We can give a probability density p1(x), such that p1(x)Δx is the probability that the particle will be found between x and x+Δx . If the particle is reasonably well localized, say near x0, the function p1(x) might be given by the graph of Fig. 6–10(a).Similarly, we must specify the velocity of the particle by means of aprobability density p2(v) , with p2(v)Δv the probability that the velocity will be found between v and v+Δv .
我们可以提供一个概率密度 p1(x)，这样，p1(x)Δx就是粒子被发现处于x和 x+Δx之间的概率。如果粒子是合理分布的，即是在x附近，那么，公式p1(x)就可以通过图6-10（a）的曲线来给予。类似地，我们可以通过概率密度p2(v)，来说明粒子的速度，p2(v)Δv就是粒子被发现处于v和 v+Δv之间的概率。


Fig. 6–10.Probability densities for observationof the position and velocity of a particle. ；图6-10 对一个粒子，观察其位置和速度，所得到的概率密度。

It is one of the fundamental results ofquantum mechanics that the two functions p1(x) and p2(v) cannot be chosen independently and, in particular, cannot both be madearbitrarily narrow. If we call the typical “width” of the p1(x) curve [Δx] , and that of the p2(v) curve [Δv] (as shown in the figure), nature demands that the product ofthe two widths be at least as big as the number ℏ/2m , where m is the mass of the particle. We may write this basic relationship as
[Δx]⋅[Δv]≥ℏ/2m. (6.22)
This equation is a statement of the Heisenberg uncertainty principlethat we mentioned earlier.
方程p1(x)和p2(v)都不能被独立地选择，特别是，不可能两个都是任意小，这是量子力学的基本结果之一。如果我们把p1(x)曲线的一般宽度，称为[Δx]，把p2(v)曲线的一般宽度，称为[Δv]，那么，自然要求，两个宽度的乘积，至少要与数字 ℏ/2m一样大，这里m是粒子的质量。我们可把这个基本关系写成：
[Δx]⋅[Δv]≥ℏ/2m. (6.22)
这个方程就是我们早先提到过的海森伯格测不准原理的表达。

Since the right-hand side of Eq. (6.22)is a constant, this equation says that if we try to “pin down” a particle byforcing it to be at a particular place, it ends up by having a high speed. Orif we try to force it to go very slowly, or at a precise velocity, it “spreadsout” so that we do not know very well just where it is. Particles behave in afunny way!
由于方程(6.22)的右边是一个常数，这个方程就是在说，如果我们想把一个粒子“钉住”，强迫它在一个具体的地点，那么，它就会有一个高的速度。或者，如果我们想让它走地非常慢，或有一个精确的速度，那么，它就会“散开”，于是，它究竟在哪里，我们就不会确切地知道。粒子的表现很有趣！

The uncertainty principle describes aninherent fuzziness that must exist in any attempt to describe nature. Our mostprecise description of nature must be in terms of probabilities.There are some people who do not like this way of describing nature. They feelsomehow that if they could only tell what is really going on with aparticle, they could know its speed and position simultaneously. In the earlydays of the development of quantum mechanics, Einstein was quite worried aboutthis problem. He used to shake his head and say, “But, surely God does notthrow dice in determining how electrons should go!” He worried about thatproblem for a long time and he probably never really reconciled himself to thefact that this is the best description of nature that one can give. There are stillone or two physicists who are working on the problem who have an intuitiveconviction that it is possible somehow to describe the world in a different wayand that all of this uncertainty about the way things are can be removed. Noone has yet been successful.
测不准原理描述了一种内在的模糊性，在任何描述自然的尝试中，它都存在。我们对自然最精确的描述，应该是用概率这种词汇来做。这种描述自然的方法，有些人不喜欢。由于某种不明原因，他们感觉到，如果他们能够准确地告诉粒子中正在发生的事情，那么，他们就可以同时知道其速度和位置。在量子力学发展的早期，爱因斯坦对这个问题非常担忧。他经常习惯性地摇着头说道：“上帝确实并没有掷骰子，以决定电子该如何走啊！”这个问题，他担忧了很长时间，他大概从来就没有真正说服他自己，同意这一事实：这是一个人所能给出的关于自然的最佳描述。研究这个问题的，还有另外一两位物理学家，他们直觉地坚信，描述这个世界，完全可能还有不同的方法，那么，这种认为事物是不确定的方法，就可被移除。但迄今为止，尚无人成功过。

The necessary uncertainty in ourspecification of the position of a particle becomes most important when we wishto describe the structure of atoms. In the hydrogen atom, which has a nucleusof one proton with one electron outside of the nucleus, the uncertainty in theposition of the electron is as large as the atom itself! We cannot, therefore,properly speak of the electron moving in some “orbit” around the proton. The mostwe can say is that there is a certain chance p(r)ΔV, of observing the electron in an element of volume ΔV at the distance r from the proton. The probability density p(r) is given by quantum mechanics. For an undisturbed hydrogen atom p(r)=Ae−2r/a. The number a is the “typical” radius, where the function is decreasing rapidly.Since there is a small probability of finding the electron at distances fromthe nucleus much greater than a , we may think of a as “the radius of the atom,” about 10−10 meter.
在我们关于粒子位置的详细说明中，测不准是必要的，当我们的希望描述原子的结构时，此必要性就变得最为重要。氢原子，由一个质子的原子核，和一个原子核外的电子组成，在氢原子中，电子位置的测不准，正如氢原子本身的大小一样！因此，我们不能合适地说，电子是在某个绕着质子的“轨道”上运行。我们最多只能说，在距离质子r处的一个体积为ΔV的元素中，观察到电子的机会为p(r)ΔV。概率密度p(r)是通过量子力学给出的。对于一个未被干扰的氢原子，p(r)=Ae−2r/a。数字a是“典型的”半经，这里函数衰减的很快。由于在距离原子核远大于a处，发现电子的概率很小，所以我们可以把a认为是“原子核的半径”，约10-10米。

Fig. 6–11.A way of visualizing a hydrogenatom. The density (whiteness) of the cloud represents the probability densityfor observing the electron. 图6-11 一种视觉化{想象}氢原子的方法。云的密度（白色部分）代表着能观察到电子的概率密度。

We can form an image of the hydrogen atom byimagining a “cloud” whose density is proportional to the probability density forobserving the electron. A sample of such a cloud is shown in Fig. 6–11. Thusour best “picture” of a hydrogen atom is a nucleus surrounded by an “electroncloud” (although we really mean a “probability cloud”). The electron isthere somewhere, but nature permits us to know only the chance offinding it at any particular place.
我们可以通过想象一团“云”，来形成氢原子的图像，此云的密度，正比于观察电子{所能得到}的概率密度。图6-11所示，就是这样一种云的例子。这样，我们关于一个原子核的最好的“图片”，就是一个原子核，被“电子云”笼罩着（虽然我们的真正意思是“概率云”）。电子就在那里的某处，但是，自然允许我们知道的，只是在某个具体地方发现它的机会。

In its efforts to learn as much as possibleabout nature, modern physics has found that certain things can never be “known”with certainty. Much of our knowledge must always remain uncertain. The mostwe can know is in terms of probabilities.
现代物理学，尽其所能，对自然进行探索，它发现，有些事物，我们永远也不可能确定地知道。我们的一些知识，应该总是测不准的。我们最多只能用概率这种词来描述。

1. After the first three games, the experiment was actually done byshaking 30 pennies violently in a box and then counting the number of headsthat showed.
脚注1、在前三局之后，这个实验实际上是这样做的，猛摇一个盒子里面的30个便士，然后数正面的数目。
2. Maxwell’s expression is p(v)=Cv2e−avv, where a is a constant related to the temperature and C is chosen so that the total probability is one.
脚注2、麦克斯韦的表达式是p(v)=Cv2e−avv，这里a是一个与温度有关的常数，选择C，以使整个概率为1。

1 Chapter7. The Theory of Gravitation 第七章 万有引力的理论 7–1Planetary motions 7-1 行星的运动
In this chapter we shall discuss one of the most far-reaching generalizations of the human mind. While we are admiring the human mind, we should take some time off to stand in awe of a nature that could follow with such completeness and generality such an elegantly simple principle as the law of gravitation. What is this law of gravitation? It is that every object in the universe attracts every other object with a force which for any two bodies is proportional to the mass of each and varies inversely as the square of the distance between them. This statement can be expressed mathematically by the equation
F=Gmm′/(r2).
在本章中，我们将讨论人类心灵的最远的概括之一。在我们敬仰人类心灵时，我们应该拿出一些时间来，敬畏自然，自然是如此完整且普遍地遵循着一个优雅简单的原理，即万有引力的规律。什么是万有引力的规律呢？那就是宇宙间的对象，都用一种力，吸引着所有其他的对象，该力对任何两个物体说，正比于其质量，反比于其距离的平方。这个陈述，可用数学公式这样表达：
F=Gmm′/(rr).

If to this we add the fact that an object responds to a force byaccelerating in the direction of the force by an amount that is inversely proportionalto the mass of the object, we shall have said everything required, for asufficiently talented mathematician could then deduce all the consequences ofthese two principles. However, since you are not assumed to be sufficientlytalented yet, we shall discuss the consequences in more detail, and not justleave you with only these two bare principles. We shall briefly relate thestory of the discovery of the law of gravitation and discuss some of itsconsequences, its effects on history, the mysteries that such a law entails,and some refinements of the law made by Einstein; we shall also discuss the relationshipsof the law to the other laws of physics. All this cannot be done in onechapter, but these subjects will be treated in due time in subsequent chapters.
如果对这个公式，我们再增加一个事实，即一个对象，会对一个力作出反应，即在这个力的方向上加速，且加速的量，与对象的质量成反比，那么，凡所需要，皆以说完，因为一个足够天才的数学家，将会推导出这两个原理的后果。然而，由于你们还没有被假定为是足够天才的，所以，我们将更仔细地讨论这个后果，而不是只告诉你这两条原理。我们将会简单地讲述万有引力规律被发现的故事，讨论它的一些后果，它在历史上的作用，这样一条规律所牵扯到的神秘，及爱因斯坦对这个规律所做的改进；我们还将他讨论，这个规律与其他物理学规律的关系。所有这些，不可能在一章中完成，但是，这些主题，将会在后续章节、合适的时间，被讨论。

The story begins with the ancients observingthe motions of planets among the stars, and finally deducing that they wentaround the sun, a fact that was rediscovered later by Copernicus. Exactly howthe planets went around the sun, with exactly what motion, took a littlemore work to discover. Beginning in the sixteenth century there were great debatesas to whether they really went around the sun or not. Tycho Brahe had an ideathat was different from anything proposed by the ancients: his idea was thatthese debates about the nature of the motions of the planets would best beresolved if the actual positions of the planets in the sky were measuredsufficiently accurately. If measurement showed exactly how the planets moved,then perhaps it would be possible to establish one or another viewpoint. Thiswas a tremendous idea—that to find something out, it is better to perform somecareful experiments than to carry on deep philosophical arguments. Pursuingthis idea, Tycho Brahe studied the positions of the planets for many years inhis observatory on the island of Hven, near Copenhagen. He made voluminoustables, which were then studied by the mathematician Kepler, after Tycho’sdeath. Kepler discovered from the data some very beautiful and remarkable, butsimple, laws regarding planetary motion.
故事开始于古人对行星在恒星之间运动的观察，最后得出，它们是绕着太阳在运动，这一事实，后来又被哥白尼重新发现。行星究竟是如何绕着太阳运动的呢？以什么样的运动方式呢？这都需要更多的工作，才能发现。从16世纪开始，关于行星是否真的绕着太阳运行，有大量的讨论。第谷的想法，与古人的都不相同，他的想法就是，如果行星在天空中的实际位置，能够被充分准确地测量到，那么，这些关于行星运动本质的辩论，就可解决。如果测量能准确地指出，行星是如何运动的，那么，就有可能建立这样或那样的观点。这个想法，非常精彩，即如果要找出某事的真相，最好是做仔细的实验，而不是去进行深刻的哲学辩论。为了实现这个想法，第谷在接近哥本哈根的汶岛上，建立他的观察站，研究行星的位置多年。他做了大量的表，在他死后，数学家开普勒研究了这些表。从这些数据出发，开普勒发现了一些关于行星运动的规律，这些规律，非常漂亮，引人注目，同时也很简单。

7–2Kepler’s laws 7-2 开普勒的规律
First of all, Kepler found that each planetgoes around the sun in a curve called an ellipse, with the sun at afocus of the ellipse. An ellipse is not just an oval, but is a very specificand precise curve that can be obtained by using two tacks, one at each focus, aloop of string, and a pencil; more mathematically, it is the locus of allpoints the sum of whose distances from two fixed points (the foci) is aconstant. Or, if you will, it is a foreshortened circle (Fig. 7–1).
首先，开普勒发现，每个行星绕太阳所走的曲线，都是一个椭圆，太阳在椭圆的一个焦点上。一个椭圆，并不仅仅是一个卵型曲线，而是一个特别的和精确的曲线，可以通过两个图钉，每一个放在一个焦点上，一段绳子，和一支铅笔得到，从数学上讲，就是所有距两个点（焦点）的距离之和为常数的点所组成的曲线。或者，如果你愿意，他是一个缩短了的圆（图7-1）。{压扁了的？}

Fig. 7–1.An ellipse. 图7-1 一个椭圆。

Kepler’s second observation was that the planets do not go around the sun at a uniform speed, but move faster when they are nearer the sun and more slowly when they are farther from the sun, in precisely this way: Suppose a planet is observed at any two successive times, let us say a week apart, and that the radius vector1 is drawn to the planet for each observed position. The orbital arc traversed by the planet during the week, and the two radius vectors, bound a certain plane area, the shaded area shown in Fig. 7–2. If two similar observations are made a week apart, at a part of the orbit farther from the sun (where the planet moves more slowly), the similarly bounded area is exactly the same as in the first case. So, in accordance with the second law, the orbital speed of each planet is such that the radius “sweeps out” equal areas in equal times.
开普勒的第二个观察，就是行星绕太阳运行的速度，并非匀速，而是近处较快，远处较慢，确切地说是这样：假设一个行星，在两个连续的时间点上被观察，比如一周，那么，可以为行星的每一个位置，都画出其半径向量（脚注1）。在这一周内，行星所经过的轨道弧，与这两个半径向量，围成了一个平面，见图7-2中的阴影区域。如果两个类似的观察，都是每隔一周的，在轨道远离太阳的部分（这里行星移动慢的多），所围成的面积与第一种的相同。于是，依据第二规律，每个行星的轨道速度就是这样：在相同的时间里，半径“扫过”的面积相同。

Fig. 7–2.Kepler’s law of areas. 图7-2 开普勒的面积规律
1. A radius vector is a line drawn from the sun to any point in a planet’s orbit.
脚注1、一个半径向量，就是从从太阳，向行星轨道上的任一点的所划的直线。

Finally, a third law was discovered byKepler much later; this law is of a different category from the other two,because it deals not with only a single planet, but relates one planet toanother. This law says that when the orbital period and orbit size of any twoplanets are compared, the periods are proportional to the 3/2 power of the orbit size. In this statement the period is the timeinterval it takes a planet to go completely around its orbit, and the size ismeasured by the length of the greatest diameter of the elliptical orbit,technically known as the major axis. More simply, if the planets went incircles, as they nearly do, the time required to go around the circle would beproportional to the 3/2 power of the diameter (or radius). Thus Kepler’s three laws are:
I. Each planet moves around thesun in an ellipse, with the sun at one focus.
II. The radius vector from the sunto the planet sweeps out equal areas in equal intervals of time.
III. The squares of the periods of anytwo planets are proportional to the cubes of the semimajor axes of theirrespective orbits: T∝a3/2 .
最后，是开普勒发现的第三定律，这是很久之后的事情了。这条规律，与另外两条，不是一类，因为它要面对的不是单个的行星，而是与两个行星有关。这条规律说，当比较任何两个行星的轨道周期和轨道尺寸时，周期正比于轨道尺寸的3/2次方。在此陈述中，周期是行星完整地绕轨道一周的时间，而尺寸，则是通过椭圆轨道的最大半径来衡量的，严格地说，就是主轴。更简单地说，如果行星的轨道是圆--它们几乎就是这样，那么，走完圆周所需的时间，正比于直径（或半径）的3/2次方。这样，就有开普勒的三条规律：
1、每个行星的都是按一条椭圆绕太阳运行，太阳在一个焦点上。
2、在相同的时间里，从太阳到行星的矢量半径所扫过的面积相同。
3、任何两个行星的周期的平方，正比于其轨道的半主轴的立方：T∝a3/2。

7–3Development of dynamics 7-3 力学的发展
While Kepler was discovering these laws,Galileo was studying the laws of motion. The problem was, what makes the planetsgo around? (In those days, one of the theories proposed was that the planetswent around because behind them were invisible angels, beating their wings anddriving the planets forward. You will see that this theory is now modified! Itturns out that in order to keep the planets going around, the invisible angelsmust fly in a different direction and they have no wings. Otherwise, it is asomewhat similar theory!) Galileo discovered a very remarkable fact aboutmotion, which was essential for understanding these laws. That is the principleof inertia—if something is moving, with nothing touching it and completelyundisturbed, it will go on forever, coasting at a uniform speed in a straightline. (Why does it keep on coasting? We do not know, but that is the wayit is.)
当开普勒在尝试发现这些规律的时候，伽利略则在研究运动的规律。这里的问题就是，究竟是什么，使得行星转着走？（在那时，有一种理论说，行星之所以转着走，乃是因为，在其后面，有看不见的天使，煽动着它们的翅膀，驱动着行星向前。你将看到这个理论，现在被修改了！结果就是，为了让行星转着走，看不见的天使，就应该在一个不同的方向飞，且它们没有翅膀，否则的话，它就是一种类似的理论了！）伽利略发现了一个关于运动的事实，非常引人注目，它对理解这些规律，具有本质性的意义。那就是惯性原理--如果某物在移动，且没有被碰到或完全没被干扰，那么，它就永远在运动，沿着一条直线，匀速滑行。（为什么它会一直滑行？我们不知道，但事情就是这样。）