物理學 1923

X 射線被輕元素散射的量子理論

亞瑟·康普頓

X 射線撞上電子後波長變長，像撞球被撞慢——光以粒子的方式攜帶動量。

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In depth · the introduction

把 X 射線照向一塊碳，射線彈回來時「顏色」卻變了——而這個小小的變化，了結了一場關於「光是波還是粒子」、持續了二十年的爭論。

核心想法

一個世紀以來，光一直被理解為波，而波在反彈時不會改變顏色。可當亞瑟·康普頓用 X 射線轟擊物質時，散射回來的射線卻更「軟」了——被拉到稍長一點的波長；而且偏折得越急，拉得越長。

他的解釋，是把 X 射線不當作波，而當作粒子：一顆攜帶動量的小小光彈。當它撞上一個電子，便把電子撞飛，恰如一顆撞球撞上另一顆，自己則帶著更少的能量彈開——對光而言，這意味著更長的波長。他甚至能把公式寫下來：拉長的量，只取決於反彈的角度，別無其他。實驗與公式相符。原來，光，是像個實物那樣撞過來的。

它是如何誕生的

早在 1905 年，愛因斯坦就提出過光是一份一份的，可在此後二十年裡，大多數物理學家都把它當作一個好用的技巧，而非字面的真理——波動理論實在太成功，讓人不忍割捨。康普頓在聖路易斯的華盛頓大學工作，一直在為散射 X 射線裡一個波動理論解釋不了的怪象而困惑。

約在 1922 年，他邁出了那一步：給每一份 X 射線的能量包一個確定的動量，並把它與電子的相遇當作一次乾脆的碰撞。數字乾淨俐落地落了出來，而他在石墨上細緻的譜儀測量證實了它們。荷蘭物理學家彼得·德拜，幾乎在同一時刻獨立地得到了同一公式。當別人很快拍到反衝電子與散射射線步調一致地飛出，此案便告了結。康普頓於 1927 年獲得諾貝爾獎。

它為何重要

這就是那個讓光子成真的實驗。光電效應曾暗示光以一塊塊的形式遞送能量，而康普頓則表明，光還攜帶動量、並在碰撞中反衝——在一切力學意義上，都像個粒子。在康普頓之後，你再也不能把光量子當作一個算起來方便的說法揮手打發掉。光，竟同時既是波又是粒子；而接受這個悖論，正是通向現代量子力學的門——僅僅兩年之後，它就來了。

一個可以想像的畫面

想像一局撞球。母球（X 射線）滾進來，撞上一顆靜止的球（電子）。母球穿不過去——它擦著彈開，被撞的那顆球帶著一些速度滾走。母球離開時，比來時慢。對光來說，「慢」不太對——光永遠以光速行進——所以它用另一種方式卸掉能量：把自己拉到更長的波長。而正如撞球裡，擦邊的一碰幾乎什麼都不改變，正面相撞卻奪走最多的速度；X 射線的拉長，在輕微偏折時最小，在徑直彈回時最大。用下面的工具，瞄準這一彈，看波長怎樣變長。

它在知識譜系裡的位置

康普頓的這次碰撞，是光的故事裡的一處合頁。在它身後，站著普朗克（1900）與愛因斯坦（1905），是他們最先提出光是一份一份的，以及波耳（1913），是他把量子放進了原子；在它身旁，站著光電效應，那是光子的另一個證據。在它身前，站著德布羅意（1924），他把康普頓的邏輯反轉過來，賦予物質一個波長，還有海森堡——他 1927 年的不確定性原理，用的正是康普頓發現的那個反衝：當你想看一個電子時，光子必然給它的那一腳。康普頓的測量，正是光的粒子從一個假說，變成一個你能秤量的事實的那一刻。

The original document

Original source text

Arthur H. Compton · Physical Review, Series 2, vol. 21, no. 5, pp. 483–502 · May 1923 · Washington University, St. Louis

The problem — scattered X-rays come back “softer”

On the wave theory of light, an X-ray passing through matter should set the electrons oscillating at its own frequency, and they should re-radiate at exactly that frequency: the scattered ray ought to have the same wavelength as the incident ray, with an intensity falling off with angle as J. J. Thomson's classical formula prescribes. Yet measurements — Compton's own among them — showed that the scattered radiation was consistently “softer” (longer in wavelength, more easily absorbed) than the primary beam, and that this softening grew steadily as the scattering angle increased. Classical electrodynamics had no room for such a shift.

The hypothesis — one quantum strikes one electron

Compton breaks with the wave picture and treats the X-ray as a quantum carrying energy hν and momentum h/λ. He pictures it making a single relativistic elastic collision with one free electron, like one billiard ball striking another, and imposes conservation of energy and of momentum on the pair. The recoiling electron carries off energy and momentum; the scattered quantum, left with less of both, must emerge with a lower frequency — a longer wavelength.

The result

Working the conservation equations through, the increase in wavelength depends only on the scattering angle θ — not on the incident wavelength, and not on the scattering material: Δλ = (h/mₑc)(1 − cos θ). The constant h/mₑc, the Compton wavelength, equals 0.0243 Å (2.43 pm). The shift vanishes straight ahead (θ = 0) and reaches its maximum, twice the Compton wavelength, for back-scattering (θ = 180°).

The test — molybdenum X-rays on graphite

Compton scattered the molybdenum Kα line (about 0.71 Å) from a block of graphite — carbon being a “light element” whose electrons are loosely bound — and measured the wavelengths of the scattered rays with a Bragg crystal spectrometer at a series of angles. At each angle the spectrum showed a shifted (“modified”) line beside an unshifted one, and the displacement of the modified line grew with angle exactly as the formula required, reaching about 0.024 Å at 90°. His verdict was plain:

This remarkable agreement between our formulas and the experiments can leave but little doubt that the scattering of X-rays is a quantum phenomenon.

[ … ]

The paper closes by drawing the consequence its title only hints at: scattering is not a wave gently re-radiating, but a corpuscle of radiation, with definite energy and momentum, deflecting a single electron — direct evidence for the reality of the light quantum.

Arthur H. Compton · Washington University, St. Louis · 1923