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We can talk about the wave function squared, the probability density, or we can talk about the radial probability distribution.
我们可以讨论它,波函数的平方，概率密度，或者可以考虑它的径向概率分布。
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But we can also think when we're talking about wave function squared, what we're really talking about is the probability density, right, the probability in some volume.
波函数平方,的时候，我们说的，是概率密度，对吧,是在某些体积内的概率，但我们有办法。
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You can square this guy and add it to the square of this guy and you will find, using the magic of trigonometry, that this is true.
你可以把这两个式子平方相加,再用一下三角函数的性质,你会发现这是正确的
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And the person we have to thank for actually giving us this more concrete way to think about what a wave function squared is is Max Born here.
需要感谢，马克思，波恩,给了我们，这个波函数平方的,具体解释，事实上。
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And when we take the wave function and square it, that's going to be equal to the probability density of finding an electron at some point in your atom.
当我们把波函数平方时，就等于在某处,找到一个电子的概率密度。
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We can't actually go ahead and derive this equation of the wave function squared, because no one ever derived it, it's just an interpretation, but it's an interpretation that works essentially perfectly.
从这个方程中,导出，波函数的平方，没有人可以这样做，这仅仅是一种解释，但这种解释，能解释的很好，自从它第一次被提出来之后。
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Ever since this was first proposed, there has never been any observations that do not coincide with the idea, that did not match the fact that the probability density is equal to the wave function squared.
从未有，任何观测，与它相抵触,从没有过，波函数的平方不等于,概率密度的情况，关于马克思，波恩。
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So again if we look at this in terms of its physical interpretation or probability density, what we need to do is square the wave function.
如果我们从物理意义或者,概率密度的角度来看这个问题，我们需要把波函数平方。
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So, at this place where it hits zero, 0 that means that the square of the wave function is also going to be zero, right.
它达到0的地方，这意味着波函数的，平方也是,如果我们看概率密度图。
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So again, we can think about the probability density in terms of squaring the wave function.
同样的，我们可以把,波函数平方考虑概率密度。
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So, the wave function at all of these points in this plane is equal to zero, so therefore, also the wave function squared is going to be equal to zero.
因此这里的,波函数平方也等于零，如果我们说在这整个平面上,任何地方找到一个ｐ电子的概率都是零。
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So the probability again, that's just the orbital squared, the wave function squared.
同样，概率密度，这就是轨道的平方，波函数的平方。
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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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But luckily we don't have to worry about how we're going to picture all this, now that I said that, no physical interpretation f or a wave function, there is a physical interpretation for what a wave function squared means.
如何想象这些图像,我这么说，虽然波函数，物理解释,但波函数的平方，有物理解释,当我们说到。
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So if we're talking about probability density that's the wave function squared.
如果我们要讨论概率密度,这是波函数的平方。
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So what is the wave function squared going to be equal to?
波函数的平方等于什么？
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But we do have an interpretation for wave function squared.
但波函数的平方我们有一个解释。
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So, what we can do to actually get a probability instead of a probability density that we're talking about is to take the wave function squared, which we know is probability density, and multiply it by the volume of that very, very thin spherical shell that we're talking about at distance r.
我们能得到一个概率,而不是概率密度的方法,就是取波函数的平方，也就是概率密度，然后把它乘以一个在ｒ处的,非常非常小的,壳层体积。
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So, also about Max Born, just to give you a little bit of a trivial pursuit type knowledge, he not only gave us this relationship between wave function squared, This is her grandfather, I don't know if you can see from the eyes, I feel like there's a little bit of a resemblance there.
这里有些，关于它的，花边新闻,他不仅带给我们，这个波函数平方的关系,还给我们带来了，他是她的外祖父，我不知道,你们能不能看出来，我觉得,他们眼睛长得很像。
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So, we can do that by using this equation, which is for s orbitals is going to be equal to dr 4 pi r squared times the wave function squared, d r.
用这个方程,对于s轨道，径向概率分布,4πr的平方,乘以波函数的平方,这很容易理解。
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So to talk about it's squared, we're going to say it's sigma 1 s squared.
要讨论它的波函数，我们说它是sigma1s的平方。
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So when we talk about a wave function squared, n l m he wave function, any one that we specify between n, l and m, at any position that we specify based on r, theta, and phi.
一个波函数,的平方时，对特定,特定位置r，theta，phi波函数，取平方,如果我们取平方。
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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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And what here is just a graph of the 1 s wave function going across some radius defined this way, and you can see that the probability - well, this is the wave function, so we would have to square it and think about the probability.
这里是,1s波函数,沿这个方向的图，你们可以看到概率,这是波函数，所以我们可以把它平方,并想成是概率。
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We call that a node, r and a node, more specifically, is any value of either r, the radius, or the two angles for 0 which the wave function, and that also means the wave function 0 squared or the probability density, is going to be equal to zero.
节点就是指对，于任何半径，或者，两个角度,波函数等于，这也意味着波函数的平方或者概率密度,等于，我们可以看到在1s轨道里。
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