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We can graph out what this is where we're graphing the radial probability density as a function of the radius.
我们可以，画出它来,这是径向概率密度,作为半径的一个函数图。
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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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We have instead what's called a probability density when we have continuous random variables.
所以我们用概率密度的概念来描述,连续型随机变量的情况
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This cartoon shows the probability density function of 1s.
下面这个动画表示了,数1s轨道的概率密度函数。
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So, doing those probability density dot graphs, we can get an idea of the shape of those orbitals, we know that they're spherically symmetrical.
概率密度点图上,我们可以对这些轨道的形状，有个大概了解,我们知道它们是球，对称的,我们今天不讲。
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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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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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So we can see if we look at the probability density plot, we can see there's a place where the probability density of is actually going to be zero.
就能看到,有些地方，找到一个电子的,概率密度,我们可以考虑。
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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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At first it might be counter-intuitive because we know the probability density at the nucleus is the greatest.
起初我们觉得这和直观感觉很不相符,因为我们知道在原子核,出的概率密度是最高的。
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So, that's probability density, but in terms of thinking about it in terms of actual solutions to the wave function, let's take a little bit of a step back here.
这就是概率密度，但作为,把它当成是，波函数的解，让我们先倒回来一点。
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And so, the radial probability density at the nucleus is going to be zero, even though we know the probability density at the nucleus is very high, that's actually where is the highest.
所以径向概率密度,在核子处等于零，虽然我们知道在，核子处概率密度很大,实际上在这里是最大的，这是因为。
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So what we should expect to see is one radial node, and that is what we see here 3s in the probability density plot.
个节点,这就是我们，在这概率密度图上所看到的,如果我们考虑。
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Anywhere where that's the case we're going to have no probability density of finding an electron.
这时面内任何地方,找到电子的概率密度都是零。
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This is the probability density map, so we're talking about the square here.
这是它的概率密度图，我们看的是平方。
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PROFESSOR: Probability density, yes.
概率密度。
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a perfectly spherical shell dr at some distance, thickness, d r, dr we talk about it as 4 pi r squared d r, so we just multiply that by the probability density.
在某个地方的完美球型壳层,厚度，我们把它叫做4πr平方，我们仅仅是把它,乘以概率密度。
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So if we talk about the probability density and we write that in, it's going to be sigma 1 s star squared, 1sb so now we're talking about 1 s a minus 1 s b, all of that being squared.
如果我们讨论概率密度，而且我们把它写出来，它等于sigma1s星的平方，现在我们说的是1sa减去，这整体再平方。
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So we talked about radial nodes when we're doing these radial probability density diagrams here.
我们画这些径向概率分布图的时候,讨论过径向节点。
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And if we go ahead and square that, then what we get is a probability density, and specifically it's the probability of finding an electron in a certain small defined volume away from the nucleus.
我们得到的是，一个概率密度，它是，在核子周围，某个很小的,特定区域,找到电子的概率,所以它是概率密度。
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And when we're looking at the probability density graphs, it doesn't make a difference, it's okay, It has no meaning for our actual plot there, because we're squaring it, so it doesn't matter whether it's negative or positive, all that matters is the magnitude.
它的概率密度图的时候,两者没什么区别，这是可以的，它对我们画这个图，没有什么意义,因为我们是取平方，所以它的正负，无所谓，只和幅值有关，但当我们说到。
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let's think about probability density.
让我们来考虑概率密度。
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So if we write out every term individually, what we end up with is essentially just the probability density for the first atom, then the probability density for the second atom, and then we have this last term here, and this is what ends up being the interference term.
如果我们把每一项都写出来，最后得到的就是,第一个原子的概率密度，然后是第二个原子的概率密度，然后是这最后一项，这就是干涉项。
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So it's a probability density.
这里很重要的一点是。
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The reason in our radial probability distributions we start -- the reason, if you look at the zero point on the radius that we start at zero is because we're multiplying the probability density by some volume, and when we're not anywhere 0 from the nucleus, that volume is defined as zero.
在径向概率密度里，我们开始,如果你们看半径的零点，我们从零点开始,因为我们用概率密,度乘以体积,而当我们，在离核子很近的地方,体积是，所以我们会在这里。
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So, again we can use these probability density plots, which are just a plot of psi squared, where the density of the dots is proportional to the density, the probability density, at that point.
同样的我们可以利用这些概率密度图，这是psi的平方的图，这里面点的密度，正比于概率密度。
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So we can think of a third case where we have the 3 s orbital, and in the 3 s orbital 0 we see something similar, we start high, we go through zero, where there will now be zero probability density, as we can see in the density plot graph.
第三个例子那就是,3s轨道,在3s轨道里，我们看到类似的现象,开始非常高，然后穿过，这里，概率密度是0，就像你们在概率密度图里看到一样，然后我们到负的。
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Then we go negative and we go through zero again, which correlates to the second area of zero, that shows up also in our probability density plot, and then we're positive again 0 and approach zero as we go to infinity for r.
并且再次经过,这和，第二块等于0的区域相关联,这也在,我们的概率密度图里反映出来了，然后它又成了正的,并且当r趋于无穷时它趋于。
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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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The important point here is it's not just a probability, it's a density, so we know that it's a probability divided by volume.
它不是概率,而是概率密度，所以我们知道，它是密，除以体积，我们。
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