CHAPTER 2  第二章

The penetrating ability of $x$$x$xx-rays enabled them to reveal the frog which this snake had swallowed. The snake’s jaws are very loosely joined and so can open widely.
$x$$x$xx -射線的穿透能力使它們能夠揭示這條蛇吞下的青蛙。蛇的下顎連接非常鬆弛，因此可以大幅度張開。
2.1 ELECTROMAGNETIC WAVESCoupled electric and magnetic oscillations thatmove with the speed of light and exhibit typicalwave behavior
2.1 電磁波 伴隨著電和磁的耦合振盪，以光速移動並展現典型的波動行為
2.2 BLACKBODY RADIATION 2.2 黑體輻射
Only the quantum theory of light can explain its origin
只有光的量子理論能解釋它的起源
2.3 PHOTOELECTRIC EFFECT 2.3 光電效應
The energies of electrons liberated by light depend on the frequency of the light
由光釋放的電子能量取決於光的頻率
2.4 WHAT IS LIGHT? 2.4 什麼是光？
Both wave and particle 波與粒子
2.5 X-RAYSThey consist of high-energy photons
2.5 X 光線它們由高能光子組成
2.6 X-RAY DIFFRACTIONHow $x$$x$xx-ray wavelengths can be determined
2.6 X 射線繞射如何確定 $x$$x$xx -射線波長
2.7 COMPTON EFFECT 2.7 康普頓效應
Further confirmation of the photon model
進一步確認光子模型
2.8 PAIR PRODUCTION 2.8 配對生產
Energy into matter 能量轉化為物質
2.9 PHOTONS AND GRAVITYAlthough they lack rest mass, photons behave asthough they have gravitational mass
2.9 光子與重力 儘管光子沒有靜止質量，但它們的行為就像擁有重力質量一樣。

In our everyday experience there is nothing mysterious or ambiguous about the concepts of particle and wave. A stone dropped into a lake and the ripples that spread out from its point of impact apparently have in common only the ability to carry energy and momentum from one place to another. Classical physics, which mirrors the “physical reality” of our sense impressions, treats particles and waves as separate components of that reality. The mechanics of particles and the optics of waves are traditionally independent disciplines, each with its own chain of experiments and principles based on their results.
在我們的日常經驗中，粒子和波的概念並沒有什麼神秘或模糊之處。一塊石頭掉入湖中，從其撞擊點擴散出的漣漪顯然只有一個共同點，就是能夠將能量和動量從一個地方傳遞到另一個地方。古典物理學反映了我們感官印象中的“物理現實”，將粒子和波視為該現實的獨立組成部分。粒子的力學和波的光學傳統上是獨立的學科，各自擁有基於其結果的實驗和原則鏈。

The physical reality we perceive has its roots in the microscopic world of atoms and molecules, electrons and nuclei, but in this world there are neither particles nor waves in our sense of these terms. We regard electrons as particles because they possess charge and mass and behave according to the laws of particle mechanics in such familiar devices as television picture tubes. We shall see, however, that it is just as correct to interpret a moving electron as a wave manifestation as it is to interpret it as a particle manifestation. We regard electromagnetic waves as waves because under suitable circumstances they exhibit diffraction, interference, and polarization. Similarly, we shall see that under other circumstances electromagnetic waves behave as though they consist of streams of particles. Together with special relativity, the wave-particle duality is central to an understanding of modern physics, and in this book there are few arguments that do not draw upon either or both of these fundamental ideas.
我們所感知的物理現實根植於原子和分子的微觀世界，電子和原子核，但在這個世界中，並不存在我們所理解的粒子或波。我們將電子視為粒子，因為它們具有電荷和質量，並且在如電視顯示管等熟悉的裝置中根據粒子力學的法則行為。然而，我們將看到，將運動的電子解釋為波的表現與將其解釋為粒子的表現同樣正確。我們將電磁波視為波，因為在適當的情況下，它們顯示出衍射、干涉和偏振。同樣，我們將看到在其他情況下，電磁波的行為就像是由粒子流組成的。與特殊相對論一起，波粒二象性是理解現代物理學的核心，在本書中，幾乎沒有論點不依賴於這些基本思想之一或兩者。

In 1864 the British physicist James Clerk Maxwell made the remarkable suggestion that accelerated electric charges generate linked electric and magnetic disturbances that can travel indefinitely through space. If the charges oscillate periodically, the disturbances are waves whose electric and magnetic components are perpendicular to each other and to the direction of propagation, as in Fig. 2.1.
在 1864 年，英國物理學家詹姆斯·克拉克·麥克斯韋提出了一個卓越的建議：加速的電荷會產生相互連結的電磁擾動，這些擾動可以無限地在空間中傳播。如果電荷周期性地振盪，這些擾動就是波，其電場和磁場成分彼此垂直，並且與傳播方向垂直，如圖 2.1 所示。

From the earlier work of Faraday, Maxwell knew that a changing magnetic field can induce a current in a wire loop. Thus a changing magnetic field is equivalent in its effects to an electric field. Maxwell proposed the converse: a changing electric field has a magnetic field associated with it. The electric fields produced by electromagnetic induction are easy to demonstrate because metals offer little resistance to the flow of charge. Even a weak field can lead to a measurable current in a metal. Weak magnetic fields are much harder to detect, however, and Maxwell’s hypothesis was based on a symmetry argument rather than on experimental findings.
從法拉第的早期研究中，麥克斯韋知道變化的磁場可以在導線圈中誘導電流。因此，變化的磁場在其效應上等同於電場。麥克斯韋提出了相反的觀點：變化的電場有一個與之相關的磁場。由電磁感應產生的電場容易演示，因為金屬對電荷的流動提供的阻力很小。即使是微弱的電場也能在金屬中產生可測量的電流。然而，微弱的磁場則難以檢測，麥克斯韋的假設是基於對稱性論證，而不是基於實驗結果。

Figure 2.1 The electric and magnetic fields in an electromagnetic wave vary together. The fields are perpendicular to each other and to the direction of propagation of the wave.
圖 2.1 電磁波中的電場和磁場一起變化。這些場彼此垂直，並且與波的傳播方向垂直。

James Clerk Maxwell (18311879) was born in Scotland shortly before Michael Faraday discovered electromagnetic induction. At nineteen he entered Cambridge University to study physics and mathematics. While still a student, he investigated the physics of color vision and later used his ideas to make the first color photograph. Maxwell became known to the scientific world at twenty-four when he showed that the rings of Saturn could not be solid or liquid but must consist of separate small bodies. At about this time Maxwell became interested in electricity and magnetism and grew convinced that the wealth of phenomena Faraday and others had discovered were not isolated effects but had an underlying unity of some kind. Maxwell’s initial step in establishing that unity came in 1856 with the paper “On Faraday’s Lines of Force,” in which he developed a mathematical description of electric and magnetic fields.
詹姆斯·克拉克·麥克斯威爾（1831-1879）出生於蘇格蘭，恰好在邁克爾·法拉第發現電磁感應之前。十九歲時，他進入劍橋大學學習物理和數學。在還是學生的時候，他研究了色彩視覺的物理學，並後來利用他的想法拍攝了第一張彩色照片。麥克斯威爾在二十四歲時因為證明土星的環不能是固體或液體，而必須由分開的小天體組成而為科學界所知。大約在這個時候，麥克斯威爾對電和磁產生了興趣，並堅信法拉第和其他人所發現的現象並不是孤立的效應，而是有某種潛在的統一性。麥克斯威爾在 1856 年發表的論文《法拉第的力線》中，首次建立了這種統一性，並發展了電場和磁場的數學描述。

Maxwell left Cambridge in 1856 to teach at a college in Scotland and later at King’s College in London. In this period he expanded his ideas on electricity and magnetism to create a single comprehensive theory of electromagnetism. The fundamental equations he arrived at remain the foundations of the subject today. From these equations Maxwell predicted that electromagnetic waves should exist that travel with the speed
麥克斯韋於 1856 年離開劍橋，前往蘇格蘭的一所大學任教，後來又在倫敦的國王學院任教。在這段期間，他擴展了對電和磁的想法，創造了一個統一的電磁學綜合理論。他所得到的基本方程式至今仍然是該學科的基礎。根據這些方程式，麥克斯韋預測應該存在以光速傳播的電磁波。
of light, described the properties the waves should have, and surmised that light consisted of electromagnetic waves. Sadly, he did not live to see his work confirmed in the experiments of the German physicist Heinrich Hertz.
光的性質，描述了波應具備的特性，並推測光是由電磁波組成的。可惜的是，他未能活到看到他的研究在德國物理學家海因里希·赫茲的實驗中得到證實。

Maxwell’s contributions to kinetic theory and statistical mechanics were on the same profound level as his contributions to electromagnetic theory. His calculations showed that the viscosity of a gas ought to be independent of its pressure, a surprising result that Maxwell, with the help of his wife, confirmed in the laboratory. They also found that the viscosity was proportional to the absolute temperature of the gas. Maxwell’s explanation for this proportionality gave him a way to estimate the size and mass of molecules, which until then could only be guessed at. Maxwell shares with Boltzmann credit for the equation that gives the distribution of molecular energies in a gas.
麥克斯威爾對於動力學理論和統計力學的貢獻與他對電磁理論的貢獻同樣深遠。他的計算顯示，氣體的黏度應該與其壓力無關，這是一個令人驚訝的結果，麥克斯威爾在妻子的幫助下在實驗室中確認了這一點。他們還發現，黏度與氣體的絕對溫度成正比。麥克斯威爾對這一比例關係的解釋使他能夠估算分子的大小和質量，而在此之前這些只能靠猜測。麥克斯威爾與玻爾茲曼共同獲得了描述氣體中分子能量分佈的方程的功勞。

In 1865 Maxwell returned to his family’s home in Scotland. There he continued his research and also composed a treatise on electromagnetism that was to be the standard text on the subject for many decades. It was still in print a century later. In 1871 Maxwell went back to Cambridge to establish and direct the Cavendish Laboratory, named in honor of the pioneering physicist Henry Cavendish. Maxwell died of cancer at the age of forty-eight in 1879, the year in which Albert Einstein was born. Maxwell had been the greatest theoretical physicist of the nineteenth century; Einstein was to be the greatest theoretical physicist of the twentieth century. (By a similar coincidence, Newton was born in the year of Galileo’s death.)
在 1865 年，麥克斯威爾回到了他在蘇格蘭的家中。在那裡，他繼續進行研究，並撰寫了一部關於電磁學的論文，這部論文在接下來的幾十年裡成為該領域的標準教材。一個世紀後仍在印刷中。1871 年，麥克斯威爾回到劍橋，建立並指導了以開創性物理學家亨利·卡文迪許命名的卡文迪許實驗室。麥克斯威爾於 1879 年因癌症去世，享年四十八歲，正是阿爾伯特·愛因斯坦出生的那一年。麥克斯威爾是十九世紀最偉大的理論物理學家；愛因斯坦則成為二十世紀最偉大的理論物理學家。（有著類似的巧合，牛頓出生於伽利略去世的那一年。）

If Maxwell was right, electromagnetic (em) waves must occur in which constantly varying electric and magnetic fields are coupled together by both electromagnetic induction and the converse mechanism he proposed. Maxwell was able to show that the speed $c$$c$cc of electromagnetic waves in free space is given by
如果麥克斯威爾是對的，電磁波必須在不斷變化的電場和磁場中發生，這些場通過電磁感應和他所提出的反向機制相互耦合。麥克斯威爾能夠顯示，在自由空間中電磁波的速度 $c$$c$cc 是由以下公式給出的。

where ${\mathit{ϵ}}_0$${\mathit{ϵ}}_0$epsilon_(0)\boldsymbol{\epsilon}_{0} is the electric permittivity of free space and ${\mu }_0$${\mu }_0$mu_(0)\mu_{0} is its magnetic permeability. This is the same as the speed of light waves. The correspondence was too great to be accidental, and Maxwell concluded that light consists of electromagnetic waves.
其中 ${\mathit{ϵ}}_0$${\mathit{ϵ}}_0$epsilon_(0)\boldsymbol{\epsilon}_{0} 是自由空間的電容率，而 ${\mu }_0$${\mu }_0$mu_(0)\mu_{0} 是其磁導率。這與光波的速度相同。這種對應關係過於顯著，無法是偶然的，馬克士威得出結論，光是由電磁波組成的。

During Maxwell’s lifetime the notion of em waves remained without direct experimental support. Finally, in 1888, the German physicist Heinrich Hertz showed that em waves indeed exist and behave exactly as Maxwell had predicted. Hertz generated the waves by applying an alternating current to an air gap between two metal balls. The width of the gap was such that a spark occurred each time the current reached a peak. A wire loop with a small gap was the detector; em waves set up oscillations in the loop that produced sparks in the gap. Hertz determined the wavelength and speed of the waves he generated, showed that they have both electric and magnetic components, and found that they could be reflected, refracted, and diffracted.
在麥克斯威爾的生涯中，電磁波的概念仍然缺乏直接的實驗支持。最終，在 1888 年，德國物理學家海因里希·赫茲證明了電磁波確實存在，並且行為完全符合麥克斯威爾的預測。赫茲通過對兩個金屬球之間的空氣間隙施加交流電來產生這些波。間隙的寬度使得每當電流達到峰值時就會產生火花。一個帶有小間隙的線圈是檢測器；電磁波在這個線圈中產生振盪，並在間隙中產生火花。赫茲測定了他所產生的波的波長和速度，顯示它們具有電和磁的成分，並發現它們可以被反射、折射和繞射。

Light is not the only example of an em wave. Although all such waves have the same fundamental nature, many features of their interaction with matter depend upon
光並不是電磁波的唯一例子。雖然所有這類波具有相同的基本特性，但它們與物質相互作用的許多特徵取決於

Figure 2.2 The spectrum of electromagnetic radiation.
圖 2.2 電磁輻射的光譜。
their frequencies. Light waves, which are em waves the eye responds to, span only a brief frequency interval, from about $4.3\times {10}^{14}\mathrm{Hz}$$4.3\times {10}^{14}\mathrm{Hz}$4.3 xx10^(14)Hz4.3 \times 10^{14} \mathrm{~Hz} for red light to about $7.5\times {10}^{14}$$7.5\times {10}^{14}$7.5 xx10^(14)7.5 \times 10^{14} Hz for violet light. Figure 2.2 shows the em wave spectrum from the low frequencies used in radio communication to the high frequencies found in x-rays and gamma rays.
它們的頻率。光波是眼睛能夠感應的電磁波，僅涵蓋一個短暫的頻率範圍，從約 $4.3\times {10}^{14}\mathrm{Hz}$$4.3\times {10}^{14}\mathrm{Hz}$4.3 xx10^(14)Hz4.3 \times 10^{14} \mathrm{~Hz} 赫茲的紅光到約 $7.5\times {10}^{14}$$7.5\times {10}^{14}$7.5 xx10^(14)7.5 \times 10^{14} 赫茲的紫光。圖 2.2 顯示了從用於無線通信的低頻率到在 X 射線和伽馬射線中發現的高頻率的電磁波譜。

A characteristic property of all waves is that they obey the principle of superposition:
所有波的特徵性質是它們遵循疊加原理：

When two or more waves of the same nature travel past a point at the same time, the instantaneous amplitude there is the sum of the instantaneous amplitudes of the individual waves.
當兩個或更多相同性質的波同時經過一個點時，該點的瞬時振幅是各個波的瞬時振幅之和。

Instantaneous amplitude refers to the value at a certain place and time of the quantity whose variations constitute the wave. (“Amplitude” without qualification refers to the maximum value of the wave variable.) Thus the instantaneous amplitude of a wave in a stretched string is the displacement of the string from its normal position; that of a water wave is the height of the water surface relative to its normal level; that of a sound wave is the change in pressure relative to the normal pressure. Since the electric and magnetic fields in a light wave are related by $E=cB$$E=cB$E=cBE=c B, its instantaneous amplitude can be taken as either $E$$E$EE or $B$$B$BB. Usually $E$$E$EE is used, since it is the electric fields of light waves whose interactions with matter give rise to nearly all common optical effects.
瞬時振幅是指在某一特定時間和位置上，構成波動的量的變化值。（未經修飾的“振幅”指的是波動變量的最大值。）因此，拉緊弦上的波的瞬時振幅是弦相對於其正常位置的位移；水波的瞬時振幅是水面相對於其正常水平的高度；聲波的瞬時振幅是相對於正常壓力的壓力變化。由於光波中的電場和磁場之間存在關聯，其瞬時振幅可以取為 $E$$E$EE 或 $B$$B$BB 。通常使用 $E$$E$EE ，因為光波的電場與物質的相互作用幾乎產生了所有常見的光學效應。

The interference of water waves. Constructive interference occurs along the line $AB$$AB$ABA B and destructive interference occurs along the line $CD$$CD$CDC D.
水波的干涉。建設性干涉發生在 $AB$$AB$ABA B 線上，而破壞性干涉發生在 $CD$$CD$CDC D 線上。

When two or more trains of light waves meet in a region, they interfere to produce a new wave there whose instantaneous amplitude is the sum of those of the original waves. Constructive interference refers to the reinforcement of waves with the same phase to produce a greater amplitude, and destructive interference refers to the partial or complete cancellation of waves whose phases differ (Fig. 2.3). If the original waves have different frequencies, the result will be a mixture of constructive and destructive interference, as in Fig. 3.4.
當兩個或更多的光波在某個區域相遇時，它們會相互干涉，產生一個新的波，其瞬時振幅是原始波振幅的總和。建設性干涉是指相同相位的波增強以產生更大的振幅，而破壞性干涉則是指相位不同的波部分或完全抵消（圖 2.3）。如果原始波的頻率不同，結果將是建設性和破壞性干涉的混合，如圖 3.4 所示。

The interference of light waves was first demonstrated in 1801 by Thomas Young, who used a pair of slits illuminated by monochromatic light from a single source (Fig. 2.4). From each slit secondary waves spread out as though originating at the slit; this is an example of diffraction, which, like interference, is a characteristic wave phenomenon. Owing to interference, the screen is not evenly lit but shows a pattern of alternate bright and dark lines. At those places on the screen where the path lengths from the two slits differ by an odd number of half wavelengths $(\lambda /2,3\lambda /2,5\lambda /2,\dots )$$(\lambda /2,3\lambda /2,5\lambda /2,\dots )$(lambda//2,3lambda//2,5lambda//2,dots)(\lambda / 2,3 \lambda / 2,5 \lambda / 2, \ldots), destructive interference occurs and a dark line is the result. At those places where the path lengths are
光波的干涉首次由托馬斯·楊於 1801 年展示，他使用了一對由單一光源照明的單色光狹縫（圖 2.4）。從每個狹縫發出的次級波似乎是從狹縫發源的；這是一個衍射的例子，衍射和干涉一樣，都是特徵性的波現象。由於干涉，屏幕並不是均勻照明的，而是顯示出交替的明暗線條。在屏幕上，當來自兩個狹縫的光程差為奇數個半波長時，會發生破壞性干涉，結果形成一條暗線。在那些光程差為...

Figure 2.3 (a) In constructive interference, superposed waves in phase reinforce each other. (b) In destructive interference, waves out of phase partially or completely cancel each other.
圖 2.3 (a) 在建設性干涉中，重疊的波相位相同，相互增強。(b) 在破壞性干涉中，波相位不同，部分或完全相互抵消。

Constructive 建設性的
interference produces bright line
干涉產生明亮的線條

Destructive interference produces dark line
破壞性干涉產生暗線

Constructive interference produces bright line
建設性干涉產生明亮的線條

Appearance of screen 螢幕外觀

Figure 2.4 Origin of the interference pattern in Young’s experiment. Constructive interference occurs where the difference in path lengths from the slits to the screen is $\theta ,\lambda ,2\lambda ,\dots$$\theta ,\lambda ,2\lambda ,\dots$theta,lambda,2lambda,dots\theta, \lambda, 2 \lambda, \ldots Destructive interference occurs where the path difference is $\lambda /2,3\lambda /2,5\lambda /2,\dots$$\lambda /2,3\lambda /2,5\lambda /2,\dots$lambda//2,3lambda//2,5lambda//2,dots\lambda / 2,3 \lambda / 2,5 \lambda / 2, \ldots.
圖 2.4 楊氏實驗中干涉圖樣的來源。當從狹縫到螢幕的路徑長度差為 $\theta ,\lambda ,2\lambda ,\dots$$\theta ,\lambda ,2\lambda ,\dots$theta,lambda,2lambda,dots\theta, \lambda, 2 \lambda, \ldots 時，發生建設性干涉；當路徑差為 $\lambda /2,3\lambda /2,5\lambda /2,\dots$$\lambda /2,3\lambda /2,5\lambda /2,\dots$lambda//2,3lambda//2,5lambda//2,dots\lambda / 2,3 \lambda / 2,5 \lambda / 2, \ldots 時，發生破壞性干涉。
equal or differ by a whole number of wavelengths $(\lambda ,2\lambda ,3\lambda ,\dots )$$(\lambda ,2\lambda ,3\lambda ,\dots )$(lambda,2lambda,3lambda,dots)(\lambda, 2 \lambda, 3 \lambda, \ldots), constructive interference occurs and a bright line is the result. At intermediate places the interference is only partial, so the light intensity on the screen varies gradually between the bright and dark lines.
當相等或相差整數個波長 $(\lambda ,2\lambda ,3\lambda ,\dots )$$(\lambda ,2\lambda ,3\lambda ,\dots )$(lambda,2lambda,3lambda,dots)(\lambda, 2 \lambda, 3 \lambda, \ldots) 時，會發生建設性干涉，結果是出現明亮的線條。在中間位置，干涉僅為部分，因此屏幕上的光強度在明亮和黑暗線條之間逐漸變化。

Interference and diffraction are found only in waves-the particles we are familiar with do not behave in those ways. If light consisted of a stream of classical particles, the entire screen would be dark. Thus Young’s experiment is proof that light consists of waves. Maxwell’s theory further tells us what kind of waves they are: electromagnetic. Until the end of the nineteenth century the nature of light seemed settled forever.
干涉和衍射只存在於波中——我們熟悉的粒子並不以這種方式行為。如果光是由一串經典粒子組成，整個屏幕將是黑暗的。因此，楊氏實驗證明了光是由波組成的。麥克斯韋的理論進一步告訴我們它們是什麼樣的波：電磁波。直到十九世紀末，光的本質似乎永遠已經確定。

Following Hertz’s experiments, the question of the fundamental nature of light seemed clear: light consisted of em waves that obeyed Maxwell’s theory. This certainty lasted only a dozen years. The first sign that something was seriously amiss came from attempts to understand the origin of the radiation emitted by bodies of matter.
隨著赫茲的實驗，光的基本性質問題似乎變得明確：光由遵循麥克斯韋理論的電磁波組成。這種確定性僅持續了十多年。第一個顯示出問題嚴重的跡象來自於對物質體所發射輻射來源的理解嘗試。

We are all familiar with the glow of a hot piece of metal, which gives off visible light whose color varies with the temperature of the metal, going from red to yellow to white as it becomes hotter and hotter. In fact, other frequencies to which our eyes do not respond are present as well. An object need not be so hot that it is luminous for it to be radiating em energy; all objects radiate such energy continuously whatever their temperatures, though which frequencies predominate depends on the temperature. At room temperature most of the radiation is in the infrared part of the spectrum and hence is invisible.
我們都熟悉一塊熱金屬的光輝，這種光輝發出可見光，其顏色隨著金屬的溫度而變化，從紅色變為黃色，再變為白色，隨著溫度的升高而變得越來越熱。事實上，還存在著我們的眼睛無法感知的其他頻率。一個物體不必熱到發光才能輻射電磁能量；所有物體無論其溫度如何，都會持續輻射這種能量，雖然主導的頻率取決於溫度。在室溫下，大部分輻射位於光譜的紅外部分，因此是不可見的。

The ability of a body to radiate is closely related to its ability to absorb radiation. This is to be expected, since a body at a constant temperature is in thermal equilibrium with its surroundings and must absorb energy from them at the same rate as it emits energy. It is convenient to consider as an ideal body one that absorbs all radiation incident upon it, regardless of frequency. Such a body is called a blackbody.
一個物體輻射的能力與其吸收輻射的能力密切相關。這是可以預期的，因為在恆定溫度下的物體與其周圍環境處於熱平衡狀態，必須以與其發射能量相同的速率從環境中吸收能量。將一個理想的物體視為能夠吸收所有入射輻射的物體是方便的，無論其頻率如何。這樣的物體稱為黑體。

The point of introducing the idealized blackbody in a discussion of thermal radiation is that we can now disregard the precise nature of whatever is radiating, since
引入理想黑體於熱輻射討論中的意義在於，我們現在可以忽略任何輻射物體的精確性質，因為

Figure 2.5 A hole in the wall of a hollow object is an excellent approximation of a blackbody.
圖 2.5 一個中空物體牆壁上的孔是黑體的極佳近似。

The color and brightness of an object heated until it glows, such as the filament of this light bulb, depends upon its temperature, which here is about 3000 K . An object that glows white is hotter than it is when it glows red, and it gives off more light as well.
一個物體加熱到發光的顏色和亮度，例如這個燈泡的燈絲，取決於它的溫度，這裡約為 3000 K。發白光的物體比發紅光的物體更熱，並且發出更多的光。
all blackbodies behave identically. In the laboratory a blackbody can be approximated by a hollow object with a very small hole leading to its interior (Fig. 2.5). Any radiation striking the hole enters the cavity, where it is trapped by reflection back and forth until it is absorbed. The cavity walls are constantly emitting and absorbing radiation, and it is in the properties of this radiation (blackbody radiation) that we are interested.
所有黑體的行為都是相同的。在實驗室中，黑體可以用一個有非常小孔通向其內部的空心物體來近似（圖 2.5）。任何射向孔的輻射都會進入腔體，在那裡它會因反射來回而被困住，直到被吸收。腔體的牆壁不斷地發射和吸收輻射，而我們感興趣的是這種輻射的特性（黑體輻射）。

Experimentally we can sample blackbody radiation simply by inspecting what emerges from the hole in the cavity. The results agree with everyday experience. A blackbody radiates more when it is hot than when it is cold, and the spectrum of a hot blackbody has its peak at a higher frequency than the peak in the spectrum of a cooler one. We recall the behavior of an iron bar as it is heated to progressively higher temperatures: at first it glows dull red, then bright orange-red, and eventually it becomes “white hot.” The spectrum of blackbody radiation is shown in Fig. 2.6 for two temperatures.
實驗上，我們可以通過檢查從腔體孔中出現的輻射來取樣黑體輻射。結果與日常經驗一致。黑體在高溫時輻射的能量比在低溫時多，熱黑體的光譜峰值頻率高於冷黑體的光譜峰值。我們回想起鐵棒在逐漸加熱到更高溫度時的行為：起初它發出暗紅色的光，然後變成明亮的橙紅色，最終變得“白熱”。黑體輻射的光譜在圖 2.6 中顯示了兩個不同的溫度。

Why does the blackbody spectrum have the shape shown in Fig. 2.6? This problem was examined at the end of the nineteenth century by Lord Rayleigh and James Jeans. The details of their calculation are given in Chap. 9. They started by considering the radiation inside a cavity of absolute temperature $T$$T$TT whose walls are perfect reflectors to be a series of standing em waves (Fig. 2.7). This is a threedimensional generalization of standing waves in a stretched string. The condition
為什麼黑體光譜呈現圖 2.6 所示的形狀？這個問題在十九世紀末由雷利勳爵和詹姆斯·珍斯進行了研究。他們計算的細節在第 9 章中給出。他們首先考慮了絕對溫度 $T$$T$TT 的空腔內的輻射，其牆壁是完美的反射體，這被視為一系列的駐波電磁波（圖 2.7）。這是拉伸弦上駐波的三維推廣。條件

Figure 2.6 Blackbody spectra. The spectral distribution of energy in the radiation depends only on the temperature of the body. The higher the temperature, the greater the amount of radiation and the higher the frequency at which the maximum emission occurs. The dependence of the latter frequency on temperature follows a formula called Wien’s displacement law, which is discussed in Sec. 9.6.
圖 2.6 黑體光譜。輻射中的能量光譜分佈僅依賴於物體的溫度。溫度越高，輻射的量越大，最大發射發生的頻率也越高。後者頻率對溫度的依賴遵循一個稱為維恩位移定律的公式，該公式在第 9.6 節中討論。
for standing waves in such a cavity is that the path length from wall to wall, whatever the direction, must be a whole number of half-wavelengths, so that a node occurs at each reflecting surface. The number of independent standing waves $G(\nu )d\nu$$G(\nu )d\nu$G(nu)d nuG(\nu) d \nu in the frequency interval between $\nu$$\nu$nu\nu and $d\nu$$d\nu$d nud \nu per unit volume in the cavity turned out to be
在這樣的腔體中，駐波的條件是牆壁之間的路徑長度，無論方向如何，必須是半波長的整數倍，以便在每個反射表面上出現一個節點。在頻率區間 $\nu$$\nu$nu\nu 和 $d\nu$$d\nu$d nud \nu 之間，每單位體積的獨立駐波數量 $G(\nu )d\nu$$G(\nu )d\nu$G(nu)d nuG(\nu) d \nu 最終為

Density of standing waves in cavity
腔體中駐波的密度

This formula is independent of the shape of the cavity. As we would expect, the higher the frequency $\nu$$\nu$nu\nu, the shorter the wavelength and the greater the number of possible standing waves.
這個公式與腔體的形狀無關。正如我們所預期的，頻率越高 $\nu$$\nu$nu\nu ，波長越短，可能的駐波數量越多。

The next step is to find the average energy per standing wave. According to the theorem of equipartition of energy, a mainstay of classical physics, the average energy per degree of freedom of an entity (such as a molecule of an ideal gas) that is a member of a system of such entities in thermal equilibrium at the temperature $T$$T$TT is $\frac{1}{2}kT$$\frac{1}{2}kT$(1)/(2)kT\frac{1}{2} k T. Here $k$$k$kk is Boltzmann’s constant:
下一步是找出每個駐波的平均能量。根據能量均分定理，這是經典物理學的一個基石，處於熱平衡狀態的系統中，每個自由度的平均能量為 $\frac{1}{2}kT$$\frac{1}{2}kT$(1)/(2)kT\frac{1}{2} k T ，該系統的成員（例如理想氣體的分子）在溫度 $T$$T$TT 下。這裡 $k$$k$kk 是玻爾茲曼常數：

Boltzmann’s constant $k=1.381\times {10}^{-23}\mathrm{J}/\mathrm{K}$$k=1.381\times {10}^{-23}\mathrm{J}/\mathrm{K}$quad k=1.381 xx10^(-23)J//K\quad k=1.381 \times 10^{-23} \mathrm{~J} / \mathrm{K}
玻爾茲曼常數 $k=1.381\times {10}^{-23}\mathrm{J}/\mathrm{K}$$k=1.381\times {10}^{-23}\mathrm{J}/\mathrm{K}$quad k=1.381 xx10^(-23)J//K\quad k=1.381 \times 10^{-23} \mathrm{~J} / \mathrm{K}
A degree of freedom is a mode of energy possession. Thus a monatomic ideal gas molecule has three degrees of freedom, corresponding to kinetic energy of motion in three independent directions, for an average total energy of $\frac{3}{2}kT$$\frac{3}{2}kT$(3)/(2)kT\frac{3}{2} k T.
自由度是一種能量擁有的方式。因此，單原子理想氣體分子具有三個自由度，對應於在三個獨立方向上的運動動能，平均總能量為 $\frac{3}{2}kT$$\frac{3}{2}kT$(3)/(2)kT\frac{3}{2} k T 。

A one-dimensional harmonic oscillator has two degrees of freedom, one that corresponds to its kinetic energy and one that corresponds to its potential energy. Because each standing wave in a cavity originates in an oscillating electric charge in the cavity wall, two degrees of freedom are associated with the wave and it should have an average energy of $2\left(\frac{1}{2}\right)kT$$2\left(\frac{1}{2}\right)kT$2((1)/(2))kT2\left(\frac{1}{2}\right) k T :
一維諧振子有兩個自由度，一個對應於其動能，另一個對應於其勢能。由於腔體中的每個駐波源於腔壁中的振盪電荷，因此與波相關聯的有兩個自由度，並且它的平均能量應為 $2\left(\frac{1}{2}\right)kT$$2\left(\frac{1}{2}\right)kT$2((1)/(2))kT2\left(\frac{1}{2}\right) k T ：

The total energy $u(\nu )d\mathit{\nu }$$u(\nu )d\mathit{\nu }$u(nu)d nuu(\nu) d \boldsymbol{\nu} per unit volume in the cavity in the frequency interval from $\nu$$\nu$nu\nu to $\nu +d\nu$$\nu +d\nu$nu+d nu\nu+d \nu is therefore
在頻率區間從 $\nu$$\nu$nu\nu 到 $\nu +d\nu$$\nu +d\nu$nu+d nu\nu+d \nu 的腔體中，每單位體積的總能量 $u(\nu )d\mathit{\nu }$$u(\nu )d\mathit{\nu }$u(nu)d nuu(\nu) d \boldsymbol{\nu} 因此為

Rayleigh-Jeans 瑞利-珍斯
formula 公式

This radiation rate is proportional to this energy density for frequencies between $\nu$$\nu$nu\nu and $\nu +d\nu$$\nu +d\nu$nu+d nu\nu+d \nu. Equation (2.3), the Rayleigh-Jeans formula, contains everything that classical physics can say about the spectrum of blackbody radiation.
這個輻射率與這個能量密度在頻率介於 $\nu$$\nu$nu\nu 和 $\nu +d\nu$$\nu +d\nu$nu+d nu\nu+d \nu 之間成正比。方程式 (2.3)，即瑞利-珍公式，包含了古典物理對黑體輻射光譜的所有描述。

Even a glance at Eq. (2.3) shows that it cannot possibly be correct. As the frequency $\nu$$\nu$nu\nu increases toward the ultraviolet end of the spectrum, this formula predicts that the energy density should increase as ${\nu }^2$${\nu }^2$nu^(2)\nu^{2}. In the limit of infinitely high frequencies, $u(\nu )d\nu$$u(\nu )d\nu$u(nu)d nuu(\nu) d \nu therefore should also go to infinity. In reality, of course, the energy density (and radiation rate) falls to 0 as $\nu \to \mathrm{\infty }$$\nu \to \mathrm{\infty }$nu rarr oo\nu \rightarrow \infty (Fig. 2.8). This discrepancy became known as the ultraviolet catastrophe of classical physics. Where did Rayleigh and Jeans go wrong?
即使僅僅看一下公式 (2.3)，也可以看出它不可能是正確的。當頻率 $\nu$$\nu$nu\nu 向光譜的紫外線端增加時，這個公式預測能量密度應該以 ${\nu }^2$${\nu }^2$nu^(2)\nu^{2} 增加。在無限高頻率的極限下， $u(\nu )d\nu$$u(\nu )d\nu$u(nu)d nuu(\nu) d \nu 因此也應該趨向無限。實際上，當 $\nu \to \mathrm{\infty }$$\nu \to \mathrm{\infty }$nu rarr oo\nu \rightarrow \infty 時，能量密度（和輻射率）當然會降至 0（圖 2.8）。這一差異被稱為古典物理學的紫外線災難。雷利和簡斯錯在哪裡？

Figure 2.7 Em radiation in a cavity whose walls are perfect reflectors consists of standing waves that have nodes at the walls, which restricts their possible wavelengths. Shown are three possible wavelengths when the distance between opposite walls is $L$$L$LL.
圖 2.7 在牆壁為完美反射體的腔體中，電磁輻射由具有在牆壁處的節點的駐波組成，這限制了它們可能的波長。顯示的是當對面牆壁之間的距離為 $L$$L$LL 時的三種可能波長。

Figure 2.8 Comparison of the Rayleigh-Jeans formula for the spectrum of the radiation from a blackbody at 1500 K with the observed spectrum. The discrepancy is known as the ultraviolet catastrophe because it increases with increasing frequency. This failure of classical physics led Planck to the discovery that radiation is emitted in quanta whose energy is $h\nu$$h\nu$h nuh \nu.
圖 2.8 1500 K 黑體輻射光譜的雷利-珍公式與觀測光譜的比較。這一差異被稱為紫外線災難，因為它隨著頻率的增加而增加。這一古典物理的失敗促使普朗克發現輻射是以量子形式發射的，其能量為 $h\nu$$h\nu$h nuh \nu 。

In 1900 the German physicist Max Planck used “lucky guesswork” (as he later called it) to come up with a formula for the spectral energy density of blackbody radiation:
在 1900 年，德國物理學家馬克斯·普朗克使用了他後來稱之為的“幸運猜測”來提出黑體輻射的光譜能量密度公式：

Here $h$$h$hh is a constant whose value is
這裡 $h$$h$hh 是一個常數，其值為

Max Planck (1858-1947) was born in Kiel and educated in Munich and Berlin. At the University of Berlin he studied under Kirchhoff and Helmholtz, as Hertz had done earlier. Planck realized that blackbody radiation was important because it was a fundamental effect independent of atomic structure, which was still a mystery in the late nineteenth century, and worked at understanding it for six years before finding the formula the radiation obeyed. He “strived from the day of its discovery to give it a real physical interpretation.” The result was the discovery that radiation is emitted in energy steps of $h\nu$$h\nu$h nuh \nu. Although this discovery, for which he received the Nobel Prize in 1918, is now considered to mark the start of
馬克斯·普朗克（1858-1947）出生於基爾，在慕尼黑和柏林接受教育。在柏林大學，他跟隨基爾霍夫和亥姆霍茲學習，正如赫茲早期所做的那樣。普朗克意識到黑體輻射的重要性，因為它是一種獨立於原子結構的基本效應，而在十九世紀末，原子結構仍然是一個謎。他花了六年時間來理解這一現象，最終找到了輻射遵循的公式。他“從發現的那一天起，就努力給它一個真正的物理解釋。”結果發現輻射是以能量步驟 $h\nu$$h\nu$h nuh \nu 發射的。儘管這一發現使他在 1918 年獲得了諾貝爾獎，但現在被認為標誌著一個新的開始。
modern physics, Planck himself remained skeptical for a long time of the physical reality of quanta. As he later wrote, “My vain attempts to somehow reconcile the elementary quantum with classical theory continued for many years and cost me great effort. . . . Now I know for certain that the quantum of action has a much more fundamental significance than I originally suspected.”
現代物理學中，普朗克本人對量子的物理現實長期持懷疑態度。正如他後來所寫的：“我徒勞的嘗試將基本量子與經典理論調和的努力持續了很多年，並耗費了我巨大的精力……現在我確信，作用量子具有比我最初懷疑的更根本的意義。”

Like many physicists, Planck was a competent musician (he sometimes played with Einstein) and in addition enjoyed mountain climbing. Although Planck remained in Germany during the Hitler era, he protested the Nazi treatment of Jewish scientists and lost his presidency of the Kaiser Wilhelm Institute as a result. In 1945 one of his sons was implicated in a plot to kill Hitler and was executed. After World War II the Institute was renamed after Planck and he was again its head until his death.
像許多物理學家一樣，普朗克是一位能幹的音樂家（他有時與愛因斯坦一起演奏），此外還喜歡登山。儘管普朗克在希特勒時代仍留在德國，但他抗議納粹對猶太科學家的待遇，結果失去了凱瑟威廉研究所的所長職位。1945 年，他的一個兒子捲入了一個刺殺希特勒的陰謀並被處決。第二次世界大戰後，該研究所以普朗克的名字重新命名，他再次成為所長，直到去世。

At high frequencies, $h\nu \gg kT$$h\nu \gg kT$h nu≫kTh \nu \gg k T and $e^{h\nu /kT}\to \mathrm{\infty }$$e^{h\nu /kT}\to \mathrm{\infty }$e^(h nu//kT)rarr ooe^{h \nu / k T} \rightarrow \infty, which means that $u(\nu )d\nu \to 0$$u(\nu )d\nu \to 0$u(nu)d nu rarr0u(\nu) d \nu \rightarrow 0 as observed. No more ultraviolet catastrophe. At low frequencies, where the RayleighJeans formula is a good approximation to the data (see Fig. 2.8), $h\nu \ll kT$$h\nu \ll kT$h nu≪kTh \nu \ll k T and $h\nu /kT$$h\nu /kT$h nu//kTh \nu / k T $\ll 1$$\ll 1$≪1\ll 1. In general,
在高頻率下， $h\nu \gg kT$$h\nu \gg kT$h nu≫kTh \nu \gg k T 和 $e^{h\nu /kT}\to \mathrm{\infty }$$e^{h\nu /kT}\to \mathrm{\infty }$e^(h nu//kT)rarr ooe^{h \nu / k T} \rightarrow \infty ，這意味著 $u(\nu )d\nu \to 0$$u(\nu )d\nu \to 0$u(nu)d nu rarr0u(\nu) d \nu \rightarrow 0 如觀察到的那樣。沒有更多的紫外線災難。在低頻率下，雷利-金斯公式對數據是一個良好的近似（見圖 2.8）， $h\nu \ll kT$$h\nu \ll kT$h nu≪kTh \nu \ll k T 和 $h\nu /kT$$h\nu /kT$h nu//kTh \nu / k T $\ll 1$$\ll 1$≪1\ll 1 。一般來說，

If $x$$x$xx is small, $e^x\approx 1+x$$e^x\approx 1+x$e^(x)~~1+xe^{x} \approx 1+x, and so for $h\nu /kT\ll 1$$h\nu /kT\ll 1$h nu//kT≪1h \nu / k T \ll 1 we have
如果 $x$$x$xx 很小， $e^x\approx 1+x$$e^x\approx 1+x$e^(x)~~1+xe^{x} \approx 1+x ，那麼對於 $h\nu /kT\ll 1$$h\nu /kT\ll 1$h nu//kT≪1h \nu / k T \ll 1 我們有

Thus at low frequencies Planck’s formula becomes
因此在低頻率下，普朗克公式變為

which is the Rayleigh-Jeans formula. Planck’s formula is clearly at least on the right track; in fact, it has turned out to be completely correct.
這就是瑞利-金斯公式。普朗克的公式顯然至少是朝著正確的方向；事實上，它已經被證明是完全正確的。

Next Planck had the problem of justifying Eq. (2.4) in terms of physical principles. A new principle seemed needed to explain his formula, but what was it? After several weeks of “the most strenuous work of my life,” Planck found the answer: The oscillators in the cavity walls could not have a continuous distribution of possible energies $ϵ$$ϵ$epsilon\epsilon but must have only the specific energies
接下來，普朗克面臨著根據物理原則證明方程式 (2.4) 的問題。似乎需要一個新的原則來解釋他的公式，但那是什麼呢？經過幾週“我一生中最艱苦的工作”，普朗克找到了答案：腔體壁中的振盪器不能有連續的可能能量分佈 $ϵ$$ϵ$epsilon\epsilon ，而必須只有特定的能量。

Oscillator energies ${ϵ}_n=nh\nu n=0,1,2,\dots$${ϵ}_n=nh\nu n=0,1,2,\dots$quadepsilon_(n)=nh nuquad n=0,1,2,dots\quad \epsilon_{n}=n h \nu \quad n=0,1,2, \ldots
振盪器能量 ${ϵ}_n=nh\nu n=0,1,2,\dots$${ϵ}_n=nh\nu n=0,1,2,\dots$quadepsilon_(n)=nh nuquad n=0,1,2,dots\quad \epsilon_{n}=n h \nu \quad n=0,1,2, \ldots

An oscillator emits radiation of frequency $\nu$$\nu$nu\nu when it drops from one energy state to the next lower one, and it jumps to the next higher state when it absorbs radiation of frequency $\nu$$\nu$nu\nu. Each discrete bundle of energy $h\nu$$h\nu$h nuh \nu is called a quantum (plural quanta) from the Latin for “how much.”
一個振盪器在從一個能量狀態降至下一個較低的狀態時，會發出頻率為 $\nu$$\nu$nu\nu 的輻射；而當它吸收頻率為 $\nu$$\nu$nu\nu 的輻射時，則會跳至下一個較高的狀態。每一個離散的能量包 $h\nu$$h\nu$h nuh \nu 被稱為量子（複數為量子）。

With oscillator energies limited to nh $\nu$$\nu$nu\nu, the average energy per oscillator in the cavity walls-and so per standing wave-turned out to be not $\overline{\mathit{ϵ}}=kT$$\overline{\mathit{ϵ}}=kT$bar(epsilon)=kT\overline{\boldsymbol{\epsilon}}=k T as for a continuous distribution of oscillator energies, but instead
由於振盪器能量限制在 nh $\nu$$\nu$nu\nu ，腔壁中每個振盪器的平均能量——因此每個駐波——結果並不是 $\overline{\mathit{ϵ}}=kT$$\overline{\mathit{ϵ}}=kT$bar(epsilon)=kT\overline{\boldsymbol{\epsilon}}=k T ，如同連續分佈的振盪器能量，而是

Actual average energy 實際平均能量
per standing wave 每個駐波

This average energy leads to Eq. (2.4). Blackbody radiation is further discussed in Chap. 9.
這個平均能量導致了方程式 (2.4)。黑體輻射在第九章中進一步討論。

Assume that a certain $660-\mathrm{Hz}$$660-\mathrm{Hz}$660-Hz660-\mathrm{Hz} tuning fork can be considered as a harmonic oscillator whose vibrational energy is 0.04 J . Compare the energy quanta of this tuning fork with those of an atomic oscillator that emits and absorbs orange light whose frequency is $5.00\times {10}^{14}\mathrm{Hz}$$5.00\times {10}^{14}\mathrm{Hz}$5.00 xx10^(14)Hz5.00 \times 10^{14} \mathrm{~Hz}.
假設某個 $660-\mathrm{Hz}$$660-\mathrm{Hz}$660-Hz660-\mathrm{Hz} 音叉可以被視為一個諧波振盪器，其振動能量為 0.04 焦耳。將這個音叉的能量量子與發射和吸收橙光的原子振盪器的能量量子進行比較，其頻率為 $5.00\times {10}^{14}\mathrm{Hz}$$5.00\times {10}^{14}\mathrm{Hz}$5.00 xx10^(14)Hz5.00 \times 10^{14} \mathrm{~Hz} 。

(a) For the tuning fork,
（a）對於音叉，

The total energy of the vibrating tines of the fork is therefore about ${10}^{29}$${10}^{29}$10^(29)10^{29} times the quantum energy $h\nu$$h\nu$h nuh \nu. The quantization of energy in the tuning fork is obviously far too small to be observed, and we are justified in regarding the fork as obeying classical physics.
因此，這把音叉的振動齒的總能量約為 ${10}^{29}$${10}^{29}$10^(29)10^{29} 倍的量子能量 $h\nu$$h\nu$h nuh \nu 。音叉中能量的量子化顯然小到無法觀察，我們有理由認為這把音叉遵循經典物理學。
(b) For the atomic oscillator,
(b) 對於原子振盪器，

In electronvolts, the usual energy unit in atomic physics,
在電子伏特中，原子物理學中常用的能量單位，

This is a significant amount of energy on an atomic scale, and it is not surprising that classical physics fails to account for phenomena on this scale.
這在原子尺度上是一個重要的能量數量，經典物理無法解釋這一尺度上的現象也就不足為奇。

The concept that the oscillators in the cavity walls can interchange energy with standing waves in the cavity only in quanta of $h\mathit{\nu }$$h\mathit{\nu }$h nuh \boldsymbol{\nu} is, from the point of view of classical physics, impossible to understand. Planck regarded his quantum hypothesis as an “act of desperation” and, along with other physicists of his time, was unsure of how seriously to regard it as an element of physical reality. For many years he held that, although the energy transfers between electric oscillators and em waves apparently are quantized, em waves themselves behave in an entirely classical way with a continuous range of possible energies.
在腔體壁中的振盪器只能以 $h\mathit{\nu }$$h\mathit{\nu }$h nuh \boldsymbol{\nu} 的量子形式與腔體中的駐波互換能量的概念，從古典物理的角度來看，是無法理解的。普朗克將他的量子假說視為一種「絕望的行為」，並且與他那個時代的其他物理學家一樣，對於應該多認真地看待它作為物理現實的一個元素感到不確定。多年來，他認為，儘管電振盪器與電磁波之間的能量轉移顯然是量子化的，但電磁波本身的行為則完全是古典的，具有連續範圍的可能能量。

During his experiments on em waves, Hertz noticed that sparks occurred more readily in the air gap of his transmitter when ultraviolet light was directed at one of the metal balls. He did not follow up this observation, but others did. They soon discovered that the cause was electrons emitted when the frequency of the light was sufficiently high. This phenomenon is known as the photoelectric effect and the emitted electrons are called photoelectrons. It is one of the ironies of history that the same work to demonstrate that light consists of em waves also gave the first hint that this was not the whole story.
在他對電磁波的實驗中，赫茲注意到當紫外線照射到其中一個金屬球時，發射器的空氣間隙中更容易產生火花。他並沒有進一步研究這一觀察，但其他人卻進行了跟進。他們很快發現，這是因為當光的頻率足夠高時會釋放出電子。這一現象被稱為光電效應，釋放出的電子稱為光電子。歷史的諷刺之一是，證明光由電磁波組成的同一工作也首次暗示了這並不是全部的故事。

Figure 2.9 shows how the photoelectric effect was studied. An evacuated tube contains two electrodes connected to a source of variable voltage, with the metal plate whose surface is irradiated as the anode. Some of the photoelectrons that emerge from this surface have enough energy to reach the cathode despite its negative polarity, and they constitute the measured current. The slower photoelectrons are repelled before they get to the cathode. When the voltage is increased to a certain value $V_0$$V_0$V_(0)V_{0}, of the order of several volts, no more photoelectrons arrive, as indicated by the current dropping to zero. This extinction voltage corresponds to the maximum photoelectron kinetic energy.
圖 2.9 顯示了光電效應的研究方式。一個抽空的管子內含有兩個電極，連接到可變電壓的電源，金屬板的表面被照射作為陽極。一些從這個表面釋放出來的光電子具有足夠的能量能夠抵達陰極，儘管陰極帶有負電，這些光電子構成了測量的電流。較慢的光電子在到達陰極之前會被排斥。當電壓增加到某個值 $V_0$$V_0$V_(0)V_{0} ，大約幾伏特的時候，將不再有光電子到達，這表明電流降至零。這個熄滅電壓對應於光電子的最大動能。

Figure 2.9 Experimental observation of the photoelectric effect.
圖 2.9 光電效應的實驗觀察。

The existence of the photoelectric effect is not surprising. After all, light waves carry energy, and some of the energy absorbed by the metal may somehow concentrate on individual electrons and reappear as their kinetic energy. The situation should be like water waves dislodging pebbles from a beach. But three experimental findings show that no such simple explanation is possible.
光電效應的存在並不令人驚訝。畢竟，光波攜帶能量，而金屬吸收的部分能量可能以某種方式集中在個別電子上，並以它們的動能重新出現。這種情況應該就像水波將卵石從海灘上沖走一樣。但三個實驗結果顯示，沒有這樣簡單的解釋是可能的。

1 Within the limits of experimental accuracy (about ${10}^{-9}\mathrm{s}$${10}^{-9}\mathrm{s}$10^(-9)s10^{-9} \mathrm{~s} ), there is no time interval between the arrival of light at a metal surface and the emission of photoelectrons. However, because the energy in an em wave is supposed to be spread across the wavefronts, a period of time should elapse before an individual electron accumulates enough energy (several eV ) to leave the metal. A detectable photoelectron current results when ${10}^{-6}$${10}^{-6}$10^(-6)10^{-6} $\mathrm{W}/{\mathrm{m}}^2$$\mathrm{W}/{\mathrm{m}}^2$W//m^(2)\mathrm{W} / \mathrm{m}^{2} of em energy is absorbed by a sodium surface. A layer of sodium 1 atom thick and $1{\mathrm{m}}^2$$1{\mathrm{m}}^2$1m^(2)1 \mathrm{~m}^{2} in area contains about ${10}^{19}$${10}^{19}$10^(19)10^{19} atoms, so if the incident light is absorbed in the uppermost atomic layer, each atom receives energy at an average rate of ${10}^{-25}\mathrm{W}$${10}^{-25}\mathrm{W}$10^(-25)W10^{-25} \mathrm{~W}. At this rate over a month would be needed for an atom to accumulate energy of the magnitude that photoelectrons from a sodium surface are observed to have.
在實驗精度的限制內（約 ${10}^{-9}\mathrm{s}$${10}^{-9}\mathrm{s}$10^(-9)s10^{-9} \mathrm{~s} ），光到達金屬表面與光電子的發射之間沒有時間間隔。然而，由於電磁波的能量被認為是分佈在波前上，因此在單個電子積累足夠的能量（幾個電子伏特）以離開金屬之前，應該會經過一段時間。當 ${10}^{-6}$${10}^{-6}$10^(-6)10^{-6} $\mathrm{W}/{\mathrm{m}}^2$$\mathrm{W}/{\mathrm{m}}^2$W//m^(2)\mathrm{W} / \mathrm{m}^{2} 的電磁能量被鈉表面吸收時，會產生可檢測的光電子電流。一層厚度為 1 個原子的鈉，面積為 $1{\mathrm{m}}^2$$1{\mathrm{m}}^2$1m^(2)1 \mathrm{~m}^{2} ，大約包含 ${10}^{19}$${10}^{19}$10^(19)10^{19} 個原子，因此如果入射光被吸收在最上面的原子層，每個原子平均以 ${10}^{-25}\mathrm{W}$${10}^{-25}\mathrm{W}$10^(-25)W10^{-25} \mathrm{~W} 的速率接收能量。以這個速率，原子需要一個月的時間才能積累到光電子從鈉表面觀察到的能量大小。
2 A bright light yields more photoelectrons than a dim one of the same frequency, but the electron energies remain the same (Fig. 2.10). The em theory of light, on the contrary, predicts that the more intense the light, the greater the energies of the electrons. 3 The higher the frequency of the light, the more energy the photoelectrons have (Fig. 2.11). Blue light results in faster electrons than red light. At frequencies below a certain critical frequency ${\nu }_0$${\nu }_0$nu_(0)\nu_{0}, which is characteristic of each particular metal, no electrons are emitted. Above ${\nu }_0$${\nu }_0$nu_(0)\nu_{0} the photoelectrons range in energy from 0 to a maximum value that increases linearly with increasing frequency (Fig. 2.12). This observation, also, cannot be explained by the em theory of light.
明亮的光比同頻率的微弱光產生更多的光電子，但電子的能量保持不變（圖 2.10）。相反，光的電磁理論預測光越強，電子的能量越大。光的頻率越高，光電子的能量越大（圖 2.11）。藍光產生的電子比紅光更快。在某個特定金屬的臨界頻率 ${\nu }_0$${\nu }_0$nu_(0)\nu_{0} 以下，沒有電子被釋放。在 ${\nu }_0$${\nu }_0$nu_(0)\nu_{0} 以上，光電子的能量範圍從 0 到隨著頻率增加而線性增長的最大值（圖 2.12）。這一觀察同樣無法用光的電磁理論解釋。

When Planck’s derivation of his formula appeared, Einstein was one of the firstperhaps the first-to understand just how radical the postulate of energy quantization
當普朗克的公式推導出現時，愛因斯坦是最早理解能量量子化假設有多麼激進的人之一，甚至可能是第一個

Figure 2.10 Photoelectron current is proportional to light intensity I for all retarding voltages. The stopping potential $V_0$$V_0$V_(0)V_{0}, which corresponds to the maximum photoelectron energy, is the same for all intensities of light of the same frequency $\nu$$\nu$nu\nu.
圖 2.10 光電子電流與所有減速電壓下的光強度 I 成正比。對應於最大光電子能量的停止電壓 $V_0$$V_0$V_(0)V_{0} ，對於相同頻率的所有光強度都是相同的 $\nu$$\nu$nu\nu 。

Figure 2.11 The stopping potential $V_0$$V_0$V_(0)V_{0}, and hence the maximum photoelectron energy, depends on the frequency of the light. When the retarding potential is $V=0$$V=0$V=0V=0, the photoelectron current is the same for light of a given intensity regardless of its frequency.
圖 2.11 停止電壓 $V_0$$V_0$V_(0)V_{0} ，因此最大光電子能量，取決於光的頻率。當反向電壓為 $V=0$$V=0$V=0V=0 時，對於給定強度的光，光電子電流與其頻率無關。

Figure 2.12 Maximum photoelectron kinetic energy ${\mathrm{KE}}_{max}$${\mathrm{KE}}_{max}$KE_(max)\mathrm{KE}_{\max } versus frequency of incident light for three metal surfaces.
圖 2.12 三種金屬表面入射光頻率對應的最大光電子動能 ${\mathrm{KE}}_{max}$${\mathrm{KE}}_{max}$KE_(max)\mathrm{KE}_{\max } 。
of oscillators was: “It was as if the ground was pulled from under one.” A few years later, in 1905, Einstein realized that the photoelectric effect could be understood if the energy in light is not spread out over wavefronts but is concentrated in small packets, or photons. (The term photon was coined by the chemist Gilbert Lewis in 1926.) Each photon of light of frequency $\nu$$\nu$nu\nu has the energy $h\nu$$h\nu$h nuh \nu, the same as Planck’s quantum energy. Planck had thought that, although energy from an electric oscillator apparently had to be given to em waves in separate quanta of $h\nu$$h\nu$h nuh \nu each, the waves themselves behaved exactly as in conventional wave theory. Einstein’s break with classical physics was more drastic: Energy was not only given to em waves in separate quanta but was also carried by the waves in separate quanta.
振盪器的描述是：「就像地面被從腳下拉走了一樣。」幾年後，在 1905 年，愛因斯坦意識到，如果光中的能量不是分散在波前上，而是集中在小包裹中，或稱為光子，那麼光電效應就可以被理解。（光子這個術語是由化學家吉爾伯特·路易斯於 1926 年創造的。）頻率為 $\nu$$\nu$nu\nu 的每個光子具有能量 $h\nu$$h\nu$h nuh \nu ，這與普朗克的量子能量相同。普朗克曾認為，儘管來自電振盪器的能量顯然必須以每個 $h\nu$$h\nu$h nuh \nu 的單獨量子形式給予電磁波，但這些波本身的行為與傳統波動理論完全相同。愛因斯坦與經典物理學的分歧更為徹底：能量不僅以單獨量子的形式給予電磁波，還由波以單獨量子的形式攜帶。

The three experimental observations listed above follow directly from Einstein’s hypothesis. (1) Because em wave energy is concentrated in photons and not spread out, there should be no delay in the emission of photoelectrons. (2) All photons of frequency $\nu$$\nu$nu\nu have the same energy, so changing the intensity of a monochromatic light beam will change the number of photoelectrons but not their energies. (3) The higher the frequency $\nu$$\nu$nu\nu, the greater the photon energy $h\nu$$h\nu$h nuh \nu and so the more energy the photoelectrons have.
上述三個實驗觀察直接源自愛因斯坦的假設。(1) 因為電磁波能量集中在光子中而不是分散開來，因此光電子的發射不應有延遲。(2) 所有頻率為 $\nu$$\nu$nu\nu 的光子具有相同的能量，因此改變單色光束的強度會改變光電子的數量，但不會改變它們的能量。(3) 頻率越高 $\nu$$\nu$nu\nu ，光子能量 $h\nu$$h\nu$h nuh \nu 越大，因此光電子擁有的能量也越多。

What is the meaning of the critical frequency ${\nu }_0$${\nu }_0$nu_(0)\nu_{0} below which no photoelectrons are emitted? There must be a minimum energy $\varphi$$\varphi$phi\phi for an electron to escape from a particular metal surface or else electrons would pour out all the time. This energy is called the work function of the metal, and is related to ${\nu }_0$${\nu }_0$nu_(0)\nu_{0} by the formula
臨界頻率 ${\nu }_0$${\nu }_0$nu_(0)\nu_{0} 的意義是什麼？在這個頻率以下不會發射光電子。電子必須具備最低能量 $\varphi$$\varphi$phi\phi 才能從特定金屬表面逃逸，否則電子會不斷流出。這個能量稱為金屬的功函數，並且與 ${\nu }_0$${\nu }_0$nu_(0)\nu_{0} 之間的關係由以下公式表示：

Work function 功函數

The greater the work function of a metal, the more energy is needed for an electron to leave its surface, and the higher the critical frequency for photoelectric emission to occur.
金屬的功函數越大，電子離開其表面所需的能量就越多，光電發射發生的臨界頻率也越高。

Some examples of photoelectric work functions are given in Table 2.1. To pull an electron from a metal surface generally takes about half as much energy as that needed
一些光電功函數的例子列在表 2.1 中。從金屬表面拉出一個電子通常需要的能量大約是所需能量的一半。

Table 2.1 Photoelectric Work Functions
表 2.1 光電功函數

to pull an electron from a free atom of that metal (see Fig. 7.10); for instance, the ionization energy of cesium is 3.9 eV compared with its work function of 1.9 eV . Since the visible spectrum extends from about 4.3 to about $7.5\times {10}^{14}\mathrm{Hz}$$7.5\times {10}^{14}\mathrm{Hz}$7.5 xx10^(14)Hz7.5 \times 10^{14} \mathrm{~Hz}, which corresponds to quantum energies of 1.7 to 3.3 eV , it is clear from Table 2.1 that the photoelectric effect is a phenomenon of the visible and ultraviolet regions.
要從該金屬的自由原子中拉出一個電子（見圖 7.10）；例如，銫的電離能為 3.9 eV，而其功函數為 1.9 eV。由於可見光譜的範圍約為 4.3 到 $7.5\times {10}^{14}\mathrm{Hz}$$7.5\times {10}^{14}\mathrm{Hz}$7.5 xx10^(14)Hz7.5 \times 10^{14} \mathrm{~Hz} ，這對應於量子能量的 1.7 到 3.3 eV，從表 2.1 中可以清楚地看出，光電效應是可見光和紫外線區域的現象。

According to Einstein, the photoelectric effect in a given metal should obey the equation
根據愛因斯坦的說法，給定金屬中的光電效應應遵循以下方程式

Photoelectric effect $h\nu ={\mathrm{KE}}_{max}+\varphi$$h\nu ={\mathrm{KE}}_{max}+\varphi$quad h nu=KE_(max)+phi\quad h \nu=\mathrm{KE}_{\max }+\phi
光電效應 $h\nu ={\mathrm{KE}}_{max}+\varphi$$h\nu ={\mathrm{KE}}_{max}+\varphi$quad h nu=KE_(max)+phi\quad h \nu=\mathrm{KE}_{\max }+\phi
where $h\nu$$h\nu$h nuh \nu is the photon energy, ${\mathrm{KE}}_{max}$${\mathrm{KE}}_{max}$KE_(max)\mathrm{KE}_{\max } is the maximum photoelectron energy (which is proportional to the stopping potential), and $\varphi$$\varphi$phi\phi is the minimum energy needed for an
其中 $h\nu$$h\nu$h nuh \nu 是光子能量， ${\mathrm{KE}}_{max}$${\mathrm{KE}}_{max}$KE_(max)\mathrm{KE}_{\max } 是最大光電子能量（與停止電壓成正比），而 $\varphi$$\varphi$phi\phi 是所需的最小能量

All light-sensitive detectors, including the eye and the one used in this video camera, are based on the absorption of energy from photons of light by electrons in the atoms the light falls on.
所有光敏感探測器，包括眼睛和這個攝影機中使用的探測器，都是基於光子能量被光照射到的原子中的電子吸收。

Figure 2.13 If the energy $h{\nu }_0$$h{\nu }_0$hnu_(0)h \nu_{0} (the work function of the surface) is needed to remove an electron from a metal surface, the maximum electron kinetic energy will be $h\nu -h{\nu }_0$$h\nu -h{\nu }_0$h nu-hnu_(0)h \nu-h \nu_{0} when light of frequency $\nu$$\nu$nu\nu is directed at the surface.
圖 2.13 如果需要能量 $h{\nu }_0$$h{\nu }_0$hnu_(0)h \nu_{0} （表面的功函數）來從金屬表面移除一個電子，當頻率為 $\nu$$\nu$nu\nu 的光照射到表面時，最大電子動能將為 $h\nu -h{\nu }_0$$h\nu -h{\nu }_0$h nu-hnu_(0)h \nu-h \nu_{0} 。
electron to leave the metal. Because $\varphi =h{\nu }_0$$\varphi =h{\nu }_0$phi=hnu_(0)\phi=h \nu_{0}, Eq. (2.8) can be rewritten (Fig. 2.13)
電子離開金屬。因為 $\varphi =h{\nu }_0$$\varphi =h{\nu }_0$phi=hnu_(0)\phi=h \nu_{0} ，方程式 (2.8) 可以重寫為 (圖 2.13)