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Our friend Schr?dinger told us that if you solve for the wave function, this is what the probability densities look like.
我们的朋友薛定谔告诉我们,如果你用波函数来解决，你就会知道这些概率密度看上去的样子。
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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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These are called population measures because they refer to the whole population of possible outcomes and they measure the probabilities.
这些是用来度量总体的变量,因为他们对应的是总体中所有的结果,度量的是所有事件的概率
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In the investment game, the more likely I was to invest, the more likely you, the more you wanted to invest.
在投资博弈中我投资的概率越大,你们也就越想要投资
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And when we define that as r being equal to zero, essentially we're multiplying the probability density by zero.
当我们定义ｒ等于０处，事实上是把概率密度乘以０．
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So the probability of having an electron at the nucleus in terms of probability per volume is very, very high.
在单位体积内发现,一个电子的概率非常非常大。
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It takes on the value 1 with the probability of 20% and the value of 0 with the probability of 80%.
等于1的概率是20%,等于0的概率是80%
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So the probability is 0 of the other guy choosing Left is, the same as, let's try it again.
同样的如果对手选左的概率是0,那也就是说,重新来
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So, one way we could look at it is by looking at this density dot diagram, where the density of the dots correlates to the probability density.
其中一个理解它的方法,就是通过看这个密度点图，这里点的密度,和概率密度想关联的。
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The highest probability now is going to be along the x-axis, so that means we're going to have a positive wave function every place where x is positive.
概率最高的地方是沿着ｘ轴，这意味着只要在ｘ,大于零的地方波函数都是正的。
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we have Nala and he meets this man, Rituparna, and this is where a probability theory apparently comes in.
有那勒,他遇到的这个人,叫睿都巴若那,这就到了讲概率论的时候了
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Right, this makes a lot of sense because if the entire atom was made up of nuclei, then we would have 100% probability of hitting one of these nuclei and having things bounce back.
因为如果整个原子,都是原子核，那我们就有100%概率,撞到一个原子核并被弹回来,所以如果我们。
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So if we actually go ahead and multiply it by the volume of our shell, then we end up just with probability, which is kind of a nicer term to be thinking about here.
乘以壳层的体积,我们就得到了概率，在这里从这个角度，理解问题更好一些,如果我们考虑的是。
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But the reality that we know from our quantum mechanical model, is that we can't know exactly what the radius is, all we can say is what the probability is of the radius being at certain different points.
我们不可能准确的知道,半径是多少，我们只能说，它在不同半径处，的概率是多少，这是,量子力学。
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We're saying the probability of from the nucleus in some very thin shell that we describe by d r.
某一非常薄的壳层dr内,一个原子的概率,你想一个壳层时。
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And again, we can define what that most probable radius is, that distance at which we're most likely to find an electron.
同样的，我们可以定义最可能距离，在这里找到电子的概率最大。
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py And finally, we can look at the 2 p y, so the highest probability is going to be along the y-axis.
最后我们来看一下，概率密度最高的是沿着ｙ轴。
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So if we superimpose our radial probability distribution onto the Bohr radius, we see it's much more complicated than just having a discreet radius.
为波尔半径,这其实比分立的轨道,要复杂很多，我们可以有任何的半径,但有些半径的概率。
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So, you should be able to generally identify and draw the general form of these radial probability distributions.
所以你们应该可以大概辨认,并且画出概率,分布的大致形式。
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Or we could just look at the radial probability distribution itself and see how many nodes there are.
或者我们可以直接,看径向概率分布图,本身看看里面有几个节点。
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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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If the average age of death was something like forty-five, maybe a fifty-fifty chance of--high risk-- of one of the parents dying.
那么这个概率其实很低,不是吗,大概有高达50%的可能性,双亲中的一人会死去
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But a real key in looking at these plots is where we, in fact, did go through zero and have this zero probability density.
是我们经历这些零值,而且有这些零概率密度,我们把它叫做节点。
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So, we can look at other radial probability distributions of other wave functions that we talked about.
我们可以来看一看我们讨论过的,其它一些波函数的径向概率分布。
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OK. So let's actually go to a clicker question now on radial probability distributions.
好，让我们来做一个关于,径向概率分布的题目。
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This is not a node because a node is where we actually have no probability density.
因为节点处是,没有概率密度的,所以。
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The mathematical theory of probability was unknown until that time and you can see that insurance suddenly made an appearance at that time.
在那之前,概率的数学理论,是不存在的,而随着概率论的出现,保险业也突然出现了
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We'll start with talking about the shape, just like we did with the s orbitals, and then move on to those radial probability distributions and compare the radial probability at different radius for p orbital versus an s orbital.
想我们对待ｓ轨道那样，我们先讨论ｐ轨道的形状，然后是径向概率密度分布,并且把ｓ轨道和ｐ轨道在,不同半径处的径向概率做一个比较。
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The idea of probability theory is that no, you can't change things, there are all these objective laws of probability out there that guide everything.
而概率论的观点是,不,你无法改变事物,世间万物遵循客观的概率,它们即是定律
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So, the quantum mechanical interpretation is that we can, in fact, have probability density here and probability density there, without having any probability of having the electron in the space between.
量子力学给出的解释是，实际上，我们可以在这有概率密度，在这里有概率密度,但在两个之间没有。
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