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玻恩定則（Born rule）是量子力學的基礎公設之一^[1]^[2]^:211ff，由物理學家馬克斯·玻恩提出，给出了量子测量得到特定結果的概率。它與海森堡測不準原理將概率的概念引入量子力學，使其展現出獨特的非決定性質。實驗中尚未發現違背玻恩定則的量子行為。^[3]

有物理學者試圖從其它假定推導出玻恩定则，但都未获学界认可。例如，有些物理學者^[谁？]聲稱從多世界詮釋可以推導出玻恩定律，但他們給出的導引被批評為循環論證^[4]。格里森定理为玻恩定則的形式給出了數學上的線索，但並沒有對於其概率行為提供物理学解释^[3]。

沃傑克·祖瑞克（英语：Wojciech Zurek）倚賴系統與環境之間的量子纏結引起的環輔不變性（英语：envariance）（環境輔助不變性，environment-assisted invariance）來推導出玻恩定則。在這裡，環境扮演了促使概率出現的關鍵角色。這也意味著祖瑞克的方法只適用於開放系統。不論如何，這是一種充滿想像力的點子，很具有未來發展的潛能。^[翻译腔]^[3]

概述

離散值譜

假設對於量子態為 $|\psi \rangle$ 的某量子系統測量可觀察量 $O$ ，而對應於可觀察量 $O$ 的厄米算符 ${\hat {O}}$ ，其值譜（英语：discrete spectrum (physics)）為離散的，所含有的本徵值與對應的本徵態分別標記為

O_{i}

與

|O_{i}\rangle

，

i=1,\ 2,\ 3,\ \cdots ,n

。

玻恩定則闡明，獲得測量值 $O_{i}$ 的概率為^[5]^:96-99^[3]

p(O_{i})=|\langle O_{i}|\psi \rangle |^{2}

。

連續值譜

假設對於量子態為 $|\psi \rangle$ 的某量子系統測量可觀察量 $X$ ，而對應於可觀察量 $X$ 的厄米算符 ${\hat {X}}$ ，其值譜為連續的，所含有的本徵值與對應的本徵態分別標記為

x

與

|x\rangle

，

-\infty \leq x\leq \infty

。

玻恩定則闡明，獲得測量值 $a\leq x\leq b$ 的概率為^[5]^:2-5

p(a\leq x\leq b)=\int _{a}^{b}|\langle x|\psi \rangle |^{2}\mathrm {d} x

。

歷史

馬克斯·玻恩最先於1926年發表論文提出玻恩定則^[6]。在這篇論文裏，玻恩解析了一個散射問題的薛定谔方程，由於受到阿爾伯特·愛因斯坦在光電效應研究的啟發^[7]，玻恩在一個角註裏總結，玻恩定則對於解答給出唯一可能的詮釋。1954年，因為「在量子力學領域的基礎研究，特別是對波函數的統計學詮釋」，玻恩與瓦爾特·博特共同榮獲諾貝爾物理學獎^[8]。約翰·馮·諾伊曼在他的1932年著作《量子力學的數學基礎》裏闡明譜理論（英语：spectrum theory）應用於玻恩定則的論述。^[9]

參閱

參考文獻

^ Zurek, Wojciech. Quantum Darwinism, Classical Reality, and the randomness of quantum jumps. Physics Today. 2014, 67 (10): 44–45.
^ Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë. Quantum Mechanics Volume 1. Hermann. ISBN 978-2-7056-8392-4.
^ ^3.0 ^3.1 ^3.2 ^3.3 Schlosshauer, Maximilian; Fine, Arthur. On Zurek’s Derivation of the Born Rule (PDF). Foundations of Physics. 2005, 35 (2): 197–213 [2015-07-09]. doi:10.1007/s10701-004-1941-6. （原始内容存档 (PDF)于2016-03-04）.
^ Landsman, N. P. The Born rule and its interpretation (PDF). Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (编). Compendium of Quantum Physics. Springer. 2008 [2024-04-18]. ISBN 978-3-540-70622-9. （原始内容存档 (PDF)于2024-06-09）. The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle
^ ^5.0 ^5.1 Griffiths, David J., Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, 2004, ISBN 0-13-111892-7
^ Zur Quantenmechanik der Stoßvorgänge, Max Born, Zeitschrift für Physik, 37, #12 (Dec. 1926), pp. 863–867 (German); English translation, On the quantum mechanics of collisions, in Quantum theory and measurement, section I.2, J. A. Wheeler and W. H. Zurek, eds., Princeton, New Jersey: Princeton University Press, 1983, ISBN 0-691-08316-9.
^ Born, Max. Born's Nobel Lecture on the statistical interpretation of quantum mechanics (PDF). Nobel Lecture. NobelPrize.org. 1954 [2015-07-09]. （原始内容存档 (PDF)于2006-05-12）. Again an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons.
^ The Nobel Prize in Physics 1954. NobelPrize.org. 1954 [2015-07-09]. （原始内容存档于2015-07-08）.
^ Mathematische Grundlagen der Quantenmechanik, John von Neumann, Berlin: Springer, 1932 (German); English translation Mathematical Foundations of Quantum Mechanics, transl. Robert T. Beyer, Princeton, New Jersey: Princeton University Press, 1955.

外部連結

Quantum Mechanics Not in Jeopardy: Physicists Confirm a Decades-Old Key Principle Experimentally （页面存档备份，存于互联网档案馆） ScienceDaily (July 23, 2010)

[1] Zurek, Wojciech. Quantum Darwinism, Classical Reality, and the randomness of quantum jumps. Physics Today. 2014, 67 (10): 44–45.

[Laloë-2] Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë. Quantum Mechanics Volume 1. Hermann. ISBN 978-2-7056-8392-4.

[Schlosshauer-3] 3.0 ^3.1 ^3.2 ^3.3 Schlosshauer, Maximilian; Fine, Arthur. On Zurek’s Derivation of the Born Rule (PDF). Foundations of Physics. 2005, 35 (2): 197–213 [2015-07-09]. doi:10.1007/s10701-004-1941-6. （原始内容存档 (PDF)于2016-03-04）.

[4] Landsman, N. P. The Born rule and its interpretation (PDF). Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (编). Compendium of Quantum Physics. Springer. 2008 [2024-04-18]. ISBN 978-3-540-70622-9. （原始内容存档 (PDF)于2024-06-09）. The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle

[Griffiths2004-5] 5.0 ^5.1 Griffiths, David J., Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, 2004, ISBN 0-13-111892-7

[6] Zur Quantenmechanik der Stoßvorgänge, Max Born, Zeitschrift für Physik, 37, #12 (Dec. 1926), pp. 863–867 (German); English translation, On the quantum mechanics of collisions, in Quantum theory and measurement, section I.2, J. A. Wheeler and W. H. Zurek, eds., Princeton, New Jersey: Princeton University Press, 1983, ISBN 0-691-08316-9.

[Born1954-7] Born, Max. Born's Nobel Lecture on the statistical interpretation of quantum mechanics (PDF). Nobel Lecture. NobelPrize.org. 1954 [2015-07-09]. （原始内容存档 (PDF)于2006-05-12）. Again an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons.

[8] The Nobel Prize in Physics 1954. NobelPrize.org. 1954 [2015-07-09]. （原始内容存档于2015-07-08）.

[9] Mathematische Grundlagen der Quantenmechanik, John von Neumann, Berlin: Springer, 1932 (German); English translation Mathematical Foundations of Quantum Mechanics, transl. Robert T. Beyer, Princeton, New Jersey: Princeton University Press, 1955.

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