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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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OK. So this is again an example, this was quadratic, and this one was quadratic.
好的，这又是一个例子了，这是平方次的，这是平方的。
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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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One over two squared minus one over n squared 3 4 5 where n takes values three, four, five, six.
除以，2，的平方再减去1除以n的平方,将n赋值为。
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If you took out somebody's cortex and flattened it out, it would be two feet square, sort of like a nice--like a rug.
如果你取出某人的大脑皮层,把它摊开,会有两平方英尺那么大,有点像一张地毯
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If it moves a lot--either way from the mean-- then this number squared is a big number.
如果距离均值的变动很大,那么这个平方数也会很大
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It varies with the square of distance so it goes - in order to go twice as far it takes four times as long.
速度是与距离的平方正相关的--,如果要扩散两倍的距离要多花四倍的时间
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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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So if we want to talk about the volume of that, we just talk about the surface area, which is 4 pi r squared, and we multiply that by the thickness d r.
如果我们要讨论它的体积，我们要用的是表面面积，也就是４πｒ的平方，乘以厚度ｄｒ
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This depiction of matter takes you right up to E=MC E=MC And, that's all he had to work with for data.
物质的叙述，让我们一直等到,的平方，天才。，squared。，Brilliant。,那是他研究数据所得出的成果。
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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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Still quadratic, right? I'm looking for the worst case behavior, it's still quadratic, it's quadratic in the length of the list, so I'm sort of stuck with that.
还是平方，对吧,我在寻找最坏的情况，它还是平方，它是列表长度的平方，我对此有点无奈了。
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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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Occasionally, you'll find you need to cancel out units, because, of course, you're always doing unit analysis as you solve your problems, and sometimes you'll need to convert joules to kilogram meters square per second squared.
偶然地，你会发现需要消除单位，因为在解题时,经常要做单位分析，所以有时候需要把,焦耳换做,一千克乘以米的平方除以秒的平方。
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And I have this, to write it out, this is order the length of the list squared, OK?
我得写下来,这是把列表的长度平方，对么？
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We knew this was trying to do squaring, so intellectually we know we can square -4, it ought to be 16, but what happens here?
我们知道程序是用来求平方数的，那么按理说我们可以来求-4的平方，也就是16，但是程序结果是怎么样的呢？
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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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If I'm running a linear algorithm, it'll take one microsecond to complete.
算法会在1微秒内完成，如果是一个平方级的方法。
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And then in the even case I've got to do a square and the divide.
如果是偶数情况的话,我还要再做一次除法和求平方。
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It weights big deviations a lot because the square of a big number is really big.
使偏离的权重更大,一个数的平方是一个更大的数
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sb So just to say that it's 1 s squared plus 1 s b, all of that together squared.
这就是说它是1sa加上，这整个的平方。
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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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Add the same thing to the y-values, squared, take the square root.
的差的平方，然后加起来开平方。
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Square of the Planck constant times pi mass of the electron.
普朗克常量的平方,乘以π再乘电子的质量。
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All right. I tried it on 2, I surely didn't expect a precise and exact answer to that but I got something, and if you square this, you'll find the answer kept pretty darn close to 2.
好，我试试求2的平方根,我当然不希望得到一个完全准确的答案了,但是我得到了一个近似值，试试将这个数平方一下,你会发现结果和2相当接近。
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All it is sigma 1 s, and then we have two electrons in it, so it's sigma 1 s squared.
所有的都是sigma1s，上面有两个电子，所以是sigma1s的平方。
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So let's actually just simplify this to the other version of the Rydberg constant, since we can use that here.
除以n初始的平方，我们把它简化成,另一种形式的Rydberg常数。
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So now we have that energy is equal to the negative of the Rydberg constant divided by n squared.
我们可以把能量方程大大简化,现在能量等于负的Rydberg常数除以n平方。
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