛而從原子核逃逸出來的現象。
### 粒子的波動性
非相對論性薛丁格方程式是波動方程式。遵守這方程式進行運動的粒子因此會顯示出波動性行為。雙縫實驗是一個範例，它能夠展示出粒子通常不會進行的波動行為。從兩條狹縫傳播出來的物質波在某些位置會相長干涉，在某些位置又會破壞性干涉，因此形成複雜的干涉圖樣。直覺而言，假設，從發射源到探測屏，每次只會出現單獨一個粒子，即每次只有一個粒子獨自通過兩條狹縫，按照微粒說，累積多次發射不應該形成干涉圖樣。但是，做實驗可以實際觀察到這干涉圖樣，如同右圖從真正實驗獲得的圖樣所展示。這意味著，雖然每次只有一個粒子通過狹縫，這粒子可以同時通過兩條狹縫，自己與自己互相干涉。光子、電子、中子、原子、甚至分子，都可以表現出這種奇異的量子行為。
## 相對論性薛丁格方程式
薛丁格方程式並沒有涉及到相對論效應。對於伽利略變換，薛丁格方程式的形式不變。 對於勞侖茲變換，薛丁格方程式的形式會改變。為了要涵蓋相對論效應，必須將薛丁格方程式加以延伸。試想能量 - 動量關係式，
: $E^{2}=p^{2}c^{2}+m^{2}c^{4}$
其中， $c$ 是光速， $m$ 是靜止質量。
將這關係式內的能量與動量改為其對應的算符，將整個關係式作用於波函數，可以得到 | [
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不含時薛丁格方程式 | : $-\hbar ^{2}{\frac {\partial ^{2}}{\partial t^{2}}}\Psi =-\hbar ^{2}c^{2}\nabla ^{2}\Psi +m^{2}c^{4}\Psi $
稍加編排，可以得到克萊因 - 戈爾登方程式：
: $(\Box ^{2}+\mu ^{2})\psi =0$
其中， $\Box ^{2}={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}$ 是達朗貝爾算符， $\mu ={\frac {mc}{\hbar }}$
對於勞侖茲變換，這方程式的形式不會改變，是個勞侖茲不變式。但是，它是時間的二階微分方程式，玻恩的統計詮釋不適用於它的解。它不適用於自旋 1/2 粒子，只適用於零自旋粒子。另外，這方程式的解擁有正頻率和負頻率。平面波波函數解的色散關係式（dispersion relation）為
: $\hbar ^{2}\omega ^{2}-\hbar ^{2}c^{2}k^{2}=m^{2}c^{4}$
其中， $\omega $ 是角頻率，可以是正值或負值。
對量子力學來說，正負角頻率或正負能量，是一個很嚴峻的問題，因為無法從底端來限制能量的最低值。雖然如此，加以適當的詮釋，這方程式仍舊能夠正確地給出零自旋粒子的相對論性波函數。
將克萊因 - 戈爾登方程式作因式分解，從所得到的兩個因子算符中的一個，可以得到整個狄拉克方程式：
: $i\hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}=\left({\frac {1}{i}}{\boldsymbol {\alpha \cdot \nabla }}+\beta m\right)\Psi (\mathbf {r} ,t)$
其中， $m$ 是自旋 -½ 粒子的質量， $\mathbf {r} $ 、 $t$ 分別是空間位置、時間， $\beta ={\begin{pmatrix}I&0\\0&-I\end{pmatrix}}$ 、 $\alpha _{i}={\begin{pmatrix}0&\sigma _{i}\\\sigma _{i}&0\end{pmatrix}}$ 是係數矩陣， $I$ 是 2×2 單位矩陣， $\sigma _{i}$ 是包立矩陣。 | [
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不含時薛丁格方程式 | 狄拉克方程式乃是時間的一階微分方程式，適用於自旋 -½ 粒子。它的解稱為旋量，擁有四個分量，因此有四個線性獨立的解，其中兩個對應於粒子，另外兩個對應於反粒子。
## 解析方法
一般來說，解析薛丁格方程式會用到下述這些方法：
* 量子微擾理論
* 變分原理
* 量子蒙特卡羅方法
* 密度泛函理論
* WKB 近似與半古典擴展
對於某些特殊的狀況，可以使用特別方法：
* 有分析解的量子力學系統列表
* 哈特里 - 福克方法與後哈特里 - 福克方法。
* 離散 Delta 位勢阱方法
## 範例
### 自由粒子
當位勢為零時，薛丁格方程式為
: $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\,\Psi (\mathbf {r} ,t)=i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)$
這薛丁格方程式有一個平面波解：
: $\Psi (\mathbf {r} ,t)=e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}$
其中， $\mathbf {k} $ 是波向量， $\omega $ 是角頻率。
將這平面波解代入薛丁格方程式，可以得到色散關係式
: ${\frac {\hbar ^{2}k^{2}}{2m}}=\hbar \omega $
由於粒子存在的機率等於 1 ，波函數 $\Psi (\mathbf {r} ,t)$ 必須歸一化，才能夠表達出正確的物理內涵。對於一般的自由粒子而言，這不是問題，因為，自由粒子的波函數，在位置空間或動量空間都是局部性的，只有在某些局部區域才呈有限值，在其它區域的數值都很微小，可以被忽略。
在量子力學裏，一個自由粒子的動量與能量不需要呈特定的數值，自由粒子的波函數以波包形式來表示：
: $\Psi (\mathbf {r} ,t)={\frac {1}{(2\pi )^{3/2}}}\int _{\mathbb {K} }A(\mathbf {k} )e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}\,\mathrm {d} \mathbf {k} $
其中，積分區域 $\mathbb {K} $ 是 $\mathbf {k} $ - 空間。
為了方便計算，只思考一維空間， | [
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不含時薛丁格方程式 | : $\Psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }A(k)~e^{i(kx-\omega (k)t)}\,\mathrm {d} k$
其中，振幅 $A(k)$ 是線性疊加的係數函數。
從在時間 $t=0$ 的波函數 $\Psi (x,0)$ ，可以得到係數函數：
: $A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }\Psi (x,0)~e^{-ikx}\,\mathrm {d} x$
已知在時間 $t=0$ 的波函數 $\Psi (x,0)$ ，通過傅立葉變換，可以推導出在任何時間的波函數 $\Psi (x,t)$ 。
### 一維諧振子
在一維諧振子問題裏，質量為 $m$ 的粒子移動於位勢 $V(x)={\frac {1}{2}}m\omega ^{2}x^{2}$ ，此粒子的哈密頓算符 ${\hat {H}}$ 為
: ${\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}$
每一個能級所對應的能量本徵態必需滿足由這哈密頓算符所形成的薛丁格方程式 ：
: ${\hat {H}}\psi _{n}=E_{n}\psi _{n}$
採用位置表現，解析這個微分方程式，使用冪級數方法。可以得到一族的解：
: $\psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}e^{\left(-{\frac {m\omega x^{2}}{2\hbar }}\right)}\cdot {\mathfrak {H}}_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right)$
: $n=0,1,2,\ldots $
其中，函數 ${\mathfrak {H}}_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}$ 為埃爾米特多項式。 | [
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不含時薛丁格方程式 | 對應於函數 ${\mathfrak {H}}_{n}$ 的能級為
: $E_{n}=\hbar \omega \left(n+{1 \over 2}\right)$
一維諧振子的能譜有以下性質：
* 能量被量子化，只能呈離散數值，即 $\hbar \omega $ 乘以 1/2, 3/2, 5/2…… 等等。這是許多種量子力學系統的特徵。
* 最低能量（當 _n_ = 0）不為零，而是 $\hbar \omega /2$ ，被稱為「基態能量」或零點能量。在基態中，根據量子力學，一振子執行所謂的「零振動」，且其平均動能是正值。這樣的現象意義重大但並不那麼顯而易見，因為通常能量的零點並非一個有意義的物理量，因為可以任意選擇；有意義的是能量差。雖然如此，基態能量有許多的意涵，特別是在量子引力學裏。
* 能級是等距的，諧振子問題的能譜與波耳模型或盒中粒子問題不同。
### 球對稱位勢
假設單獨粒子移動於球對稱位勢 ，描述這量子系統運動的薛丁格方程式為
: $-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\psi +V(r)\psi =E\psi $
其中， $\mu $ 是粒子的質量， $\psi $ 是粒子的波函數， $V(r)$ 是位勢， $r$ 是徑向距離。
採用球坐標 $(r,\,\theta ,\,\phi )$ ，將拉普拉斯算子 $\nabla ^{2}$ 展開：
: $-{\frac {\hbar ^{2}}{2\mu r^{2}}}\left\{{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial }{\partial r}}\right)+{\frac {1}{\sin ^{2}\theta }}\left[\sin \theta {\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)+{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right]\right\}\psi +V(r)\psi =E\psi $
滿足薛丁格方程式的本徵函數 $\psi $ 的形式為： | [
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不含時薛丁格方程式 | : $\psi (r,\,\theta ,\,\phi )=R(r)\Theta (\theta )\Phi (\phi )$
其中， $R(r)$ ， $\Theta (\theta )$ ， $\Phi (\phi )$ ，都是函數。 $\Theta (\theta )$ 與 $\Phi (\phi )$ 時常會合併為一個函數 $Y_{lm}(\theta ,\,\phi )=\Theta (\theta )\Phi (\phi )$ ，稱為球諧函數。這樣，本徵函數 $\psi $ 的形式變為：
: $\psi (r,\,\theta ,\,\phi )=R(r)Y_{lm}(\theta ,\,\phi )$
#### 角部分解答
參數為天頂角 $\theta $ 、方位角 $\phi $ 的球諧函數 $Y_{lm}$ ，滿足角部分方程式
: $-{\frac {1}{\sin ^{2}\theta }}\left[\sin \theta {\frac {\partial }{\partial \theta }}{\Big (}\sin \theta {\frac {\partial }{\partial \theta }}{\Big )}+{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right]Y_{lm}(\theta ,\phi )=l(l+1)Y_{lm}(\theta ,\phi )$
其中，非負整數 $l$ 、 $m$ 分別是角量子數、磁量子數。
磁量子數遵守關係式 $-l\leq m\leq l$ 。不同的 $l$ 與 $m$ 對應於不同的球諧函數解答 $Y_{lm}$ ：
: $Y_{lm}(\theta ,\,\phi )=(i)^{m+|m|}{\sqrt {{(2l+1) \over 4\pi }{(l-m)! \over (l+m)!}}}\,P_{lm}(\cos {\theta })\,e^{im\phi }$
其中， $i$ 是虛數單位， $P_{lm}(\cos {\theta })$ 是伴隨勒讓德多項式，以方程式表示為
: $P_{lm}(x)=(1-x^{2})^{|m|/2}\,{\frac {d^{|m|}}{dx^{|m|}}}P_{l}(x)$
而 $P_{l}(x)$ 是 $l$ 階勒讓德多項式，以羅德里格公式表示為 | [
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不含時薛丁格方程式 | : $P_{l}(x)={1 \over 2^{l}l!}{d^{l} \over dx^{l}}(x^{2}-1)^{l}$
#### 徑向部分解答
將角部分解答代入薛丁格方程式，則可得到一維二階微分方程式：
: $\left\{-{\hbar ^{2} \over 2\mu r^{2}}{d \over dr}\left(r^{2}{d \over dr}\right)+{\hbar ^{2}l(l+1) \over 2\mu r^{2}}+V(r)\right\}R(r)=ER(r)$
設定函數 $u(r)=rR(r)$ ，代入方程式，經過一番繁雜的運算，可以得到
: $-{\hbar ^{2} \over 2\mu }{d^{2}u(r) \over dr^{2}}+{\hbar ^{2}l(l+1) \over 2\mu r^{2}}u(r)+V(r)u(r)=Eu(r)$
徑向方程式變為
: $-{\hbar ^{2} \over 2\mu }{d^{2}u(r) \over dr^{2}}+V_{\mathrm {eff} }(r)u(r)=Eu(r)$
其中，有效位勢 $V_{\mathrm {eff} }(r)=V(r)+{\frac {\hbar ^{2}l(l+1)}{2\mu r^{2}}}$
這正是函數為 $u(r)$ ，有效位勢為 $V_{\mathrm {eff} }$ 的薛丁格方程式。徑向距離 $r$ 的定義域是從 $0$ 到 $\infty $ 。新加入有效位勢的項目，稱為離心位勢。為了要更進一步解析，必須知道位勢的形式。不同的位勢有不同的解答。
## 參見
* 量子數
* 類氫原子
* 薛丁格繪景
* 薛丁格貓
* 薛丁格 - 牛頓方程式
## 註釋
## 參考文獻
## 外部連結
* 理查・費曼應用路徑積分方法推導出薛丁格方程式：Feynman's derivation of the Schrödinger equation。（英文）
* Hazewinkel, Michiel (編), Schrödinger equation, 数学百科全书，Springer, 2001, ISBN 978-1-55608-010-4 （英文）
* 量子物理學 （頁面存檔備份，存於網際網路檔案館） — 教科書有處理與時間無關的薛丁格方程式（英文） | [
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不含時薛丁格方程式 | * 線性薛丁格方程式 （頁面存檔備份，存於網際網路檔案館） EqWorld: 數理方程式的世界。（英文）
* 非線性薛丁格方程式 （頁面存檔備份，存於網際網路檔案館） EqWorld: 數理方程式的世界。（英文）
* 一維的薛丁格方程式 以及 書的目錄.（英文）
* 所有關於三維的薛丁格方程式 （頁面存檔備份，存於網際網路檔案館）（英文） | [
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量子疊加 | 在量子力學裏，**態疊加原理**（superposition principle）表明，假若一個量子系統的量子態可以是幾種不同量子態中的任意一種，則它們的歸一化線性組合也可以是其量子態。稱這線性組合為「疊加態」。假設組成疊加態的幾種量子態相互正交，則這量子系統處於其中任意量子態的機率是對應權值的絕對值平方。
從數學表述，態疊加原理是薛丁格方程式的解所具有的性質。由於薛丁格方程式是個線性方程式，任意幾個解的線性組合也是解。這些形成線性組合（稱為「疊加態」）的解時常會被設定為相互正交（稱為「基底態」），例如氫原子的電子能級態；換句話說，這幾個基底態彼此之間不會出現重疊。這樣，對於疊加態測量任意可觀察量所得到的期望值，是對於每一個基底態測量同樣可觀察量所得到的期望值，乘以疊加態處於對應基底態的機率之後，所有乘積的總和。
更具體地說明，假設對於某量子系統測量可觀察量 $A$ ，而可觀察量 $A$ 的本徵態 $|a_{1}\rangle $ 、 $|a_{2}\rangle $ 分別擁有本徵值 $a_{1}$ 、 $a_{2}$ ，則根據薛丁格方程式的線性關係，疊加態 $|\psi \rangle =c_{1}|a_{1}\rangle +c_{2}|a_{2}\rangle $ 也可以是這量子系統的量子態；其中， $c_{1}$ 、 $c_{2}$ 分別為疊加態處於本徵態 $|a_{1}\rangle $ 、 $|a_{2}\rangle $ 的機率幅。假設對這疊加態系統測量可觀察量 $A$ ，則測量獲得數值是 $a_{1}$ 或 $a_{2}$ 的機率分別為 $|c_{1}|^{2}$ 、 $|c_{2}|^{2}$ ，期望值為 $\langle \psi |A|\psi \rangle =|c_{1}|^{2}a_{1}+|c_{2}|^{2}a_{2}$ 。
舉一個可直接觀察到量子疊加的實例，在雙縫實驗裏，可以觀察到通過兩條狹縫的光子相互干涉，造成了顯示於偵測屏障的明亮條紋和黑暗條紋，這就是雙縫實驗著名的干涉圖樣。
再舉一個案例，在量子運算裏，量子位元是的兩個基底態 $|0\rangle $ 與 $|1\rangle $ 的線性疊加。這兩個基底態 $|0\rangle $ 、 $|1\rangle $ 的本徵值分別為 $0$ 、 $1$ 。
## 理論 | [
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量子疊加 | 在數學裏，疊加原理表明，線性方程式的任意幾個解所組成的線性組合也是這方程式的解。由於薛丁格方程式是線性方程式，疊加原理也適用於量子力學，在量子力學裏稱為態疊加原理。假設某量子系統的量子態可以是 $|f_{1}\rangle $ 或 $|f_{2}\rangle $ ，這些量子態都滿足描述這量子系統物理行為的薛丁格方程式。則這量子系的量子態也可以是它們的線性組合 $|f\rangle =c_{1}|f_{1}\rangle +c_{2}|f_{2}\rangle $ ，也滿足同樣的薛丁格方程式；其中， $c_{1}$ 、 $c_{2}$ 是複值係數，為了歸一化 $|f\rangle $ ，必須讓 $|c_{1}|^{2}+|c_{2}|^{2}=1$ 。
假設 $\theta $ 為實數，則雖然 $e^{i\theta }|f_{2}\rangle $ 與 $|f_{2}\rangle $ 標記同樣的量子態，他們並無法相互替換。例如， $|f_{1}\rangle +|f_{2}\rangle $ 、 $|f_{1}\rangle +e^{i\theta }|f_{2}\rangle $ 分別標記兩種不同的量子態。但是， $|f_{1}\rangle +|f_{2}\rangle $ 和 $e^{i\theta }(|f_{1}\rangle +|f_{2}\rangle )$ 都標記同一個量子態。因此可以這樣說，整體的相位因子並不具有物理意義，但相對的相位因子具有重要的物理意義。這種相位因子固定不變的量子疊加稱為「相干量子疊加」。
### 電子自旋範例
設想自旋為 $1/2$ 的電子，它擁有兩種相互正交的自旋本徵態，上旋態 $|\uparrow \rangle $ 與下旋態 $|\downarrow \rangle $ ，它們的量子疊加可以用來表示量子位元：
: $|\psi \rangle =c_{\uparrow }|\uparrow \rangle +c_{\downarrow }|\downarrow \rangle $ ；
其中， $c_{\uparrow }$ 、 $c_{\downarrow }$ 分別是複值係數，為了歸一化 $|\psi \rangle $ ，必須讓 $|c_{\uparrow }|^{2}+|c_{\downarrow }|^{2}=1$ 。 | [
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] |
量子疊加 | 這是最一般的量子態。係數 $c_{\uparrow }$ 、 $c_{\downarrow }$ 分別給定電子處於上旋態或下旋態的機率：
: $p_{\uparrow }=|c_{\uparrow }|^{2}$ 、
: $p_{\downarrow }=|c_{\downarrow }|^{2}$ 。
總機率應該等於 1： $p=p_{\uparrow }+p_{\downarrow }=|c_{\uparrow }|^{2}+|c_{\downarrow }|^{2}=1$ 。
這電子也可能處於這兩個量子態的疊加態：
: $|\psi \rangle ={3i \over 5}|\uparrow \rangle +{4 \over 5}|\downarrow \rangle $ 。
電子處於上旋態或下旋態的機率分別為
: $p_{\uparrow }=\left|\;{\frac {3i}{5}}\;\right|^{2}={\frac {9}{25}}$ 、
: $p_{\downarrow }=\left|\;{\frac {4}{5}}\;\right|^{2}={\frac {16}{25}}$ 。
再次注意到總機率應該等於 1：
: $p={\frac {9}{25}}+{\frac {16}{25}}=1$ 。
### 非相對論性自由粒子案例
描述一個非相對論性自由粒子的含時薛丁格方程式為
: $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\ \Psi (\mathbf {r} ,t)=i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)$ ；
其中， $\hbar $ 是約化普朗克常數， $\Psi (\mathbf {r} ,t)$ 是粒子的波函數， $\mathbf {r} $ 是粒子的位置， $t$ 是時間。
這薛丁格方程式有一個平面波解：
: $\Psi (\mathbf {r} ,t)=e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}$ ；
其中， $\mathbf {k} $ 是波向量， $\omega $ 是角頻率。
代入薛丁格方程式，這兩個變數必須遵守關係式 | [
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] |
量子疊加 | : ${\frac {\hbar ^{2}k^{2}}{2m}}=\hbar \omega $ 。
由於粒子存在的機率等於 1，波函數 $\Psi (\mathbf {r} ,t)$ 必須歸一化，才能夠表達出正確的物理意義。對於一般的自由粒子而言，這不是問題。因為，自由粒子的波函數，在位置或動量方面，都是局部性的。在量子力學裏，一個自由粒子的動量與能量不必須擁有特定的值。自由粒子的波函數可以表示為很多平面波的量子疊加：
: $\Psi (\mathbf {r} ,t)={\frac {1}{(2\pi )^{3/2}}}\int _{\mathbb {K} }A(\mathbf {k} )e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}\mathrm {d} \mathbf {k} $ ；
其中，積分區域 $\mathbb {K} $ 是 $\mathbf {k} $ - 空間。
為了方便計算，只思考一維空間，
: $\Psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }A(k)~e^{i(kx-\omega (k)t)}\ \mathrm {d} k$ ；
其中，振幅 $A(k)$ 是量子疊加的係數函數。
逆反過來，係數函數表示為
: $A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }\Psi (x,0)~e^{-ikx}\,\mathrm {d} x$ ；
其中， $\Psi (x,0)$ 是在時間 $t=0$ 的波函數。
所以，知道在時間 $t=0$ 的波函數 $\Psi (x,0)$ ，通過傅立葉變換，可以推導出在任何時間的波函數 $\Psi (x,t)$ 。
## 參見
* 疊加原理
* 波函數
* Delta 位勢阱
* Delta 位勢壘
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镜面反射_(物理) | **鏡面反射**或**規則反射**，是波來自表面的反射，例如來自鏡子表面的光。
**反射定律**指出，反射光的射線從反射面以與入射線對法線相同的角度從入射面上射出，但在由入射光線和反射光線形成平面內的法線另一側。這種行為最初由亞歷山大港的希羅（ADc.10–70）描述。
鏡面反射可與漫反射形成對比，後者的光線在一系列方向上從表面散射出去。
## 反射定律
當光遇到物資的邊界時，它會受到材料對電磁波的光學和電子回應函數的影響。由反射和折射組成的光學過程用邊界兩側的折射率差表示，而反射與吸收是由於材料的電子結構引起響應的實部和虛部。這些過程在傳輸中的參與程度是光的頻率或波長、偏振度和入射角的函數。一般而言，反射隨入射角的增加和邊界上吸收率的增加而減少。菲涅耳方程式描述了光學邊界處的物理。
反射可以是鏡面反射或類似鏡子的反射和漫反射。鏡面反射以相同的角度反射從給定方向到達的所有光，而漫反射則以廣泛的方向反射光。這種區別可以用塗有光澤油漆和啞光油漆的表面來說明。啞光塗料基本上表現出完全的漫反射，而光澤塗料顯示出更多的鏡面反射。由非吸收性粉末（如石膏）製成的表面可以是近乎完美的漫射器，而拋光的金屬物體可以非常有效地鏡面反射光線。鏡子的反射材料通常是鋁或銀。
光以電磁場的波前形式在空間傳播。射線的特徵是垂直於波前的方向（「波法線」）。當射線遇到表面時，波法線相對於表面法線的角度稱為入射角，由這兩個方向定義的平面稱為入射平面。入射光線的反射也發生在入射面上。
**反射定律**指出，光線的反射角等於入射角，入射方向、表面法線和反射方向為共面。
當光線垂直於表面照射時，它會直接反射回光源的方向。
反射現象源於平面波在平坦邊界上的繞射。當邊界尺寸遠大於波長時，邊界處的電磁場僅在鏡面方向上以同相位振盪。
### 向量公式
反射定律也可以用線性代數等效地表示。反射線的方向由入射向量和表面法線向量確定。給定從光源到表面的入射方向 $\mathbf {\hat {d}} _{\mathrm {i} }$ ，以及表面法線方向 $\mathbf {\hat {d}} _{\mathrm {n} }$ ，鏡面反射方向 $\mathbf {\hat {d}} _{\mathrm {s} }$ （三個向量都是單位向量）可表為： | [
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镜面反射_(物理) | : $\mathbf {\hat {d}} _{\mathrm {s} }=2\left(\mathbf {\hat {d}} _{\mathrm {n} }\cdot \mathbf {\hat {d}} _{\mathrm {i} }\right)\mathbf {\hat {d}} _{\mathrm {n} }-\mathbf {\hat {d}} _{\mathrm {i} },$
此處 $\mathbf {\hat {d}} _{\mathrm {n} }\cdot \mathbf {\hat {d}} _{\mathrm {i} }$ 是用向量內積得出的純量。不同的作者可能會以不同的符號定義入射和反射方向。假設這些歐基里德向量用列向量表示，則等式可以等效地表示為矩陣向量乘法：
: $\mathbf {\hat {d}} _{\mathrm {s} }=\mathbf {R} \;\mathbf {\hat {d}} _{\mathrm {i} },$
此處 $\mathbf {R} $ 是所謂的豪斯霍爾德變換矩陣，定義如下：
: $\mathbf {R} =\mathbf {I} -2\mathbf {\hat {d}} _{\mathrm {n} }\mathbf {\hat {d}} _{\mathrm {n} }^{\mathrm {T} };$
以單位矩陣 $\mathbf {I} $ 和 $\mathbf {\hat {d}} _{\mathrm {n} }$ 外積的兩倍來表示.
## 反射率
_反射率_是反射波的功率與入射波的功率之比。它是輻射波長的函數，與材料的折射率有關，可用菲涅耳方程式表示。在電磁波譜中，材料吸收顯著的波段，它通過複折射率的虛分量與電子吸收光譜相關。不透明材料的電子吸收光譜很難或不可能直接測量，因此可以通過克拉莫 - 克若尼關係式變換間接確定。反射光的偏振取決於入射探測光相對於材料中吸收躍遷偶極矩的排列對稱性。
鏡面反射的測量，是使用正常或可變入射反射分光光度計（「反射計」）使用掃描可變波長光源進行的。較低量級的量測可使用光澤計，以光澤單位量化表面的光澤外觀。
## 後果
### 內部反射
當光線在材料中傳播並與折射率較低材質的介面接觸時，會反射一些光線。如果入射角大於臨界角，則發生全內反射：所有光線都被反射。臨界角可以由下式給出： | [
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镜面反射_(物理) | : $\theta _{\text{crit}}=\arcsin \!\left({\frac {n_{2}}{n_{1}}}\right)\!.$
### 極化
當光線照射到兩種材料之間的介面時，反射光通常是部分偏振。但是，如果光線以布魯斯特角照射介面，反射光將「完全」平行於介面的線性偏振。布魯斯特角由下式給出：
: $\theta _{\mathrm {B} }=\arctan \!\left({\frac {n_{2}}{n_{1}}}\right)\!.$
### 反射的影像
平面鏡中的影像具有以下特徵：
* 它在鏡子後面的距離與物體在前面的距離相同。
* 它與物件的大小相同。
* 這是正立的影像（直立）。
* 它是相反的。
* 它是「虛像」，這意味著影像似乎在鏡子後面，不能投影到螢幕上。
平面鏡的影像反轉視情況而定。在許多情況下，鏡子中的影像似乎從左到右反轉。如果一面平面鏡子安裝在天花板上，則一個人站在它下面抬頭看它，它可能會看起來是「上下」反轉。類似的，對於前方車輛的駕駛員而言，後方「左」轉彎的車輛，在後視鏡中仍顯示為「左」轉。方向的反轉與否，取決於方向的定義。更具體地說，鏡子會改變座標系中的手性，座標系的一個軸看起來被反轉，並且圖像的手性可能會發生變化。例如，右腳鞋的圖像將看起來像左腳鞋。
## 例子
鏡面反射的一個典型例子是鏡子，它是專門為鏡面反射而設計的。
除了可見光之外，還可以在無線電波的電離層反射和飛行物體對無線電或微波雷達信號的反射中觀察到鏡面反射。X 射線反射率量測技術利用鏡面反射率，利用現代實驗室光源或同步加速器 X 射線，研究次奈米分辯率的薄膜和介面。
非電磁波也可以表現出鏡面反射，如反射聲音的聲鏡和反射中性原子的原子反射鏡。對於原子從固態鏡的有效反射，使用極冷原子和 / 或掠射，以提供顯著的量子反射；脊柱鏡用於增強原子的鏡面反射。中子反射法與 x 射線反射率類似，使用鏡面反射來研究材料表面和薄膜介面。
## 相關條目
* 幾何光學
* 哈密頓光學
* 反射係數
* 鏡射 (數學)
* 鏡面反射高光
* 鏡面反射率
## 註解
## 參考資料
* Hecht, Eugene. Optics 2nd. Addison Wesley. 1987. ISBN 0-201-11609-X. | [
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反物質 | **反物質**（英語：antimatter）在粒子物理學中是反粒子概念的延伸，反物質是由反粒子構成的，如同普通物質是由普通粒子所構成的。例如一顆反質子和一顆反電子（正電子）能形成一個反氫原子，如同電子和質子形成一般物質的氫原子。此外，物質與反物質的結合，會如同粒子與反粒子結合一般，導致兩者湮滅，且因而釋放出高能光子（伽瑪射線）或是其他能量較低的正反粒子對。正反物質湮滅所造成的粒子，賦予的動能等同於原始正反物質對的動能，加上原物質靜止質量與生成粒子靜質量的差，後者通常佔大部分。（愛因斯坦相對論指出，質量與能量是等價的。）
反物質無法在自然界找到，僅有稍縱即逝的少量存在（例如因放射衰變或宇宙射線等現象）。這是由於反物質若非存在於像物理實驗室的人工環境下，則無可避免地隨即與自然界的物質發生碰觸並湮滅。反粒子和一些穩定的反物質（例如反氫）可以人工製造出極少量，但卻基本不足以達到可對這些物質驗證其理論性的程度（但大量的正電子可由正子斷層照影等儀器產生)。
在科學或科幻方面，焦點環繞在為何所見的宇宙幾乎充滿了物質、是否有其他地方則是幾乎充滿了反物質以及是否能夠駕馭反物質，在現今可見的宇宙範圍中明顯的正反物質不對稱性成了物理的最大難題之一。許多可能的物理過程都是在探究重子時所發現。
## 歷史
1927 年 12 月，英國物理學家保羅・狄拉克提出了電子的相對論方程式，即狄拉克方程式。有趣的是，等式中發現除了一般正能量之外的負能量結果。這顯示出一個問題，當電子趨向於朝著最低可能的能階躍遷時；負無限大的能量是毫無意義的。但為了要彌補這條件，狄拉克提出真空狀態中是充滿了負能量電子的「海」，稱作狄拉克之海。任何真實的電子因此會填補這些海中具有正能量的部分。
衍伸這個想法，狄拉克發現海中的這些「洞」則具有正電荷。起初他認為這是質子，但赫爾曼・外爾指出這些洞應該是具有和電子相同的質量。1932 年由美國物理學家卡爾・安德森在實驗中證實了正電子的存在。在此期間，反物質有時也常被稱作「**反地物質**」。雖然狄拉克自己沒有使用反物質這個術語，但是後來的科學家將反質子等粒子稱呼為反物質。完整的反物質元素週期表由夏爾・讓內於 1929 年完成。
## 性質
反質子、反中子和反電子如果像質子、中子、電子那樣結合起來就形成了反原子。 | [
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反物質 | 反物質和物質一旦相遇，就相互吸引、碰撞並完全轉化為光並釋放出的巨大的能量，這個過程叫做**湮滅**。湮滅過程會釋放出正、反物質中蘊涵的所有靜質量能，根據愛因斯坦著名的質能關係式──E=mc²，一種在科學界受到普遍認同的理論認為，宇宙大爆炸早期曾產生了數量相當的物質和反物質，隨後發生的物質和反物質的湮滅消耗掉了絕大部分的正、反物質，遺留下的少部分正物質構成了現如今的物質世界。理論上宇宙大爆炸時所產生的粒子與反粒子應該數量相同，但是為什麼現今所遺留下來的絕大多數都是正粒子，這即所謂的「正反物質對稱性破壞」（對稱破缺），雖然在幾個粒子對撞試驗中，都發現了正粒子與反粒子的衰變略有不同，即所謂的電荷宇稱不守恆（CP 破壞），但在數量上仍不足以解釋為何現今反物質消失的問題，這在粒子物理學上仍是一大未解決的問題。
## 應用
因為物質與反物質的湮滅時質量可完全轉換成能量，帶來最大的能源效率，且單位產量是核能的千百倍或常規燃料的億兆倍。根據愛因斯坦的質能關係式 E=mc2。其中 E 為湮滅產生能量，m 為參與的正物質和反物質湮滅前總靜止質量，c 為光速≈3x108 米 / 秒。舉例來說，二分之一克反物質湮滅所產生的能量大約與廣島市原子彈爆炸所產生的能量相當（即是一克反物質湮滅所產生的能量約為 2-3 萬噸 TNT 當量，或者是大約 200 億千卡），所以一直有人研究其作為新能源的可行性，主要用於很難補給燃料的航太用，甚至作為反物質武器。但是由於目前人為製造反物質的方式，是由加速粒子打擊固定靶產生反粒子，再減速合成的。此過程所需要的能量遠大於湮滅作用所放出的能量，且生成反物質的速率極低，因此尚不具有經濟價值。此外，反物質與物質相遇會發生湮滅，保存上也是一大問題。
## 參閱
* 反粒子
* 量子場論
* 粒子物理學
## 參考資料
## 延伸閱讀
* G. Fraser. Antimatter, The Ultimate Mirror. Cambridge University Press. 2000. ISBN 978-0-521-65252-0.
## 外部連結
* Freeview Video 'Antimatter' by the Vega Science Trust and the BBC/OU（頁面存檔備份，存於網際網路檔案館）
* CERN Webcasts (RealPlayer required) | [
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反物質 | * What is Antimatter? (from the Frequently Asked Questions at the Center for Antimatter-Matter Studies)
* FAQ from CERN with lots of information about antimatter aimed at the general reader, posted in response to antimatter's fictional portrayal in Angels & Demons
* What is direct CP-violation?
* Animated illustration of antihydrogen production at CERN（頁面存檔備份，存於網際網路檔案館） from the Exploratorium. | [
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麥克斯韋-玻爾茲曼分佈 | **馬克士威 - 波茲曼分布**是一個描述一定溫度下微觀粒子運動速度的機率分布，在物理學和化學中有應用。最常見的應用是統計力學的領域。任何（宏觀）物理系統的溫度都是組成該系統的分子和原子的運動的結果。這些粒子有一個不同速度的範圍，而任何單個粒子的速度都因與其它粒子的碰撞而不斷變化。然而，對於大量粒子來說，處於一個特定的速度範圍的粒子所占的比例卻幾乎不變，如果系統處於或接近處於平衡。馬克士威 - 波茲曼分布具體說明了這個比例，對於任何速度範圍，作為系統的溫度的函數。它以詹姆斯・馬克士威和路德維希・波茲曼命名。
這個分布可以視為一個三維向量的大小，它的分量是獨立和常態分布的，其期望值為 0，標準差為 $a$ 。如果 $X_{i}$ 的分布為 $\ X\sim N(0,a^{2})$ ，那麼
: $Z={\sqrt {X_{1}^{2}+X_{2}^{2}+X_{3}^{2}}}$
就呈馬克士威 - 波茲曼分布，其參數為 $a$ 。
## 馬克士威 - 波茲曼分布的物理應用
馬克士威 - 波茲曼分布形成了分子運動論的基礎，它解釋了許多基本的氣體性質，包括壓力和擴散。馬克士威 - 波茲曼分布通常指氣體中分子的速率的分布，但它還可以指分子的速度、動量，以及動量的大小的分布，每一個都有不同的機率分布函數，而它們都是聯繫在一起的。
馬克士威 - 波茲曼分布可以用統計力學來推導（參見馬克士威 - 波茲曼統計）。它對應於由大量不交互作用的粒子所組成、以碰撞為主的系統中最有可能的速率分布，其中量子效應可以忽略。由於氣體中分子的交互作用一般都是相當小的，因此馬克士威 - 波茲曼分布提供了氣體狀態的非常好的近似。
在許多情況下（例如非彈性碰撞），這些條件不適用。例如，在電離層和空間電漿的物理學中，特別對電子而言，重組和碰撞激發（也就是輻射過程）是重要的。如果在這個情況下應用馬克士威 - 波茲曼分布，就會得到錯誤的結果。但如果系統在恆溫槽中且處於熱力學平衡，即使發生非彈性碰撞，其以熱的形式失去的動能仍然可由恆溫槽再以熱的形式補償回來，使得馬克士威 - 波茲曼分布依然適用。另外一個不適用馬克士威 - 波茲曼分布的情況，就是當氣體的量子熱波長與粒子之間的距離相比不夠小時，由於有顯著的量子效應也不能使用馬克士威 - 波茲曼分布。另外，由於它是基於非相對論的假設，因此馬克士威 - 波茲曼分布不能做出分子的速度大於光速的機率為零的預言。
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麥克斯韋-玻爾茲曼分佈 | 馬克士威最初的推導假設了三個方向上的表現都相同，但後來在波茲曼的一個推導中利用分子運動論去掉了這個假設。現在，馬克士威 - 波茲曼分布可以輕易地從能量的波茲曼分布推出：
: ${\frac {N_{i}}{N}}={\frac {g_{i}\exp \left(-E_{i}/kT\right)}{\sum _{j}^{}g_{j}\,{\exp \left(-E_{j}/kT\right)}}}\qquad \qquad (1)$
其中 _N_i 是平衡溫度 _T_ 時，處於狀態 _i_ 的粒子數目，具有能量 _E_i 和簡併度 _gi_ ，_N_ 是系統中的總粒子數目，_k_ 是波茲曼常數。（注意有時在上面的方程式中不寫出簡併度 _g_i。在這個情況下，指標 _i_ 將指定了一個單態，而不是具有相同能量 _E_i 的 _g_i 的多重態。）由於速度和速率與能量有關，因此方程式 1 可以用來推出氣體的溫度和分子的速度之間的關係。這個方程式中的分母稱為正則配分函數。
### 動量向量的分布
下列所述的推導，與詹姆斯・克拉克・馬克士威描述的推導和後來由路德維希・波茲曼描述的具有較少假設的推導都有很大不同。它與波茲曼在 1877 年的探討比較接近。
對於「理想氣體」（由基態的非交互作用原子所組成）的情況，所有能量都是動能的形式。宏觀粒子的動能與動量的關係為：
: $E={\frac {p^{2}}{2m}}\qquad \qquad (2)$
其中 _p_2 是動量向量 **p** = [_p_x, _p_y, _p_z] 的平方。因此，我們可以把方程式 1 寫成：
: ${\frac {N_{i}}{N}}={\frac {1}{Z}}\exp \left[-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right]\qquad \qquad (3)$
其中 _Z_ 是配分函數，對應於方程式 1 中的分母。在這裡，_m_ 是氣體的分子質量，_T_ 是熱力學溫度，_k_ 是波茲曼常數。這個 _N_**i**/_N_ 的分布與找到具有這些動量分量值的分子的機率密度函數 _f_**p** 成正比，因此： | [
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麥克斯韋-玻爾茲曼分佈 | : $f_{\mathbf {p} }(p_{x},p_{y},p_{z})={\frac {c}{Z}}\exp \left[-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right].\qquad \qquad (4)$
歸一化常數 _c_ 可以通過認識到分子具有_任何_動量的機率必須為 1 來決定。因此，方程式 4 在所有 _p_x、_p_y 和 _p_z 上的積分必須是 1。
可以證明：
: $c={\frac {Z}{(2\pi mkT)^{3/2}}}.\qquad \qquad (5)$
把方程式 5 代入方程式 4，得出：
: $f_{\mathbf {p} }(p_{x},p_{y},p_{z})=\left({\frac {1}{2\pi mkT}}\right)^{3/2}\exp \left[-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right].\qquad \qquad (6)$
可以看出，這個分布是三個獨立、呈常態分布的變量 $p_{x}$ 、 $p_{y}$ 和 $p_{z}$ 的乘積，其方差為 $mkT$ 。此外，可以看出動量的大小呈馬克士威 - 波茲曼分布，其中 $a={\sqrt {mkT}}$ 。
### 能量的分布
利用 _p_² = 2_mE_，以及動量的大小的分布函數（參見以下速率分布的章節），我們便得出能量的分布：
: $f_{E}\,dE=f_{p}\left({\frac {dp}{dE}}\right)\,dE=2{\sqrt {\frac {E}{\pi (kT)^{3}}}}~\exp \left[{\frac {-E}{kT}}\right]\,dE.\qquad \qquad (7)$
由於能量與三個呈常態分布的動量分量的平方和成正比，因此這個分布是具有三個自由度的卡方分布：
: $f_{E}(E)\,dE=\chi ^{2}(x;3)\,dx$
其中
: $x={\frac {2E}{kT}}.\,$
馬克士威 - 波茲曼分布還可以通過把氣體視為量子氣體來獲得。
### 速度向量的分布
認識到速度的機率密度函數 _f_**v** 與動量的機率密度函數成正比： | [
0.16464869678020477,
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麥克斯韋-玻爾茲曼分佈 | : $f_{\mathbf {v} }d^{3}v=f_{\mathbf {p} }\left({\frac {dp}{dv}}\right)^{3}d^{3}v$
並利用 **p** = m**v**，我們便得到：
: $f_{\mathbf {v} }(v_{x},v_{y},v_{z})=\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left[-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}\right],\qquad \qquad $
這就是馬克士威 - 波茲曼速度分布。在速度相空間（_v_x, _v_y, _v_z）的一塊無窮小區域 [_dv_x, _dv_y, _dv_z] 內找到具有特定速度 **v** = [_v_x, _v_y, _v_z] 的氣體分子的機率為
: $f_{\mathbf {v} }\left(v_{x},v_{y},v_{z}\right)\,dv_{x}\,dv_{y}\,dv_{z}.$
像動量一樣，這個分布是三個獨立、呈常態分布的變量 $v_{x}$ 、 $v_{y}$ 和 $v_{z}$ 的乘積，但方差為 ${\frac {kT}{m}}$ 。還可以看出，對於速度向量 [_v_x, _v_y, _v_z]，馬克士威 - 波茲曼速度分布是三個方向上的分布的乘積：
: $f_{v}\left(v_{x},v_{y},v_{z}\right)=f_{v}(v_{x})f_{v}(v_{y})f_{v}(v_{z})$
其中一個方向上的分布為：
: $f_{v}(v_{i})={\sqrt {\frac {m}{2\pi kT}}}\exp \left[{\frac {-mv_{i}^{2}}{2kT}}\right].\qquad \qquad $
這個分布具有常態分布的形式，其方差為 ${\frac {kT}{m}}$ 。正如所預料的，對於靜止的氣體，在任何方向上的平均速度都是零。
### 速率的分布
通常，我們更感興趣於分子的速率，而不是它們的速度分量。馬克士威 - 波茲曼速率分布為： | [
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