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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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But you can write it down It's a concept that links itself to a literary or verbal written expression.
但是你们可以将负二的平方用笔写下来,那是种概念，能将其自身,与文学或文字书面的表达连接起来。
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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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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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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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They go up at rates squared.
它们以平方的速度增长
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If your plan is to accelerate a 3 kg mass with an acceleration of 2 meters per second, you better have a rope that can furnish that force and it can take the tension of 6 Newtons.
如果你想用 2 米每秒平方的加速度,去拉一个 3 千克的物体,那么你就需要一根能提供拉力的绳子,并且它能够承受 6 牛的张力
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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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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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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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We have seen log, linear, quadratic, and exponential.
平方级的和指数级复杂度的方法，再说一遍，可能会有些常量。
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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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Right, It's just squaring an integer, is what it's doing.
对了，这个程序就是，用来求一个整数的平方的。
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It's the square root of /100.
就是/100的开平方
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What we want to do now is compute the mean and variance of the portfolio-- or the mean and standard deviation, since standard deviation is the square root of the variance-- for different combinations of the portfolios.
我们现在要做的是,计算这个投资组合的均值和方差-,或者均值和标准差，因为标准差的平方就等于方差-,这对任何投资组合都是一样的。
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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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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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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 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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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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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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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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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, 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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