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If I pick an angle of 60 degrees, these are some numbers like half and root 3 over 2.
要是我选择一个 60°的角,上面的系数就会是二分之一和二分之根号三
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You put a quarter in and you look through and you can see all through New York
你只要在望远镜里投入两角五分钱，整个纽约的景色就一览无遗了。
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They want to get as far away from each other possible, the ideal angle is 120. But what we have here is a four-membered ring, so what angle does 90° that have to be, that bond? 90 degrees.
它们想要尽量远离彼此，最理想的是形成120°键角,但现在是个四元环,所以这键角应该多大？
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He sees weird kinds of beasts and animals that had like--they're bodies of lambs, but they've got horns and they're bleeding all over the place.
看见了奇怪的动物与野兽,有着羔羊的身体，却长着角，浑身是血。
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I'm going to give you the data set of the scattering angles of the atoms.
我会把,这些原子散射角的数据发给你。
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Isn't the reason I think unicorns are logically coherent is because I can imagine them so easily?
我认为,独角兽在逻辑上不矛盾的原因,不正是因为我能够轻易的想象它们吗
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Why do you like to measure angle in the peculiar way?
为什么要用这种特殊的方式衡量一个角呢
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You can go ahead and use that equation, or you could figure it out every time, because if you know the total number of nodes, and you know the angular node number, then you know how many nodes you're going to have left.
你们可以直接用这个方程，或者每次都自己算出来，因为如果你们知道了总的节点数，又知道角向节点数，就知道剩下的节点数是多少。
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Angular nodes, we're not going to have any of those, we'll have zero, l equals 0, so we have zero angular nodes.
角向节点，当然,是没有的，０个，ｌ等于０，所以是０个角向节点。
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Remember, we're talking about angular nodes here, so you need to read the question carefully.
要知道，我们这里讨论的是角向节点，大家需要看清题目问的是什么。
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And so now I have hydrogens at the four corners of a tetrahedron.
放在碳正四面体,的四个角上。
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It seems to follow that unicorns are logically possible.
可以推测,独角兽在逻辑上是可能的
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No matter where you specify your electron is in terms of those two angles, it doesn't matter the angular part of your wave function is going to be the same.
不论你将，这两个角度,取成什么值，波函数的角向部分,都是，相同的。
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So, you remember from last time radial nodes are values of r at which the wave function and wave function squared are zero, so the difference is now we're just talking about the angular part of the wave function.
你们记得上次说径向节点在,波函数和波函数的平方,等于零的ｒ的处，现在的区别是我们讨论的是,角向波函数。
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So again, if we think about that shape of that carbon atom, it's going to be trigonal planar, 120° it's going to have bond angles of 120 degrees, because we have this set up of having three hybrid orbitals.
如果我们考虑碳原子的形状，它是平面三角形，键角是，因为我们有这三个杂化轨道。
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So, in a settling, what is the bond angle here?
这里的键角是多少？
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I could imagine a world with unicorns.
我可以想象一个有独角兽的世界
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And as a reminder, hopefully I don't need to remind any of you, but exam 1 is on Wednesday, so rather than our clicker question being on something from last class, which is exam 2 material, let's just make sure everyone remembers some small topic from exam 1 material, which is the idea of angular nodes.
而作为一个提醒--我希望大家都不需要提醒，但是这周三就要进行第一次考试了，因此我们的这个选择题,不是关于上节课内容的，那是第二次考试的内容，这里只是想确认大家都记得,第一次考试所要求的一些小问题，也就是关于角向节点的概念。
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So, let me get a little bit more specific about what we mean by nodal plane and where the idea of nodal plane comes from, and nodal planes arise from any place you have angular nodes.
关于节面的意义,或者节面概念的起源，让我们讲的更具体一点，节面起源于角向节点。
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His idea was, the formula written for the opposite one will have the opposite angle; That turns out to be the correct answer.
他的方法是,反过来的方程,以相反的角来表示,那就能推出正确的答案了
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I can imagine a world with unicorns.
我可以想象一个有独角兽的世界
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And when we talk about angular nodes, the number of angular nodes we have in an orbital is going to be equal to l.
当我们谈到角向节点时，一个轨道的,角向节点数等于ｌ
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Yup, zero radial nodes. So, for a 2 p orbital, all the nodes actually turn out to be angular nodes.
没有，对于2p轨道，所有的节点都是角向节点。
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We'll adapt the convention that we'll measure angle in a radian.
我们要适应它,以后都用弧度来表示角的大小
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For an angular node, we're just talking about what the l value is, so whatever l is equal to is equal to the number of angular nodes you have.
对于角向节点，我们其实就是在讨论l，的值是多少，因此不管，l，的值等于几,它就等于你所有的角向节点的数目。
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You can also have angular notes, and when we talk about an anglar node, what we're talking about is values of theta or values of phi at which the wave function, and therefore, the wave function squared, or the probability density are going to be equal to zero.
我们也可以有角向节点，当我们说道一个角向节点时，我们指的是在某个theta的值,或者phi的值的地方，波函数以及波函数的平方，或者概率密度等于零。
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For an f orbital, what is the quantum number l equal to?
对于一个，f，轨道，它的角量子数，l，等于几？
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In contrast when we're looking at a p orbital, so any time l is equal to 1, and you look at angular part of the wave function here, what you see is the wave function either depends on theta or is dependent on both theta and phi.
相反当我们看ｐ轨道时，任何时候ｌ等于１,你们看它的角向波函数，你们可以看到它要么是和theta有关,要么是和theta和phi都有关。
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We'll introduce in the next course angular nodes, but today we're just going to be talking about radial nodes, psi and a radial node is a value for r at which psi, and therefore, 0 also the probability psi squared is going to be equal to zero.
将会介绍角节点,但我们今天讲的是，径向节点，径向节点就是指，对于某个r的值,当然，也包括psi的平方，等于,当我们说到s轨道时。
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The more important thing that I want you to notice when you're looking at this wave equation for a 1 s h atom, is the fact that if you look at the angular component of the wave function, you'll notice that it's a constant.
我要你们注意的,更重要的一点是,当你们看到,这个氢原子1s轨道方程的时候，如果你们看,波函数，的角向部分，你们会发现它是一个常数。
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