物理學 1924

量子理論研究（量子論研究）

路易·德布羅意

若光可以是粒子，那麼物質便可以是波。

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In depth · the introduction

一位年輕的貴族提出了一個簡單而大膽的問題：如果光——一種波——能表現得像粒子，那粒子為什麼不能表現得像波呢？

核心想法

到了 1924 年，物理學家們已經勉強接受：光是兩面的——它既是一列會擴散、會干涉的波，又是一陣被稱作光子的微小能量包。路易·德布羅意卻注意到，這筆交易是單方面的。光，本是波，卻被賦予了粒子般的行為；而物質，本由粒子構成，卻從未被賦予波動般的行為。他提議把這本帳平一平。

他的主張是：每一個運動的物體，都有一個波長，由一個簡潔得驚人的公式給出——波長，等於普朗克常數除以動量。一樣東西動得越快、越重，它的波就越短。對於一個棒球，這波小到無法想像，你永遠察覺不到。可對於電子——又輕又快——這波大約只有一個原子那麼大，大到足以舉足輕重，也大到足以被測量。

它是如何誕生的

路易·德布羅意出身於法國一個顯赫的大家族，起初學的是歷史。他被新生的量子理論中的種種謎題吸引而轉向物理，又受到哥哥莫里斯——一位研究 X 射線的實驗物理學家——的影響，把光的雙重身份這個問題，在腦中反覆琢磨了許多年。

他的答案，成了他 1924 年在索邦口試的博士論文。這想法太不尋常，以至於口試委員們都拿不準它是否可能為真。他們徵詢了阿爾伯特·愛因斯坦的意見；愛因斯坦讀後回信說，德布羅意「掀起了那道巨大帷幕的一角」。這一句話，改變了一切——不出三年，實驗便把電子射向晶體，看見它們如波一般泛起漣漪、彼此干涉，分毫不差。

它為何重要

德布羅意波長，是現代物理轉身時所憑的那道樞軸。它告訴了埃爾溫·薛丁格該去尋找一個什麼樣的方程，而他在 1926 年找到的那個波動方程，成了整個化學與量子力學的引擎——讓我們得以解釋：原子為何成鍵、材料為何導電或絕緣、元素週期表為何長成那個樣子。物質有波長這一想法，也正是電子顯微鏡得以存在的緣由，讓我們看見了病毒，乃至一個個單獨的原子。

一個可以想像的畫面

想像一根長長的跳繩。握住兩端，猛地一抖：波會沿著繩子傳過去，可繩子本身只是上下擺動——那道能量的凸起，跑得比繩上任何一個點都快。德布羅意的物質波，正是這樣運作的。電子，就是那道移動得較慢的凸起——「波包」——它把電子從這裡帶到那裡；而內部那些細小、飛快的波紋（相位波），則一路搶先飛奔。他證明了，這兩者永遠步調一致，就像一隻滾動的車輪，輻條與輪輞一同轉動。

它的位置

這一想法是一座橋。它的身後，站著馬克斯·普朗克與阿爾伯特·愛因斯坦——是他們最先發現，光以量子化的能量包出現；它的身前，站著薛丁格與維爾納·海森堡——他們 1926 年的波動力學與矩陣力學，把德布羅意的一個暗示，變成了一套完整的理論。德布羅意憑這篇論文，獲得了 1929 年的諾貝爾獎。他晚年一直在為一種「導波」圖景辯護，那圖景為多數物理學家所擱置——然而數十年後，它又被戴維·玻姆重新拾起；而關於這列波究竟是什麼的爭論，也從未真正合上。

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Original source text

導言——一種有待恢復的對稱

Louis de Broglie · Recherches sur la théorie des quanta · Thesis, Paris, 1924 · Annales de Physique (10) 3 (1925): 22–128

[Light, since Einstein's 1905 light-quanta, had been granted a particle aspect alongside its wave aspect. De Broglie's thesis asks whether the converse holds for matter.]

After long reflection in solitude and meditation, I suddenly had the idea, during the year 1923, that the discovery made by Einstein in 1905 should be generalised by extending it to all material particles and notably to electrons.

第一章——相位波

An internal periodic phenomenon

We shall assume the existence of a certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated parcel of energy, and which depends on its proper mass through the Planck–Einstein equation.

[For a particle of rest mass m₀, de Broglie sets the rest-frame frequency by hν₀ = m₀c². Seen from the laboratory, the moving particle's internal clock runs slow (time dilation) at ν₁ = ν₀√(1−β²), yet an accompanying wave runs at the higher frequency ν = ν₀ ⁄ √(1−β²) and at phase velocity V = c²/v.]

We are then inclined to admit that any moving body may be accompanied by a wave and that it is impossible to disjoin motion of body and propagation of wave.

The theorem of the harmony of phases

[De Broglie proves that the slow internal clock of the particle and the fast external wave, though they have different frequencies, stay perpetually in step: at the particle's position the phase of the wave always agrees with the phase of the internal vibration. This is the harmony of phases that the wave's velocity V = c²/v was chosen precisely to secure.]

[ … ]

The phase wave guides the displacement of energy; its group velocity is equal to the velocity of the particle, while its phase velocity exceeds the velocity of light without, since it carries no energy, contradicting relativity.

物質的波長

[Combining the relations gives the result the thesis is remembered for: a moving body of momentum p has an associated wavelength.]

λ = h / p

[For an electron this wavelength is comparable to the spacing of atoms in a crystal — which is why de Broglie foresaw that a stream of electrons ought to be diffracted by a crystal lattice, just as X-rays are.]

重新解讀玻爾的軌道

[De Broglie shows that Bohr's mysterious quantum condition for the allowed electron orbits is simply the requirement that the phase wave close on itself — that a whole number of wavelengths fit around the orbit, a standing wave rather than a wave that interferes with itself and dies away.]

The stability conditions of the trajectories in Bohr's theory are interpretable as the resonance condition of the phase wave along the closed path.