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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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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 we reasoned that these two eventually reach some kind of an equilibrium separation which we are using lowercase r to represent.
我们推导,这两个最终能达到,某种平衡分离,我们用小写的r代表。
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If you do it in monthly terms your interest rate is r/12 because there are twelve months in a year.
如果是每月还款那么月利率是r/12,因为一年有12个月
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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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And so, you know from your Newtonian mechanics, as you were learning in 8.01, the dynamic force here mv^2/r is mv squared over r.
在8。01节对牛顿动力学系统的学习中，我们可以知道这里的运动受力,就是。
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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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And now the force, in its most general term / is q1q2 over 4 pi epsilon zero, which is the conversion factor r squared.
库仑力的最基本形式,就是，其中r是一个变量。
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q1*q2/ That's simply q1, q2 over 4 pi epsilon zero R.
那只是简单的。
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r So the circumference is 2 pi r.
周长=2π
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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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If you get them backwards, logr you will integrate one over r and will get log r.
如果你逆推的话，对1/r积分得到。
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You have to discount something that's one period in the future by dividing it by 1+r.
你必须把未来一段时间内的资金通过,除以1+r来贴现
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So, what's one over R to the 8th going to look like on this scale?
所以R分之一的8次方,在这种情况下会是什么样？
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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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And we can find r naught by looking for the minimum.
我们可以通过寻找最小值得到r圈。
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And this equilibrium spacing is denoted r sub zero or r naught.
这个平衡间距,用r下标0或者r圈表示。
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When you get one over R, you get a gradual fall.
当你用1除以R，你得到逐渐的下降。
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Generally, if it pays c dollars for every period, the present value is c/r.
推广开来,如果在每段时期内支付c美元,那么它的现值就是c/r
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Then, I was talking it over with another graduate student and he said, well if you want to find out why don't you ask J.R. Hicks?
随后,我和另一位研究生讨到了此问题,他说,如果你想知道,干嘛不直接去问J·R·希克斯
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If you get one over R squared, it'll be a little steeper.
如果你用1除以R的平方，它会变得稍微陡峭一些。
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When you vary time a little bit and ask, "How does R change?"
当你把时间改变一个微元,然后问,"位矢 R 会怎样变化"
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I'm going to write the particular case of R and take derivatives.
我会写出 R 的一个特例,然后对其求导
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l But now we need to talk about l and m as well. So now when we talk about a ground state in terms of wave function, we need to talk about the wave function of 1, 0, 0, and again, as a function of r, theta and phi.
但我们现在需要讨论,和m，现在当我们讨论,波函数的基态时，我们讨论的，是1，0，0的波函数，同样的，它也是r，theta和phi的函数。
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Energy goes as one over r.
能量和1/r成正比。
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If it's going in a circle, you will say from now on, that it, indeed, has an acceleration, even though no one's stepping on the accelerator, of amount v^2 over R.
如果它在一个圆周上运动,你会说从现在起它其实有加速度,即使没有人去踩油门,加速度大小为 v^2 / R
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and a radius R and that's where your particle is.
半径为 R,这就是质点的位置
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That's what I meant. This is called R.
这就是我说的 R
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The size of that is v^2 over R.
它的大小是 v^2 / R
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take the derivative of this, get the velocity vector and you notice his magnitude is a constant Whichever way you do it, you can then rewrite this as v square over r.
对这个式子求一次导,就能得到速度矢量,你会发现其模长是常数,不管用什么方法,加速度也可以写成 v^2 / r
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