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In an orbital is remember that this area right here at r equals zerio, that is not a node.
例如对于１ｓ轨道,记住这里ｒ等于０处不是一个节点。
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We notice that the value of E at r naught is negative, as it should be. It's a negative number.
我们主要到在r圈时E的值为负，和它本来的值一致，是一个负数。
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So, let's say we start off at the distance being ten angstroms. We can plug that into this differential equation that we'll have and solve it and what we find out is that r actually goes to zero at a time that's equal to 10 to the negative 10 seconds.
也就大约是这么多,所以我们取初始值10埃,我们把它代入到,这个微分方程解它,可以发现，r在10的,负10次方秒内就衰减到零了。
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RT2 So it's R T2, right, now we're at a lower temperature times log the log of V4 over V3.
等于，这时温度比刚才低,乘以。
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er is a vector at each point of length one pointing radially away from the center.
r 是一个模长恒为 1 的矢量,方向沿半径向外
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dE/dr=0 We take dE by dr equals zero at r equals r naught.
当r等于r圈时。
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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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And orbiting around this is a lone electron out at some distance r.
有一个单电子,在环原子核的轨道上运行。
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So if we take this term, which is a volume term, and multiply it by probability over volume, what we're going to end up with is an actual probability of finding our electron at that distance, r, from the nucleus.
如果我们取这项，也就是体积项然后，乘以概率除以体积，我们能得到的就是真正在距离,原子核ｒ处找到电子的概率。
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And at a distance of 2 r naught, I have a positive repulsive term.
在这2r圈的距离上，这里有一个不可忽略的排斥场。
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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 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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At very, very low values of r, one over r to the 10th dominates.
在很小很小的r是，1/r的10次方占主导。
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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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See, at very, very high values of r, one over r dominates.
看，很大很大r，1/r占主导。
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Whenever you see a particle moving in a circle, even if it's at a constant speed, it has an acceleration, v square over r directed towards the center.
只要看到质点做圆周运动,即使是匀速圆周运动,也存在一个加速度,大小为 v^2 / r,方向指向圆心
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e The charge on the anion times minus e, so there is the minus e squared, 0R0 and divided by 4 pi epsilon zero r naught, because now I am evaluating this function at r naught, one minus one over n where n is the Born exponent.
阴离子的电荷乘以，因此会有-e的频繁，除以4πε,因为现在我用r圈评估这个函数，1-1/n，n是波恩指数。
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OK, so R minimum, the minimum separation occurs when the energy is at its minimum.
好的，所以R处是最小值，间距最小值出现在,当能量为最小的时候。
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That was Cartesian space. When I plot r as a distance out from the nucleus that is sort of our simple-minded planetary model. Now let's look at energy.
笛卡尔坐标系，当我用r表示,离原子核的距离时，那只是我们头脑中简单的,类似行星的模型，现在我们看一下能量问题。
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And this applies only at, obviously, r equals r naught.
这表明，r=r圈。
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You don't need Born repulsion at 2 r naught.
我们不需要在r等于2r圈处的Born力。
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At every instant, it's got a location given by the vector R; R itself is contained in a pair of numbers, x and y, and they vary with time.
在每一个瞬时,它的位置由位矢 R 给出,R 本身包含了一对坐标值 x 和 y,并且它们都随着时间的变化而变化
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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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We know it's going around in a circle because if I find the length of this vector, which is the x-square part, plus the y-square part, I just get r square at all times, because sine square plus cosine square is one.
我们之所以知道它做圆周运动,是因为我求出了这个矢量的模长,也就是 x 的平方加上 y 的平方,我就得到了它在任意时刻的模长平方,因为正弦平方加余弦平方始终等于1
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