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But the point I want to make about it is much more general and we don't have to look at particular cases to make it.
但是我想说的是这是很普遍的,我们不用看很多特例去研究它。
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And what they said was, look, you can't put this in, we're starting to notice it does some strange things.
而且他们说，看，你不能放这个进去，我们开始注意到了它会引发一些奇怪的事情。
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And, following a test, a lot of people write in pencil: don't try to erase something and bring it back and say that, oh, actually, this is what I had because we look at that and go, this is perfect.
还有，在测试的时候，很多人用铅笔：,不要试图擦掉什么，把它带回,说，事实上，这是我的,因为我们看了一下就走了，这很完美。
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It's not going to be zero this time because we have non zero heat exchange between the system and the environment, right.?
热力学量，但是现在让我们看一下,我们的特殊函数，这次它不会是零,因为系统与外界的热量交换并不是零，对吧？
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And the thing you should notice here is that it's doing a lot of the same things over 2 1 and over again. So, for example, we'll see 2, 1 2 1 here. And 2, 1 here.
在这里你要注意的是,它不断在重复同样的工作,例如我们看这里的1,还有这里的。
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So we'll look at and play another game, and we'll call this game, "Going to the Movies."
请大家看下一个博弈,我们就称它为,一起看电影吧
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We won't come back to it at least for the moment, but you see that it's mercifully short, and as time passes we will do some rather interesting tricks with it.
至少我们这节课用不上，看，这篇故事很短，过一段时间我们会针对它做一些很有趣的练习。
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Now, let's go back to a theory which has at least got the virtue of being very old, although hardly anybody believes it anymore.
现在,让我们再回去看一个理论,它的优势在于非常古老,虽然已经没人再相信它
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So now let's take a look at how else we could approach this.
现在让我们看一下我们还可以怎么处理它。
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No, the most immediately shocking aspect of the poem was its style, and we have to look at the actual poetic form by which this poem is constructed because the poetic form is absolutely integral to its meaning.
不是这样的，此诗最先让人感到震惊的是它的格式,我们需要看一下这首诗遵循的诗体,因为诗体与其意是一体的。
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Harmony was invisible when we mean definition No. 1- can't be seen.
我们取第一种定义的时候,和声是无形因为它是看不见的。
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And when we don't see the seed of greatness, when we don't water it and shed a light on it, it withers and dies, which is unfortunately the fate of most human potential.
如果我们看不见学生的潜能，不去栽培它，它就会枯萎而死，遗憾的是，大部分人类潜能都因此而被磨灭。
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So if we look at its length, it's actually shorter than a single bond, but longer than a double bond.
因此如果我们看它的键长的话，它实际上比单键更短，却比双键更长。
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Also, when we're looking at the Schrodinger equation, it allows us to explain a stable hydrogen atom, which is something that classical mechanics did not allow us to do.
当我们看一个薛定谔方程的时候，它给出一个稳定的氢原子,这是在经典力学中做不到的。
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So we can't actually see any of that, it's too high energy for us to see. So everything we see is going to be where we have the final energy state being n equals 2.
所以我们是看不见它的,它能量太高了，我们能看见的,都是终态等于2的情况。
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We see that it varies monotonically from left to right as we move along a row.
当我们沿着表格横栏一一看过去时,它会从左到右单调变化。
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The other thing that we know, which is what we were just discussing when we were going through the table is how this all relates to energy.
另外一个我们知道的事情就是，我们看这张图表的时候,我们讨论了,它和能量之间的关系。
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But it will let us illustrate a few points.
但是它能证明给我们看很多道理。
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Or if we look at how strong it is, it's actually stronger than a single bond, but weaker than a double bond.
或者如果我们看键的强度的话，它实际上比单键更强，而比双键更弱。
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All right. So we kind of saw what was happening here, you saw it move a little bit.
好的，我们看稍微看一下,这里发生了什么，你们看到它移动了一点儿。
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OK. How can we check, for example, if it does or if it doesn't seem reasonable.
我们如何检查呢？举个例子，看它是否合理或者不合理。
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So you can think of it, if we were to just think of it as a straight line that we were going across, essentially what we're saying is that we're getting from point a to point c without ever getting through point b.
你们可以来想想看，如我我们把它想成是一条直线,我们要沿着这条线经过，这就是说我们要从ａ点,不经过ｂ点而到ｃ点。
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So, let's, for example, look at nitrogen. So n 2, we can do the chart here in green, so it's the green dotted line, and what we see is that we have now defined this energy as where the 2 nitrogen atoms are separated.
那么，让我们举个例子，看一下氮，那么氮分子，我们可以把它用绿色曲线画在这，这是绿色的虚线，可以看到,我们已经定义为零点能，当两个氮原子分离时。
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So, for example, if we were looking at the actual wave function, we would say that these parts here have a positive amplitude, and in here we have a negative amplitude.
我们看，一个波函数,我们说，它这部分幅值，为正,这部分幅值为负，当我们看。
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So, if we look at the molecular orbital, that's actually going to be lower in energy than either of the two atomic orbitals.
如果我们看分子轨道的话，它实际上要比,两个原子轨道都要低。
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All right, I want to show you one last example of using recursion because we're going to come back to this. This is a classic example of using recursion. And that is dating from the 1200s and it is due to Fibonacci.
并且返回答案,我想给你看看递归的最后一个例子,因为我们还要再看一遍,这是一个递归的经典案例,它可以追溯到13世纪。
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When we look at this angular part, we see that it's always the square root of 1 over 4 pi, it doesn't matter what the angle is, it's not dependent on the angle.
当我们看这角向部分，可以看到它总是等于１除以４pai开根号，这和是什么角度没有关系，它和角度无关。
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- We know from Wednesday if -- briefly -- that there's this thing called a "char" or "char," depending on how you want to pronounce it, which is just a single character but where there's also an int.
我们知道从周三起--简单说下-,我们有个叫做“char“或“char“,看你们怎么读它了，那代表一个单一的字符,但那里会有个整型数与之对应。
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We're going to see variations of this, we're going to see a variation of it called recursion, a little later on, but for now we're just going to talk about how do we do iterations.
我们稍后会去，看它的变量,它的所谓的递归数的变量，但是现在我得先讲讲,怎么来实现迭代。
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So, nitrogen is shown here, we know that because it has 7 electrons.
那么这里显示的是氮，我们能看出来是因为它有七个电子。
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