04 Sommerfeld Free Electron Theory of Electrons in Metals.srt

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好吧，这是第四次讲座，孩子们当然要离开我们了
all right this is a fourth lecture the
kids matter of course where we left off

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上一次我们谈论自由电子
last time
we were talking about the free electron

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或金属自由电子的Sommerfeld理论金属和
or the Sommerfeld theory of metals free
electron Sommerfeld theory of metals and

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00:00:24,289 --> 00:00:29,970
 Sommerfeld在做什么，他或多或少地遵循了金属设计
what Sommerfeld was doing he's more or
less following drew design that a metal

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只是电子的气体，他试图应用动力学理论
is just a gas of electrons and he was
trying to apply kinetic theory the only

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他做的不同的事情是他尊重费米的统计数据
thing he was doing differently is he was
respecting Fermi statistics he was

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跟踪您不能将两个电子放在一起的事实，我可以这样说： 
keeping track of the fact that you can't
put two electrons in one I can state and

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在上一堂课中，我们得出了费米波矢与密度相关的信息。 
last lecture we derived that the Fermi
wave vector is related to the density of

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电子由三个PI平方n到三分之一，其中n是电子
electrons by three PI squared n to the
one third where n is the electron

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密度，不幸的是，在上一讲中，我错误地犯了一个错误
density and unfortunately in the last
lecture I made an error I mistakenly

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称此为费米动量实际上是费米波矢
call this thing the fermi momentum it's
actually the Fermi wave vector and I

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甚至将其错误地写在板上h-bar乘以波矢KF为
even wrote it on the board incorrectly
h-bar times the wave vector K F is the

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费米的势头对此感到抱歉
fermi momentum sorry about that

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从费米波矢量中，我们可以用通常的方法EF获得费米能量
from the Fermi wave vector we can get
the Fermi energy in the usual way EF is

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 H bar平方KF平方超过2m，我们可以用以下表达式代替
H bar squared K F squared over 2m which
we can substitute in our expression for

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自由波矢量H bar平方2m 3 PI平方n等于2/3这是
the free wave vector H bar squared over
2m 3 PI squared n to the 2/3 this is a

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非常重要的关系，我们将再次使用，但重要的是
rather important relationship that we'll
use again but what's important to

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意识到这里的电子密度越大，电子越大
realize here is the bigger the density
of electrons you have the bigger the

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费米能，在像铁或铅这样的典型金属中，电子的密度为
fermi energy and in a typical metal like
iron or lead the density of electrons is

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每个原子真的有几个电子，而你有很多原子
really big a couple of electrons per
atom and you have a whole lot of atoms

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原子和原子每两埃的极高密度，所以费米
the very high density of atoms and atom
every couple angstroms so the fermi

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能量变得巨大，达到80,000开尔文甚至更高
energy gets to be enormous on the order
of 80,000 Kelvin or even bigger

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有时在本讲课中，我们现在打算瞄准
sometimes now in in this lecture what
we're going to aim to

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是我们之前讨论的热容量，从实验中我们
is something that we discussed earlier
the heat capacity and from experiment we

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知道低温下金属的热容量采用这种形式的t立方
know the heat capacity for metals at low
temperature takes this form of t cubed

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加号T，这个T立方项来自振动或德拜理论，我们
plus plus T where this T cubed term
comes from vibrations or Debye theory we

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已经讨论过，这个伽马T术语对金属来说是特殊的，实际上是
discussed that already and this gamma T
term is special to metals and in fact is

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电子的热容量，所以我们要使用的是伽马T项
the heat capacity of the electrons so
it's this gamma T term that we're going

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要对今天感兴趣，在我们实际执行此操作之前，我们需要做一些操作
to be interested in today now before we
actually do this we need to do a little

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特别是一些预备代数，我们将需要一些
bit of preparatory algebra in particular
we're going to need to take some over

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并将其置于比本征态更可行的形式
eigenstates and put it into a more
workable form than the eigenstates in

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这种情况将是e，你知道平面波e到IK点R和
this case are going to be e to the you
know plane waves e to the I K dot R and

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再一次，因为我们将其写为指数式就是我们隐式的
again since we're writing this as
exponential is what we're implicitly

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我们要做的是将事物放置在Borna von Karman边界框内
doing is we're putting the thing in a
periodic box Borna von Karman boundary

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条件在所有方向上都是周期性的，因此我们可以处理指数平面波
conditions periodic in all directions so
we can work with exponential plane waves

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而不是正弦和余弦，我们想要做的是
instead of sines and cosines what we'd
like to do is we'd like to take this sum

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超过本征态，我们想将其转换成能量的积分
over eigenstates and we would like to
convert it into an integral over energy

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能量g，其中g现在是状态密度，这与我们非常相似
g of energy where g is now a density of
states this is very similar to what we

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从理论上讲，当我们做了两个时，我们会做些什么？ 
did when we did two by theory how are we
going to do something slightly different

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在这里稍有不同，我们将从
here slightly different we're going to
remove a factor of the volume from the

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状态密度，所以现在是每单位体积的状态密度，我们这样做
density of states so this is now density
of states per unit volume and we do this

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因为这样做很传统，而且这样做很方便，如果您
because it's conventional to do so and
it's convenient to do so and if you

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在传统和方便来自同一个词之前没有注意到它
didn't notice it before conventional and
convenient come from the same word so

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这就是所有这些事情，恰好这样做很方便，所以我们
it's it's both of those things it just
happens to be handy to do so so we're

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要去做，然后写出这种状态密度的定义是
going to do it and then write the
definition of this density of states is

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 ede的G是州的数目
that G of e d e is the number of states

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在这种情况下，每单位体积的能量在ε和
per unit volume in this case volume with
energies energies between epsilon and

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 epsilon plus D Epsilon非常类似于我们为Debye所拥有的
epsilon plus D Epsilon
very similar to what we had for Debye

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我们以前在思考每个频率的密度状态之前的理论
theory before when we were thinking
about density States per frequency now

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总体思路还是要对所有本征态求和
the general idea again is we're going to
take the sum over all the eigenstates

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我们实际上将把各个本征态的总和转换为
and we're actually going to convert that
sum over individual eigenstates to an

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能量的积分乘以每个能量的状态数
integral over energies times the number
of states at each energy just a

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不同的写作方式可以让您的生活更轻松，好吗
different way of writing it that makes
your life a lot easier all right so what

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我们要在这里做的事情是从总和算起第一件事
are we going to do here to get from the
sum into the integral well first thing

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我们要对本征态做一些运算，实际上是对K求和，但实际上
we're going to do some over eigenstates
is really a sum over K but it actually

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前面有2倍的因子，因为每个K自旋电子有两个自旋
has a factor of 2 out front because
there are two spins per K spins electron

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可以用相同的波矢旋转或旋转
can be spin up or it can be spin down
with the same wave vector then we're

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进行与理论上相同的操作，从而使
going to do the same manipulation we did
with the by theory leave the factor of

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前面两个，我们将用2上的整数D 3k替换K上的和
two out front we're going to replace the
sum over K with an integral D 3k over 2

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 PI立方体，这是将总和转换为积分的方式，我们将得出
PI cubed this is the way sums get
converted into integrals and we'll make

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今年替换很多次，K的总和变成了体积乘以
that replacement many times this year a
sum over K becomes a volume times

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积分D 3k超过2 PI立方，然后由于我们正在考虑各向同性
integral D 3k over 2 PI cubed and then
since we're thinking about an isotropic

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系统可以在三个笛卡尔方向和两个笛卡尔方向上转换积分
system we can convert the integral over
three Cartesian directions and two

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球形极坐标，所以我们要V超过2 PI立方，然后有一个
spherical polar coordinates so we've to
V over 2 PI cubed and then we have an

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整数0到无穷大4 PI K平方DK，其中4 PI K平方是通常的4 
integral 0 to infinity 4 PI K squared DK
where the 4 PI K squared is the usual 4

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 pi是球面上的方向，所以它是通常的球面极性
pi is the directions on the sphere so
it's the usual spherical polar

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坐标，这是一个很好的结果，但实际上我们很想写
coordinates and this is a pretty good
result but really we'd like to write

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这是根据能量而不是波矢，因此我们将使用epsilon 
this in terms of energies not in terms
of wave vectors so we'll use epsilon is

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 H bar平方K平方超过2m，否则我可以写成K平方
H bar squared K squared over 2m or I
guess we could write that as K is square

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 H bar的2m根乘以ε到一半，尤其是
root of 2m over H bar times epsilon to
the one half and in particular that

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会给我们DK同样的事实，是相同因子平方的1/2倍
would give us DK is the same fact well
it's 1/2 times the same factor square

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 H bar上2m的根乘以epsilon到负1/2 D Epsilon，然后如果我们插入
root of 2m over H bar times epsilon to
the minus 1/2 D Epsilon then if we plug

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这些东西放到这里，然后我们得到的就可以了，所以我们现在有两个音量
these things into here what we then get
is ok so we now have two volume I'll

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拉出4pi，我们在楼下有2个PI立方体，然后有一个
pull out the 4pi and we have the 2 PI
cubed downstairs then we have an

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将0积分到无穷大D Epsilon，然后将这些因子放入K中
integral 0 to infinity D Epsilon and
then putting in those factors for the K

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平方DK我们得到1/2，其中三个因子2m平方根2m 
squared DK we get 1/2 there's three of
these factors 2m square root of 2m over

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将H条切成立方体，然后将其ε化为1/2，这看起来几乎就像我们想要的
H bar cubed and then epsilon to the 1/2
so this looks almost like what we want

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它几乎是e的G的整数，所以我们确定好吧，所以我只是
it's almost the integral of G of e de so
we then identify well ok so I'll just

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再次写出来，所以这个东西对于e de的无穷大G是整数0 
write it out again so this thing is
integral 0 to infinity G of e de where

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我们已经定义了e的G到D的M到H bar立方时间的三分之二
we've defined G of e to then D to M to
the three-halves over H bar cubed times

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 2 pi平方中的1乘以epsilon至1/2，因此密度保持与
1 over 2 pi squared times epsilon to the
1/2 so density stays proportional to

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 epsilon到1/2，这是一个很好的答案，但实际上
epsilon to the 1/2 and this is a
perfectly good answer but it's actually

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方便，因此通常可以在H bar上转换2 m的因数
convenient and therefore conventional to
to convert this factor of 2 m over H bar

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切成看起来更好的东西，而我们的方法是使用
cubed into something that looks a little
nicer and the way we do that is by using

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这个方程在这里，那方程确保如果我把它写在这里
this equation here that equation there
make sure maybe write it over here if I

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将该等式带到三等分的幂中，我得到EF的三等分
take that equation to the three halves
power I get EF to the three halves

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等于H bar超过2m的立方，等于三分之二乘以3 PI平方
equals H bar cubed over 2m
to the three-halves times 3 PI squared

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我做对了吗我想我做对了
and did I do that right I think I do
that right

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好的，然后您会注意到我可以解决这个问题
ok and then you'll notice that I can
turn this around

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或将其倒置至M到三分之三的h bar的立方，则为3 
or make it upside down to M to the
three-halves over h bar cubed is then 3

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00:09:00,090 --> 00:09:08,090
 pi在EF上将n平方为三分之二，这里的系数就是这里的系数
pi squared n over EF to the three-halves
and this factor here is this factor here

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所以插入，我们得到e的e的G就是我知道的3 PI平方密度
so plugging that in we get G of e of e
is then what 3 PI squared density I know

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这是很多代数，这是重要的代数日，因为这是星期一
this is a lot of algebra it's a big
algebra day because it's a monday

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将三分之二的π平方乘以1到二分之一，然后取消一些
three-halves 1 over 2 pi squared epsilon
to the one-half and then canceling a few

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00:09:33,750 --> 00:09:41,180
我们得到最终结果的结果g epsilon的密度是EF的三分之二
things we get our final result g of
epsilon is three-halves density over EF

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倍于EF的能量达到一半，没有犯任何错误，看起来
times energy over EF to the one-half
didn't make any mistakes does that look

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正确的任何对象看起来都很好，这将是相当公平的
right anyone object look good all right
this is going to be something fairly

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有用的，尤其是查看状态的密度是有用的
useful and in particular it's useful to
look at the density of states at the

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00:10:01,560 --> 00:10:07,950
费米能量仅是EF的三分之二
Fermi energy which is just three-halves
density over EF which i think is

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00:10:07,950 --> 00:10:10,680
您被要求从他们的家庭作业中得到的东西以及第一个
something that you're asked to derive in
their homework as well the first

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00:10:10,680 --> 00:10:15,720
作业，我会给你一个快速提示，我认为这是一种更简单的方法
homework set and I'll give you a quick
hint that I think there's a easier way

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00:10:15,720 --> 00:10:19,860
比我刚刚做的要去那儿，但你知道如果你不明白
to get there than what I just did but
you know if you can't figure it out you

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00:10:19,860 --> 00:10:23,850
可以跟随这个，但是有一种更便宜的方法，但是这个是
can just follow this but there's there's
a cheaper way but this one's this is

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00:10:23,850 --> 00:10:28,140
您知道的越多，路线越直接，另一种方式是
sort of the more you know it's the more
direct route the other way is sort of

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00:10:28,140 --> 00:10:33,839
无论如何，运动鞋看看现在您是否能以任何速度解决问题， 
sneakier anyway see if you can figure it
out ok at any rate now we're going to

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00:10:33,839 --> 00:10:38,190
尝试使用这个结果找出我们知道状态的密度
try to use this result figure out we
know the density of states here density

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00:10:38,190 --> 00:10:41,220
说单位体积，我们将尝试使用它来计算热量
say for unit volume we're going to try
to use this to figure out the heat

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00:10:41,220 --> 00:10:48,450
电子在低温下的容量现在总是超过
capacity of the of the electrons at low
temperature now there's always more than

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00:10:48,450 --> 00:10:53,610
一种做某事的方法，有正确的方法，有作弊的方法
one way to do something there's the
right way and there's the cheating way

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00:10:53,610 --> 00:10:56,940
我要做的是我实际上
and
what I'm going to do is I'm actually

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00:10:56,940 --> 00:11:01,500
将解释如何正确地做，然后我们将作弊， 
going to explain how the right way is
done and then we're going to cheat and

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00:11:01,500 --> 00:11:05,340
我们之所以作弊是因为正确的方法是代数
the reason we're gonna cheat is because
the right way is algebraically really

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00:11:05,340 --> 00:11:10,410
太可怕了，仅仅通过做代数我们很难获得任何直觉
horrible and it's really hard to get any
intuition just by doing algebra we just

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00:11:10,410 --> 00:11:14,010
做了足够的代数，我保证你用正确的方法做三遍
did enough algebra and I promise you
doing it the right ways is three times

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00:11:14,010 --> 00:11:19,530
更重要的是，实际的计算是，我真的
more so furthermore the actual
calculation is so I was really

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00:11:19,530 --> 00:11:23,340
很复杂，您在牛津大学的任何考试中都不会被问到，那就是我
complicated that you'll never be asked
it on any exam at Oxford and that is I

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意思是我不能100％保证它，但我可以99％保证它是这样
mean I can't 100% guarantee it but I can
99% guarantee it so so in that so

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00:11:29,370 --> 00:11:31,860
因此，我们将采取欺骗手段，使您
because of that we're gonna just do it
the cheating way which gives you the

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00:11:31,860 --> 00:11:35,970
对正在发生的事情的直觉，避免了很多代数，但这是值得知道的
intuition for what's going on and avoids
a lot of algebra but it's worth knowing

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00:11:35,970 --> 00:11:40,260
至少如果你真的想诚实的话你会怎么做，如果你
at least how you would go about it if
you really want to be honest so if you

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00:11:40,260 --> 00:11:44,460
老实说你首先要做的是写一个方程式
really want to be honest what you would
do first is you would write an equation

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00:11:44,460 --> 00:11:48,360
系统中的电子数，我们之前写了这个方程
for the number of electrons in the
system and we wrote this equation before

123
00:11:48,360 --> 00:11:52,920
上次是所有本征态的总和
last time it's the sum over all
eigenstates of the probability that each

124
00:11:52,920 --> 00:11:55,740
本征态被填充，本征态的概率为
eigenstate is filled and that
probability of an eigenstate being

125
00:11:55,740 --> 00:12:02,730
填充的是β的费米函数，是时间的逆温度乘以能量
filled is the Fermi function of beta the
time inverse temperature times energy of

126
00:12:02,730 --> 00:12:06,680
本征态减去mu的化学势
the eigenstate minus mu the chemical
potential and being that we just derived

127
00:12:06,680 --> 00:12:12,990
状态密度我们可以将其重写为体积乘积
the density of states we can rewrite
that as an integral while volume times

128
00:12:12,990 --> 00:12:18,710
每单位体积乘以状态密度从0到无穷大的整数
the integral from 0 to infinity of the
density of states per unit volume times

129
00:12:18,710 --> 00:12:26,190
 Fermi函数，所以它与ε表达式完全相同
a Fermi function so it's exactly the
same the same expression the epsilon

130
00:12:26,190 --> 00:12:30,360
完全相同的表达式，除了在本征态上写和
exactly the same expression except
instead of writing sum over eigenstates

131
00:12:30,360 --> 00:12:35,370
您在所有能量上积分每种能量的状态数和
you integrate over all energies the
number of states at each energy and of

132
00:12:35,370 --> 00:12:38,220
当然，您总是在整合状态被填充的概率为
course you're always integrating the
probability that a state is filled is

133
00:12:38,220 --> 00:12:45,930
每个人都很好，是的，是的，现在很好，您可以想到的这个方程式
everyone good with this yes yeah ok good
now this equation here you can think of

134
00:12:45,930 --> 00:12:49,380
它以两种不同的方式之一是您是否可以修复化学势， 
it in two different ways one way is if
you could fix the chemical potential and

135
00:12:49,380 --> 00:12:53,310
你知道温度会告诉你你有多少电子，但是更多
you knew the temperature it would tell
you how many electrons you have but more

136
00:12:53,310 --> 00:12:57,120
通常它会绕着相反方向知道电子数
often than not it goes the other way
around you know the number of electrons

137
00:12:57,120 --> 00:12:59,790
您在系统中拥有的，因为您知道您拥有多少原子或其他什么东西
you have in your system because you know
how many atoms you have or something

138
00:12:59,790 --> 00:13:03,360
这样，您就知道了温度，它使您能够找出
like that and you know the temperature
and it enables you to figure out the

139
00:13:03,360 --> 00:13:08,580
化学势，所以有点像你知道逆关系
chemical potential so it's sort of a you
know an inverse relationship you know

140
00:13:08,580 --> 00:13:10,630
这个你知道这个，所以你可以找出
this
you know this and so you can figure out

141
00:13:10,630 --> 00:13:14,529
原则上来说，这样做虽然在代数上很麻烦，但是
this in principle although it's
algebraically messy to do so but in

142
00:13:14,529 --> 00:13:17,440
原则上，您可以算出化学势
principle it would allow you to figure
out the chemical potential given that

143
00:13:17,440 --> 00:13:20,710
你知道粒子的数量，知道温度后， 
you know the number of particles and you
know the temperature once you have the

144
00:13:20,710 --> 00:13:25,200
化学势，您可以为系统中的能量写一个表达式
chemical potential you could write an
expression for the energy in the system

145
00:13:25,200 --> 00:13:31,060
再次积分所有状态的积分，但现在您积分了能量
integral again integrating over all
states but now you integrate the energy

146
00:13:31,060 --> 00:13:39,430
乘以一个状态被填充的概率
times the probability that a state is
filled okay

147
00:13:39,430 --> 00:13:42,850
因此，您不仅要对粒子进行计数，还要对粒子次数进行计数
so instead of just counting the
particles you count the particles times

148
00:13:42,850 --> 00:13:46,360
它们的能量来获取系统中的总能量，以便您找到化学物质
their energy to get the total energy in
the system so you find the chemical

149
00:13:46,360 --> 00:13:49,900
势首先，您一旦知道化学势便找到能量
potential first you then find the energy
once you know the chemical potential

150
00:13:49,900 --> 00:13:53,350
因此，您将知道能量是温度的函数，您可以
then therefore you would know the energy
as a function of temperature you can

151
00:13:53,350 --> 00:13:56,589
因此从这种原理上区分得到热容量
differentiate that to get the heat
capacity so in principle from this kind

152
00:13:56,589 --> 00:14:03,100
从理论上讲，如果您可以进行这些积分，就可以获得热容量
of argument you could get the heat
capacity if you could do these integrals

153
00:14:03,100 --> 00:14:06,460
问题是这些积分是否真的很讨厌，这就是为什么我们不
the problem is if these integrals are
really nasty and that's why we're not

154
00:14:06,460 --> 00:14:10,810
将以这种方式进行，相反，我们将做出一些假设
going to do it this way instead we're
going to make some assumptions which

155
00:14:10,810 --> 00:14:15,070
不太正确，但先看看假设是什么，让我先画一个
aren't quite right but to see what the
assumptions are let me first draw a

156
00:14:15,070 --> 00:14:23,140
再次这是我们上次绘制的F 0的费米函数
diagram this is the Fermi function again
which we drew last time an F 0

157
00:14:23,140 --> 00:14:30,850
温度和F从1到右边，这里F 1到0，所以这是T 
temperature and F goes from 1 to right
here the F 1 to 0 so this is this is T

158
00:14:30,850 --> 00:14:35,830
等于0，然后有限e Fermi函数会像
equals 0 and then a finite e the Fermi
function smears out a little bit like

159
00:14:35,830 --> 00:14:41,860
这是T大于0好吧，这一切看起来都很熟悉，我希望好
this is T greater than 0
ok this all looks familiar I hope ok

160
00:14:41,860 --> 00:14:45,130
顺便说一句，我相信这个完整的计算，我认为您实际上已经做到了
incidentally I believe this full
calculation I think you actually did it

161
00:14:45,130 --> 00:14:49,360
去年在您的统计机修课程中，或者至少是讲师，您知道
last year in your stat mech course or at
least lecturer did it and you know you

162
00:14:49,360 --> 00:14:52,900
可能记得那很可怕，很难记住
probably remember that it was pretty
awful and it's hard to actually remember

163
00:14:52,900 --> 00:14:56,020
关于代数真的很糟糕的一切直觉
anything about the intuition of what's
going on if the algebra is really awful

164
00:14:56,020 --> 00:14:59,230
因此，我们将尝试以Chiti方式进行操作，从而为您提供
so we're going to try to do this in the
Chiti way that is going to give you the

165
00:14:59,230 --> 00:15:03,459
直觉好多了，所以我们要假设的一件事是
intuition a lot better so the one thing
we're going to assume which isn't quite

166
00:15:03,459 --> 00:15:06,790
是的，但是非常接近的是，化学势不会
right but it's pretty close to right is
that the chemical potential doesn't

167
00:15:06,790 --> 00:15:09,700
实际上是随温度变化的，现在它确实改变了
actually change as a function of
temperature now it does change the

168
00:15:09,700 --> 00:15:12,970
由于这个快速方程，温度的函数，但它只会改变
function of temperature because of this
quick equation but it only changes a

169
00:15:12,970 --> 00:15:16,630
有点为什么，如果看一看费米，它只会改变得很好
little bit why is it it only changes a
little bit well if you look at the Fermi

170
00:15:16,630 --> 00:15:21,550
您可以想象这里的功能随着温度的升高而发生了什么
function here you can imagine as you
raise the temperature what's happening

171
00:15:21,550 --> 00:15:25,089
这是我们现在填补的一些州
here is some of
the states here that we're filled now

172
00:15:25,089 --> 00:15:28,809
移到这里，所以这里有些电子被清空了
move to here so some there were some
electrons here they empty out and they

173
00:15:28,809 --> 00:15:33,309
现在将这些州填满吧
fill these states up here okay now if
the number of states that empty out and

174
00:15:33,309 --> 00:15:36,549
这个数字说它充满了彼此的化学势
the number says it fills are equal to
each other the chemical potential

175
00:15:36,549 --> 00:15:40,749
完全不需要移动，您可以保持化学物质不变
doesn't have to move at all you would
keep in constant keeping the chemical

176
00:15:40,749 --> 00:15:44,470
实际上，您必须调整化学势a 
potential constant now in truth you have
to adjust the chemical potential a

177
00:15:44,470 --> 00:15:47,739
一点点随温度变化的函数，以保持粒子总数
little bit as a function of temperature
to keep the total number of particles

178
00:15:47,739 --> 00:15:51,759
固定，但基本上不必调整很多化学势
fixed but you don't have to adjust it a
lot basically the chemical potential

179
00:15:51,759 --> 00:15:56,290
保持几乎完全相同，所以我们要做出这个假设
stays almost exactly the same so we're
going to make this assumption where do I

180
00:15:56,290 --> 00:16:07,839
写出来，也许这里假设mu的深度与T无关
write it maybe here
assume mu is in depth independent of T

181
00:16:07,839 --> 00:16:14,879
这不是很正确，但是一旦我们有了一个近似值，还算不错
it's not quite right but it's not too
bad an approximation once we have that

182
00:16:14,879 --> 00:16:19,839
好吧，一旦有了，我们可以在这里取这个方程，然后说
well actually once we had that we could
take this equation here and then say

183
00:16:19,839 --> 00:16:23,589
好吧，让我们插入固定的mu作为温度的函数，然后我们
okay let's plug in mu fixed as a
function of temperature and then we

184
00:16:23,589 --> 00:16:27,639
假设mu是固定的，则可以计算能量随温度的变化
could calculate the energy as a function
of temperature here assuming mu is fixed

185
00:16:27,639 --> 00:16:31,360
区分它并获得热容量，但即使那样太复杂
differentiate it and get the heat
capacity but even that's too complicated

186
00:16:31,360 --> 00:16:34,689
因为这个积分真的很讨厌，所以我们甚至都不会这样做
because this integral is just really
nasty so we're not even going to do that

187
00:16:34,689 --> 00:16:38,769
我们将做一些更简单的事情，所以我们要做的是
we're going to do something even simpler
so what we're going to do is we're going

188
00:16:38,769 --> 00:16:45,309
写某个温度T处的能量等于T处的能量等于零
to write the energy at some temperature
T is the energy at T equals zero plus

189
00:16:45,309 --> 00:16:52,169
两个非常近似的事物近似事物之一是电子数量
two very approximate things approximate
thing one is number of electrons that

190
00:16:52,169 --> 00:17:07,199
会变得兴奋，会变得兴奋，而事物2是x的能量
can get excited that can get get excited
and thing 2 is x amount of energy of

191
00:17:07,199 --> 00:17:19,480
能量各自吸收，所以您可以想象从零温度开始，然后
energy each absorbs so you imagine
starting at zero temperature and then

192
00:17:19,480 --> 00:17:23,019
你打开温度，然后一些电子会被激发
you turn on the temperature and then the
some number of electrons can get excited

193
00:17:23,019 --> 00:17:27,099
每个人吸收的能量总量，一些能量和
each one absorbs a total amount of
energy some amount of energy and the

194
00:17:27,099 --> 00:17:30,490
这两个的乘积大致为您提供了您已经拥有的总能量
product of these two roughly gives you
the total amount of energy that you've

195
00:17:30,490 --> 00:17:36,160
通过合理的声音来增加您的系统是的，是的，这是否合理
increased your system by sound
reasonable yeah yes yes is it reasonable

196
00:17:36,160 --> 00:17:42,549
大概是对的，你知道这大概是对的。 
it's roughly true you know it's roughly
right okay bear with me okay so now we

197
00:17:42,549 --> 00:17:49,210
我们要做的就是找出这两个因素是什么
all we have to do is we have to figure
out what are these two factors so the

198
00:17:49,210 --> 00:17:52,059
可以激发的电子数，所以这是
number of electrons that can get excited
so this is this is sort of the

199
00:17:52,059 --> 00:17:56,730
有趣的是，被激发的电子数量大致
interesting piece here the number of
electrons that are excited is roughly

200
00:17:56,730 --> 00:18:08,740
在EF y的KBT内的电子数，因此必须在这个范围内
number of electrons within within KBT of
EF y so you have to be so this range

201
00:18:08,740 --> 00:18:16,960
这是关于KBT的信息，您必须在EF的范围内才能获得
here is about KBT and you have to be
within this range of EF in order to get

202
00:18:16,960 --> 00:18:21,669
兴奋，因为如果您在这里更远的地方，您将不会兴奋
excited because if you're farther down
here somewhere you can't get excited

203
00:18:21,669 --> 00:18:24,850
很难兴奋到正确的地步，所以那是你无法得到的
hard to get excited right so it's you
can't get it that was supposed to be

204
00:18:24,850 --> 00:18:31,179
有趣的是，您的电子处于这种状态，我根本无法兴奋，因为
funny you an electron in the state down
here I can't get excited at all because

205
00:18:31,179 --> 00:18:35,110
充满兴奋的所有状态，所以无法移动，它只是卡住了
all the states it would get excited into
re filled so can't move it's just stuck

206
00:18:35,110 --> 00:18:37,809
在那种状态下，除非你把
in that state and it's going to be
frozen there unless you turn the

207
00:18:37,809 --> 00:18:43,480
你知道温度很高，所以它可能一路跳到这里好，所以
temperature you know huge so it could
jump all the way up to here okay so it's

208
00:18:43,480 --> 00:18:48,900
 EF的KBT内的电子数量以及它的数量
the number of electrons within KBT of EF
and how many is that well it's basically

209
00:18:48,900 --> 00:18:55,780
费米能量乘以单位体积的状态密度乘以体积
the density of states per unit volume at
the Fermi energy times the volume so

210
00:18:55,780 --> 00:18:59,919
这样就可以得到费米每单位能量的状态总数
this gives you the total number of
states per unit energy at the Fermi

211
00:18:59,919 --> 00:19:06,490
表面乘以KB T，因此我们在此采用每单位体积的状态数
surface times KB T so we're taking the
number of states per unit volume at this

212
00:19:06,490 --> 00:19:09,909
能量，我们将其乘以该范围即可得出
energy and we're multiplying it by this
range to give you the total number of

213
00:19:09,909 --> 00:19:16,450
 EF的KBT中的状态让我考虑该因子，然后将其乘以
states within KBT of EF let me take that
factor and we multiply it by the amount

214
00:19:16,450 --> 00:19:23,520
每个电子吸收的能量大约为KBT 
of energy each electron absorbs which
has got to be roughly KBT

215
00:19:24,380 --> 00:19:30,420
 Gabey G乘以KB T，所以费米的KBT内从这里下来的一些电子
Gabey G times KB T here so some electron
from down here within KBT of the Fermi

216
00:19:30,420 --> 00:19:37,110
通过吸收大约KBT的能量，表面激发到了这里
surface got excited up to here by
absorbing about KBT of energy and then

217
00:19:37,110 --> 00:19:41,700
你知道要尝试对我们
you know to be to try to be a little bit
more honest about the fact that we're

218
00:19:41,700 --> 00:19:48,540
作弊完全作弊在这里，我要添加一个软糖因素，所以
cheating completely cheating here well
I'm going to add a fudge factor so e

219
00:19:48,540 --> 00:19:56,100
 t处的e等于0加上我们刚刚得出的因子
total is e at t equals zero plus those
factors they are which we just derived

220
00:19:56,100 --> 00:20:04,650
就是G @ EF乘以KB T的平方
so to be G @ EF times KB T squared and
then the fudge factor is we'll call

221
00:20:04,650 --> 00:20:11,370
伽玛twiddle超过2和伽玛twiddle只是我们承认内
gamma twiddle over 2 and gamma twiddle
is just our admission of guilt that we

222
00:20:11,370 --> 00:20:16,800
并没有进行真正的真实计算，我们知道我们将得到答案
didn't do a real real calculation we and
we know we're going to get the answer

223
00:20:16,800 --> 00:20:20,790
由于阶数为1的某种因素而引起的错误，所以您可能知道伽玛旋转-它
wrong by some factor of order 1
so gamma twiddle could be you know - it

224
00:20:20,790 --> 00:20:24,870
可能是1/2，可能是馅饼，可能是2 pi，但不会是100 
could be 1/2 it could be pie it could be
2 pi but it's not going to be a hundred

225
00:20:24,870 --> 00:20:28,380
不会是一千，不会是0.01，所以这是一些
it's not gonna be a thousand it's not
going to be 0.01 okay so it's some

226
00:20:28,380 --> 00:20:31,350
一阶的数字，这是我们承认的内感，而我们没有
number of order one which is our
admission of guilt that we didn't

227
00:20:31,350 --> 00:20:37,290
有了这个表达式，我们就可以真正进行真正的计算了
actually do the real calculation okay
once we have this expression we can of

228
00:20:37,290 --> 00:20:43,140
当然要区分CV de DT，这就是为什么我将2的因数
course differentiate it CV de DT which
is this is why I put the factor of 2 in

229
00:20:43,140 --> 00:20:52,910
因此，当您区分下一个聪明的KB，然后区分EF和G时，它就消失了
so it goes away when you differentiate
next clever the KB and then G at EF and

230
00:20:52,910 --> 00:20:59,460
您剩下KBT，实际上我们已经在那派生了G @ EF，所以
you're left with KBT and in fact we
already derived G @ EF over there so

231
00:20:59,460 --> 00:21:03,900
让我们将其插入三等分和EF，我们还将使用
let's plug it in three halves and over
EF and we're also going to use maybe

232
00:21:03,900 --> 00:21:08,490
我在这里将其作为一种方便的常规
I'll put it here
a convenient conventional that the

233
00:21:08,490 --> 00:21:14,070
密度乘以体积就是粒子数，所以当我乘以
density times the volume is the number
of particles so when I multiply this

234
00:21:14,070 --> 00:21:21,420
很好地在这里写下来，所以我在这里乘以三半并超过EF 
factor by well write it out here so I
multiplied three halves and over EF here

235
00:21:21,420 --> 00:21:29,460
用V表示V和较小的端点给您较大的端点，我们将让这个伽玛旋转
by V the V and the small end give you a
big end and we'll get this gamma twiddle

236
00:21:29,460 --> 00:21:43,159
因数乘以三倍大n KB乘以EF的KB T好吧，这就是我们的
factor times three halves big n
KB times KB T over EF okay so that's our

237
00:21:43,159 --> 00:21:49,220
最终结果，实际上如果您真的进行了计算，您会
final result and actually if you did the
calculation really honestly you would

238
00:21:49,220 --> 00:21:56,539
发现伽玛旋转是PI乘以3的平方，我确定你是
discover that gamma twiddle is is PI
squared over three and I'm sure you're

239
00:21:56,539 --> 00:22:00,759
不会为知道这一点而负责，只是为了您知道我们
not going to be held responsible for
knowing that but just for you know our

240
00:22:00,759 --> 00:22:04,610
如果您是普通的教育者，那么数字实际上就是
general edification that's what the
number actually happens to be if you

241
00:22:04,610 --> 00:22:07,700
真的很想看看如何进行此计算您可以返回统计信息
really want to see how to do this
calculation you can go back to your stat

242
00:22:07,700 --> 00:22:11,749
去年的机械笔记，或者您可以知道吗？ 
mech notes from last year or you can you
know this it's in a lot of a lot of

243
00:22:11,749 --> 00:22:14,749
书本等等，当人们说你知道书本很多时，你知道我讨厌它
books and so forth you know I hate it
when people say you know it's in a lot

244
00:22:14,749 --> 00:22:17,720
书，但请相信我，您知道如果
of books but trust me you know the
algebra doesn't really teach you much if

245
00:22:17,720 --> 00:22:21,350
你真的知道，你想知道它是如何完成的吗
you really you know want to see how it's
done you can you can work through it

246
00:22:21,350 --> 00:22:26,029
 Sommerfeld确实通过了这项工作，他得到了正确的答案，但我们没有去
Sommerfeld did work through it he got
the right answer but we were not going

247
00:22:26,029 --> 00:22:28,789
反正你对此负有责任
to be you know held responsible for it
anyway

248
00:22:28,789 --> 00:22:32,809
关于此结果的几个重要事项首先是线性关系
a couple of important things about this
result first of all it's linear in

249
00:22:32,809 --> 00:22:36,470
温度及其温度呈线性的原因基本上来自
temperature and the reason it's linear
in temperature comes fundamentally from

250
00:22:36,470 --> 00:22:41,059
只有费米表面附近的电子才能吸收能量的事实
the fact that only electrons near the
Fermi surface can absorb energy okay

251
00:22:41,059 --> 00:22:45,919
当温度降至零时，可以激发的电子范围接近
when temperature goes to zero the range
of electrons that can get excited near

252
00:22:45,919 --> 00:22:49,549
费米表面降到零，而您的热容量完全消失了
the Fermi surface drops to zero and you
lose your heat capacity altogether okay

253
00:22:49,549 --> 00:22:54,019
所以实际上这是我们想要通过实验获得的热量
so in fact this is what we wanted
experimentally we wanted to get a heat

254
00:22:54,019 --> 00:22:57,169
温度是线性的，这是第一件事
capacity that is linear in temperature
so that's the first good thing about

255
00:22:57,169 --> 00:23:00,440
这个表达真的很好的另一件事是
this another thing that's really nice
about this expression is you'll

256
00:23:00,440 --> 00:23:05,830
在这里认识到这件作品在这件作品中，它是古典作品
recognize this piece here this piece
here as the classical result classical

257
00:23:05,830 --> 00:23:11,869
来自单原子气体的三分之二和kb听起来很熟悉，然后
result from a monatomic gas three-halves
and kb sounds familiar right and then

258
00:23:11,869 --> 00:23:18,190
它乘以这个系数，因此KBT超过EF，这是很小的
it's multiplied by this factor here KBT
over EF which is what it's tiny it's

259
00:23:18,190 --> 00:23:24,200
例如室温300开尔文超过80,000开尔文
like room temperature 300 Kelvin over
80,000 Kelvin tiny tiny tiny amount so

260
00:23:24,200 --> 00:23:27,470
你只有一个事实，那就是只有几个电子
you have only again coming from the fact
that only a few electrons are

261
00:23:27,470 --> 00:23:33,980
参与热容，所以索默费尔德的这一结果过去了，他
participating in the heat capacity so
this result in Sommerfeld went and he

262
00:23:33,980 --> 00:23:37,999
说好的，这是我对费米气体的热容量的新预测
said okay this is my new prediction for
what the heat capacity of a Fermi gas

263
00:23:37,999 --> 00:23:43,309
应该是，所以让我们将其与实验进行比较好吧，让我们写
should be and so let's compare it to
experiment so well okay so let's write

264
00:23:43,309 --> 00:23:47,809
当所有这些常量都被伽玛t吸收时， 
it as gamma t all those constants get
absorbed into the overall

265
00:23:47,809 --> 00:23:56,840
伽玛，如果您想知道多少热量的真实理论，他会没事的
gamma and he's going to okay if you if
you want a real theory of how much heat

266
00:23:56,840 --> 00:24:00,259
容量特定的金属应该具有您需要知道的密度
capacity particular metal should have
you need to know what the density of the

267
00:24:00,259 --> 00:24:03,919
金属中的电子是，所以您可以计算EF，但他做了平常的事情
electrons in the metal is so you can
calculate EF but he did the usual thing

268
00:24:03,919 --> 00:24:08,210
说好吧，让我们假设每个原子有一个电子
and said okay let's assume one electron
per atom the same way we've done before

269
00:24:08,210 --> 00:24:16,009
如果这样做，则为实验划分出伽玛C伽玛
and if you do that then you write out
gamma C gamma for the experiment divided

270
00:24:16,009 --> 00:24:21,909
根据理论中的伽马，并且假设每个原子有一个电子
by gamma from the theory and with the
Theory assuming one electron per atom

271
00:24:21,909 --> 00:24:34,549
对于锂，我们得到2.3，对于钠，我们得到1.3钾1.2铜1.5，这是
for lithium we get 2.3 for sodium we get
1.3 potassium 1.2 copper 1.5 this is a

272
00:24:34,549 --> 00:24:38,059
相当不错的协议，考虑到
pretty good agreement it's extremely
good agreement considering that the

273
00:24:38,059 --> 00:24:43,460
古典理论犹大所做的三分之二的nkb结果太大了
classical Theory what Judah had done the
three-halves nkb result is too big by a

274
00:24:43,460 --> 00:24:47,629
因数是100，突然之间我们得到了非常漂亮的结果
factor of 100 and all of a sudden we're
getting results which are really pretty

275
00:24:47,629 --> 00:24:50,749
接近正确答案以及我们获得接近结果的原因
close to the right answer and the reason
we're getting results that are close to

276
00:24:50,749 --> 00:24:55,549
正确的答案是因为现在更诚实地检索费米统计信息还可以
the right answer is because retrieving
Fermi statistics more honestly now okay

277
00:24:55,549 --> 00:24:58,489
好吧，我们之前根本没有对待他们，现在索默菲尔德说你把它放进去
well we didn't treat them at all before
and now Sommerfeld said you put it in

278
00:24:58,489 --> 00:25:05,149
费米统计，你现在就可以得到正确的热容量
the Fermi statistics you get the heat
capacity right okay now he went on and

279
00:25:05,149 --> 00:25:09,529
说，事实上，有了对发生的事情的新发现
said in fact with this newfound
understanding of what's going on

280
00:25:09,529 --> 00:25:12,559
你知道费米统计很重要，我们可以解决一些
you know Fermi statistics being
important we can fix some of the

281
00:25:12,559 --> 00:25:15,619
犹大理论的问题，然后是你回想起犹大的事情之一
problems with judah theory and then one
of the things that you recall from judah

282
00:25:15,619 --> 00:25:22,669
理论上回想犹大，我们计算出的东西之一，实际上使我们变得很漂亮
theory recall judah one of the things we
calculated which we actually got pretty

283
00:25:22,669 --> 00:25:32,419
靠近右边的是导热系数
close to right was the thermal
conductivity thermal conduct the

284
00:25:32,419 --> 00:25:38,479
导热系数密度/每cv热容量
the thermal conductivity one third
density cv heat capacity per per

285
00:25:38,479 --> 00:25:46,429
电子然后是b平方乘以tau散射时间，并在此画出了de 
electron then it was b squared times tau
scattering time and drew de in this

286
00:25:46,429 --> 00:25:53,769
犹大（Judah）曾经用过这些表情，这些表情我们不再喜欢
expression judah used used these things
that we don't like anymore the CV is

287
00:25:53,769 --> 00:25:59,570
三分之二kb经典结果，他也将经典结果用于V 
three-halves kb the classical result and
he also used the classical result for V

288
00:25:59,570 --> 00:26:12,229
在pi上平方8k BTW KBT，因此组合CV 
squared 8k
BTW KBT over pi m so the combination CV

289
00:26:12,229 --> 00:26:19,309
 V平方乘以实际进入导热系数的乘积
times V squared which actually enters in
the thermal conductivity it has the

290
00:26:19,309 --> 00:26:30,169
比琥珀色的pi k kb t值高12，现在好了，Sommerfeld说这两个
value 12 over pi c kb t over amber right
okay now Sommerfeld said both of those

291
00:26:30,169 --> 00:26:37,279
结果是错误的，所以让我们使用正确的结果，这样他的结果就是简历
results are wrong so let's use the right
results so his result was CV is while he

292
00:26:37,279 --> 00:26:43,549
 PI取3平方，然后将Kb减为经典结果的三等分，然后
has the PI squared over 3 and then 3
halves Kb the classical result and then

293
00:26:43,549 --> 00:26:56,119
 KBF超过EF，然后他说ok和V平方不是由
KBT over EF and then he said ok and V
squared is not not given by the

294
00:26:56,119 --> 00:27:00,469
经典动力学理论结果，但应由VF平方给出
classical kinetic theory result but
should instead be given by VF squared

295
00:27:00,469 --> 00:27:04,639
费米速度平方，使事情变得简单一些，我们可以写
the Fermi velocity squared and just make
things a little bit simpler we can write

296
00:27:04,639 --> 00:27:10,940
 EF是质量的1/2乘以VF的平方，那么我们得到了一些抵消
EF as 1/2 mass times VF squared put
those together we get some cancellation

297
00:27:10,940 --> 00:27:19,419
再次，如果我们看cv乘以B平方，我们得到pi平方乘以kb t 
again if we look at the factor cv times
B squared we get pi squared times kb t

298
00:27:19,419 --> 00:27:27,289
超过m，所以实际上我们发现与Juda的预测相距不远
over m so in fact what we discover it's
not too far from the juda prediction in

299
00:27:27,289 --> 00:27:39,369
事实上Garuda Sommerfeld Sommerfeld是12块PI立方体之类的东西
fact Garuda Sommerfeld Sommerfeld is is
what it's 12 over PI cubed or something

300
00:27:39,369 --> 00:27:44,479
大约是1/2，所以我得到正确的12个结果非常接近相同的结果
about 1/2 so it gets pretty close to the
same result did I get the right 12 are

301
00:27:44,479 --> 00:27:49,399
 PI的立方体是的，我想是的，所以您很接近同一个人
PI's cubed yeah I think so so it's
pretty you get pretty close to the same

302
00:27:49,399 --> 00:27:53,389
结果很好，因为我们喜欢导热系数的答案
result which is good because we like the
answer for the thermal conductivity that

303
00:27:53,389 --> 00:27:57,709
我们进入了犹大理论，它满足了Vietminh Franz定律， 
we got in judah theory it satisfied this
Vietminh Franz law that was known to

304
00:27:57,709 --> 00:28:01,489
实验性地存在，所以我们不想破坏它，而实际上Sommerfeld 
exist experimentally and so we didn't
want to ruin that and in fact Sommerfeld

305
00:28:01,489 --> 00:28:07,549
理论不会毁了它两个错误热容量和速度抵消
theory doesn't ruin it two mistakes the
heat capacity and the velocity cancel

306
00:28:07,549 --> 00:28:10,399
彼此很好，不是很好，但是很漂亮
each other out
exactly well not exactly but pretty

307
00:28:10,399 --> 00:28:15,050
完全接近，但还有其他一些事情
close to exactly but there were other
things that

308
00:28:15,050 --> 00:28:20,650
犹大犯了错，其中之一就是珀尔帖系数珀尔帖
juda got completely wrong one of them
was the Peltier coefficient Peltier

309
00:28:20,650 --> 00:28:26,840
让我们看看什么完全是系数我想我想我已经离开了
which let's see what was totally a
coefficient I think I think I left it

310
00:28:26,840 --> 00:28:34,180
是的，这是珀耳帖系数pi是CV乘以T 
over here yeah okay it was a Peltier
coefficient pi is a CV times T over

311
00:28:34,180 --> 00:28:40,460
减去E的三倍，在朱迪思理论中得出的结果是一百倍
three times minus E and in Judith theory
this came out a hundred times too big

312
00:28:40,460 --> 00:28:44,330
或多或少，但现在如果我们插入新的CV值，这是一百倍
more or less but now if we plug in the
new value of CV which is a hundred times

313
00:28:44,330 --> 00:28:48,500
较小的我们突然间变得看起来不错，所以
smaller we're all of a sudden getting
things that start to look right okay so

314
00:28:48,500 --> 00:28:55,490
这很好，所以Sommerfeld对这个结果非常满意，但是
this is this is good so Sommerfeld was
pretty happy with this result but

315
00:28:55,490 --> 00:29:01,330
不幸的是，与我们所有人一样，他提出了一个新问题
unfortunately as with all us with all
things he introduced a new problem

316
00:29:01,570 --> 00:29:06,440
新问题是什么好吧，让我们回过头来记住电导率
what's the new problem well okay let's
go back and remember the conductivity

317
00:29:06,440 --> 00:29:14,120
在m上的平方tau表达式仍然与Sommerfeld中的结果相同
expression n e squared tau over m that's
still the same result in in Sommerfeld

318
00:29:14,120 --> 00:29:17,810
理论给您相同的预测，最好是相同的预测，因为我们
theory give you the same prediction it
better be the same prediction because we

319
00:29:17,810 --> 00:29:23,090
有比例的您知道我们想获得正确的散热比例
have the ratio of you know we wanted to
get the the right ratio of thermal

320
00:29:23,090 --> 00:29:27,800
从电导率到电导率，我们没有改变热
conductivity to electrical conductivity
and we didn't change the thermal

321
00:29:27,800 --> 00:29:30,980
电导率很大，所以最好不要改变电导率
conductivity much so we better not
change the electrical conductivity much

322
00:29:30,980 --> 00:29:35,090
或多或少取决于您知道PI立方体的12倍
either so more or less up to you know
maybe a factor of 12 over PI cubed we

323
00:29:35,090 --> 00:29:38,690
期望电导率应由以下表达式给出
expect that the conductivity should be
given by this expression

324
00:29:38,690 --> 00:29:43,640
也许这个表达是一个值得坚持的表达，我们不知道
probably this expression is a good one
to stick with and we don't we don't know

325
00:29:43,640 --> 00:29:51,280
 tau但我们可以测量Sigma来测量Sigma来获取tau 
tau but we can measure Sigma measure
this measure Sigma to get tau get this

326
00:29:51,280 --> 00:29:58,310
得到tau然后从tau中我们可以计算出平均自由程lambda均值
get tau and then from tau we can
calculate the mean free path lambda mean

327
00:29:58,310 --> 00:30:07,370
通过散射长度，该长度现在是Sommerfeld理论中的V乘以tau 
pass scattering lengths which would be V
times tau now in Sommerfeld theory we

328
00:30:07,370 --> 00:30:11,540
可能用VF乘以tau代替它，因为电子是
would probably replace this with VF
times tau because the the electrons are

329
00:30:11,540 --> 00:30:16,670
以接近VF的速度移动，问题在于VF接近VF 
moving around at speeds close to VF and
the problem is that VF is close to the

330
00:30:16,670 --> 00:30:20,060
光速不是很接近，但是是光速的1％ 
speed of light
well not close to but it's 1% the speed

331
00:30:20,060 --> 00:30:24,380
的光非常快，这意味着平均路径非常长，所以
of light is extremely fast which means
the mean path is extremely long so

332
00:30:24,380 --> 00:30:31,040
 lambda太大了，房间有多大
lambda is huge
unreasonably big how big well at room

333
00:30:31,040 --> 00:30:40,040
茶室lambda的茶室温度可以说是一百埃
temperature at tea room at tea room room
lambda can be say a hundred angstroms

334
00:30:40,040 --> 00:30:47,630
听起来可能不大，但在低T时λ可能又是一毫米
may not sound enormous but at low T
lambda can be a millimeter and again

335
00:30:47,630 --> 00:30:51,290
这对您来说听起来可能不算什么，但您必须考虑一下
that may not sound huge to you but you
have to think about how many things the

336
00:30:51,290 --> 00:30:56,750
电子必须经过它才能很好地散射到每埃
electron has to go past before it
scatters well every angstrom there's

337
00:30:56,750 --> 00:30:59,720
另一个原子或每两个埃是电子可以
another atom or every two angstroms is
another atom that the electron could

338
00:30:59,720 --> 00:31:03,380
碰到了这里，它就走了一百个
bump into so here it goes by a hundred
of them here it goes by the million of

339
00:31:03,380 --> 00:31:07,640
他们撞到东西之前，所以撞到什么东西可以
them before it bumps into something so
what can it have bumped into it could

340
00:31:07,640 --> 00:31:11,060
撞到原子核，它可能撞到核心电子
bumped into the nucleus it could have
bumped into the core electrons it could

341
00:31:11,060 --> 00:31:14,000
已经撞到自由电子气中的其他自由电子
have bumped into the other free
electrons in the free electron gas that

342
00:31:14,000 --> 00:31:17,870
跑来跑去，由于某种原因，它并没有撞到任何一个
are running around and for some reason
it doesn't bump into any of them really

343
00:31:17,870 --> 00:31:22,070
真的很奇怪，这是我们要回答的问题
really strange and this is something
that we're not going to answer until

344
00:31:22,070 --> 00:31:26,650
在我们研究固体的能带理论之后， 
much later in the term when we study
band theory of solids so for now the

345
00:31:26,650 --> 00:31:30,950
漫长的平均自由路径只是我们必须要解决的难题
unreasonably long mean free path is just
a puzzle that we're going to have to

346
00:31:30,950 --> 00:31:35,780
处理，这使索默费尔德感到困扰，然后困扰了
deal with so this was something that
troubled Sommerfeld back then troubled a

347
00:31:35,780 --> 00:31:42,050
当时有很多人，但是你知道索默菲尔德很勇敢，他决定
lot of people back then but you know
Sommerfeld was brave and he decided what

348
00:31:42,050 --> 00:31:45,500
否则，您知道我们在了解热容量方面做得很好
else you know we did pretty well over
here understanding the heat capacity the

349
00:31:45,500 --> 00:31:50,420
导热系数珀耳帖系数我们还能计算出什么
thermal conductivity Peltier coefficient
what else can we calculate that we might

350
00:31:50,420 --> 00:31:55,460
能够正确无须担心这个特殊的小问题，因此
be able to get right not worrying about
this particular little problem and so

351
00:31:55,460 --> 00:31:59,750
现在，我们将进行一些乱序绕行，并讨论
now we're going to take a little bit of
a out-of-order detour and discuss a

352
00:31:59,750 --> 00:32:04,130
现在有点磁性，所以这周是今年的最后几次讲座
little bit of magnetism now the last
couple lectures of the year so week

353
00:32:04,130 --> 00:32:08,930
七个完全与磁性有关，所以这有点混乱，但是我
seven are entirely about magnetism so
this is a little bit out of order but I

354
00:32:08,930 --> 00:32:13,580
认为它很适合这里，因为它实际上基于完全相同的
think it fits in here well because it
really is based on exactly the same same

355
00:32:13,580 --> 00:32:18,680
我们刚刚经历的业务以及特定类型的
business that we just we just went
through and the particular type of

356
00:32:18,680 --> 00:32:25,670
我们将要研究的磁学是所谓的pauli副磁学
magnetism we're going to study is what's
known as pauli paramagnetism para

357
00:32:25,670 --> 00:32:37,390
现在自由电子的自由电子的磁性Holly当然是你
magnetism of free electrons of
free electrons now Holly of course you

358
00:32:37,390 --> 00:32:41,710
所有人都知道他是排斥原则的人，他也是骄傲的人
all know him he was the exclusion
principle guy he's also proud people

359
00:32:41,710 --> 00:32:45,370
说他是有史以来最傲慢的人，你知道他曾经告诉过他
said he was the most arrogant man who
ever lived you know he used to tell his

360
00:32:45,370 --> 00:32:48,160
学生，他们犯错是可以的
students that it was okay for them to
make a mistake

361
00:32:48,160 --> 00:32:50,920
当然他从来没有犯错，但是对他的学生来说还可以
of course he never made a mistake
himself but it was okay for his students

362
00:32:50,920 --> 00:32:55,270
犯了一个错误，所以他很典型，但是他是一个非常
to make a mistake so he was sorted that
was typical of him but he was a very

363
00:32:55,270 --> 00:33:00,190
伟大的科学家，他实际上在Sommerfeld进行计算之前就进行了计算
great scientist and he actually did this
calculation before Sommerfeld did it did

364
00:33:00,190 --> 00:33:03,970
他的工作如此之多，应该被称为多元论而不是索默菲尔德
his work so a lot of this maybe should
be called poly theory not Sommerfeld

365
00:33:03,970 --> 00:33:10,720
无论如何，顺磁理论是什么，所以一个人做的就是应用一个
theory at any rate paramagnetism what is
that so what one does is one applies a

366
00:33:10,720 --> 00:33:14,980
磁场进入系统，然后测量产生的磁化强度
magnetic field to your system and you
measure the magnetization that comes out

367
00:33:14,980 --> 00:33:20,200
零零零碎的天空上的比例常数就是
the proportionality constant it sky over
mu naught mu naught here is just the

368
00:33:20,200 --> 00:33:24,070
通常常数渗透率渗透率正确拼写
usual constant the permeability
permeability properly spelling this

369
00:33:24,070 --> 00:33:30,130
错误的渗透性是正确的，无论如何都要正确地进行修正
wrong permeability is that right make
this right anyway that's a constant that

370
00:33:30,130 --> 00:33:34,750
显示在您的数据表上，您知道一些基本常数， 
shows up on your data sheet you know
some fundamental constants this Chi here

371
00:33:34,750 --> 00:33:42,220
易感性被称为可接受性易感性
is the susceptibility is known as
acceptability susceptibility so this

372
00:33:42,220 --> 00:33:50,440
方程定义了Chi顺磁功率磁势意味着Chi更大
equation defines Chi paramagnetism
power magnetism means Chi is greater

373
00:33:50,440 --> 00:33:54,970
大于零，换句话说，您向物理系统施加了磁场
than zero so in other words you apply a
magnetic field to the physical system

374
00:33:54,970 --> 00:33:59,650
它会在与磁场相同的方向上产生磁化
and it develops a magnetization in the
same direction as the field that you

375
00:33:59,650 --> 00:34:04,050
应用，所以我们要如何解决这个问题，首先要做的是
applied so how are we going to address
this well first thing we have to do is

376
00:34:04,050 --> 00:34:10,030
我们必须为电子写一个哈密顿量，好吧，P平方超过2m 
we have to write a Hamiltonian for our
electrons all right P squared over 2m

377
00:34:10,030 --> 00:34:17,440
通常的动力学术语加上电子自旋与磁场的耦合
the usual kinetic term plus a coupling
of the electron spins to a magnetic

378
00:34:17,440 --> 00:34:23,320
场您可能在上学期的原子物理学课程中看到了这一点，因此Sigma 
field you may have seen this in the
atomic physics course last term so Sigma

379
00:34:23,320 --> 00:34:30,639
这是Poway自旋算子可能是自旋跳算子
here is the Poway spin operators
probably spin hop operators and it has

380
00:34:30,639 --> 00:34:38,019
重要的是它的特征值是正负1/2 B是磁场mu 

381
00:34:38,020 --> 00:34:49,010
这是玻尔磁子玻尔磁子的基本常数是eh bar超过2 M 
here is the Bohr Magneton Bohr Magneton
fundamental constant is eh bar over 2 M

382
00:34:49,010 --> 00:34:54,770
从数字上讲，记住该数字在某处很有用
and numerically it's useful to keep in
mind that that number is somewhere

383
00:34:54,770 --> 00:35:05,330
大约每G特斯拉的每G特斯拉的能量G是G因子，对于
around a Tesla per Kelvin per Tesla of
energy G here is the G factor and for

384
00:35:05,330 --> 00:35:09,890
电子G因子通常是自由电子，您在太空中处于G之外
electrons the G factor is typically well
free electron you are out in space the G

385
00:35:09,890 --> 00:35:13,610
因数是2，所以我们只使用2，这将阻止我们
factor is 2 so we're just going to use 2
and that's going to prevent us from

386
00:35:13,610 --> 00:35:17,270
不得不写g和g旋转而感到困惑，因为我们已经在使用g 
having to write g and g twiddle and get
confused because we're already using g

387
00:35:17,270 --> 00:35:21,950
用于状态密度，因为它是常规的，因此方便
for density of states because it's
conventional therefore convenient ok

388
00:35:21,950 --> 00:35:29,060
无论如何，如果您在此时，您可能想知道您是否服用了
anyway and if you're at this point you
might be wondering if you took the

389
00:35:29,060 --> 00:35:35,630
相对论课程上学期我为什么写P平方超过200万
relativity course last term why is it
that I wrote P squared over 2m if we

390
00:35:35,630 --> 00:35:39,980
在您可能期望有磁场的地方，应该
have magnetic fields where you might
have expected that instead you should

391
00:35:39,980 --> 00:35:47,000
 P加EA平方超过2m，其中a是矢量势
have P plus EA squared over 2m where a
is the vector potential does this look

392
00:35:47,000 --> 00:35:52,970
是的，很熟悉，所以在动能中出现的EA 
familiar from yeah ok so the EA that
shows up in the kinetic energy this is

393
00:35:52,970 --> 00:35:56,750
如果将电子排除在外，会使电子弯曲的那一部分
the piece that makes the electrons
curved if you leave it out the electrons

394
00:35:56,750 --> 00:36:01,220
不要弯腰好吧，我们将其遗漏的原因是我们将其遗漏的原因是
don't curve ok we're going to leave it
out the reason we're leaving it out is

395
00:36:01,220 --> 00:36:06,500
首先是双重的，因为很难治疗它实际上是一个相当
well twofold first of all because it's
hard to treat it's actually a rather

396
00:36:06,500 --> 00:36:12,350
复杂的计算来处理多少电子弯曲以及
complicated calculation to deal with how
much the electrons curved and what that

397
00:36:12,350 --> 00:36:16,520
确实对磁化强度有影响，但是将其省略的更好原因是因为
does to the magnetization but the better
reason to leave it out is because the

398
00:36:16,520 --> 00:36:21,380
此EA术语的影响实际上小于
influence of this term this EA term is
actually less than the influence of the

399
00:36:21,380 --> 00:36:24,860
与自旋耦合与自旋耦合实际上比
coupling to the spin the coupling to the
spin is actually more important than the

400
00:36:24,860 --> 00:36:29,030
耦合到电子的实际物理运动，所以我们将要处理
coupling to the actual physical motion
of the electron so we're going to treat

401
00:36:29,030 --> 00:36:33,110
这个学期，如果您真的想知道
this term we're going to throw out this
term ok if you really want to know the

402
00:36:33,110 --> 00:36:37,370
实际上，这个词会做什么，它将使答案改变三分之一。 
answer in fact what this term will do is
it will change the answer by a third in

403
00:36:37,370 --> 00:36:40,910
如果把它放进去，另一个会得到相反的结果
the other its own will get if you put
this in it actually has the opposite

404
00:36:40,910 --> 00:36:44,480
结果我们将到达这里，只有三分之一很大，所以它将使
sign as the result we'll get here and
it's only a third is big so it will make

405
00:36:44,480 --> 00:36:48,950
 2/3的最终结果最终结果并不重要，我们将忽略
the final result of 2/3 the final result
it's not important we're going to ignore

406
00:36:48,950 --> 00:36:55,070
目前，我们将保持此状态，因此忽略EA术语
this for now we're going to keep this
alright so ignoring the EA term just

407
00:36:55,070 --> 00:36:59,570
保持旋转项在这里，我们拥有的是
keeping the
the spin term here what we have is that

408
00:36:59,570 --> 00:37:09,740
波矢量为K且自旋向上的电子的能量为零
the energy for an electron with wave
vector K and spin up is ek naught plus

409
00:37:09,740 --> 00:37:20,600
 mu BB，而能量下降则为零，而mu BB的能量为H 
mu B B whereas energy spin down is ek
naught minus mu B B where ek naught is H

410
00:37:20,600 --> 00:37:27,620
 bar平方K平方超过2m的自由电子能，所以这个想法是
bar squared K squared over 2m the usual
free electron energy so the idea is that

411
00:37:27,620 --> 00:37:32,090
当您施加磁场时，自旋电子变得更昂贵
when you apply a magnetic field the up
spin electrons become more expensive the

412
00:37:32,090 --> 00:37:35,300
向下旋转的电子变得越来越便宜，所以将会发生什么
down spin electrons become less
expensive and so what's going to happen

413
00:37:35,300 --> 00:37:38,000
是一些向上自旋的电子会翻转过来试图变得向下
is some of the up spin electrons are
going to flip over to try to become down

414
00:37:38,000 --> 00:37:41,540
自旋电子，因为这会降低其能量，但是它们不能全部翻转
spin electrons because that would lower
their energy however they can't all flip

415
00:37:41,540 --> 00:37:45,380
结束，因为许多州已经满员，他们无法翻身
over because a lot of the states are
already filled and they can't flip over

416
00:37:45,380 --> 00:37:47,990
进入已经充满的状态，所以您将获得其中的一些
into states that are already filled so
you're going to get some of them

417
00:37:47,990 --> 00:37:53,570
翻过来，但不是很多，所以我要在这里用光
flipping over but not a lot of them okay
so I'm going to run out of room here

418
00:37:53,570 --> 00:37:58,850
很快，所以让我们从B等于0开始进行整个计算
pretty quickly so let's start with B
equals 0 and do the whole calculation

419
00:37:58,850 --> 00:38:04,070
我们将在T等于0时进行操作，因为T比TF小得多，所以可以
we'll do it at T equals 0 that's ok
because T is much much less than TF so

420
00:38:04,070 --> 00:38:10,250
它非常接近T等于0，如果我们计算了旋转次数
it's pretty close to T equals 0 and if
we calculated the number of spin up

421
00:38:10,250 --> 00:38:13,580
电子或自旋向上的电子密度
electrons or the density of spin up
electrons as the number of spin up

422
00:38:13,580 --> 00:38:17,450
电子除以B处的体积等于0，它应该与
electrons divided by the volume at B
equals 0 it should be the same as the

423
00:38:17,450 --> 00:38:22,280
在这种情况下，自旋下降电子的数量上下波动是对称的，我们
number of spin down electrons ups and
down are symmetric in that case B and we

424
00:38:22,280 --> 00:38:32,150
可以将整数0写入无穷大de，实际上将截止整数
can write that is integral 0 to infinity
de actually will cut off the integral at

425
00:38:32,150 --> 00:38:35,390
如果我们只计算电子的状态， 
EF we're going to count only the
electrons that are the states that are

426
00:38:35,390 --> 00:38:41,270
将e的G填满2，然后将其除以2，因为在这里
filled G of e over 2 and it's I put in
the divided by 2 because here we're

427
00:38:41,270 --> 00:38:45,110
仅针对旋转上升或下降而不是两者都编写表达式
writing expression for only the spin ups
or the spin downs not both of them

428
00:38:45,110 --> 00:38:49,970
 e的状态密度G是我们计算的状态密度
G of e the density of states that we
calculated was the total density of

429
00:38:49,970 --> 00:38:54,210
旋转和旋转的状态都可以，所以让我们
states of both spin ups and spin downs
okay so let's

430
00:38:54,210 --> 00:39:04,320
让我们在这里画出自旋运算的e的状态密度G这里是e，你会
let's draw here density of states G of e
for spin ops here this is e and you'll

431
00:39:04,320 --> 00:39:09,090
回想一下董事会上的某个地方，我想我已经将其向下滚动到顶部了，也许是
recall somewhere on the board I think I
scrolled it off the top well maybe it's

432
00:39:09,090 --> 00:39:16,890
是的，这里是E的G与e成正比
over here yeah here it is
the G of E is proportional to e to the

433
00:39:16,890 --> 00:39:22,530
一半，所以这东西看起来像抛物线，然后我们也可以
one-half so this thing looks like a
parabola that way and then we can also

434
00:39:22,530 --> 00:39:31,380
画出e的G代表下降， 
plot G of e for the spin downs it's
going to look exactly the same for the

435
00:39:31,380 --> 00:39:39,660
这样旋转下来，它们都充满了费米能量EF 
spin downs like this and they both get
filled up to the Fermi energy EF this

436
00:39:39,660 --> 00:39:48,000
是不是零磁场现在添加磁场的方式
isn't zero magnetic field yes
now when we add the magnetic field way

437
00:39:48,000 --> 00:39:52,590
在那里，旋转的电子将变得更加昂贵，这些家伙
up there the spin up electrons are going
to become more expensive these guys are

438
00:39:52,590 --> 00:39:57,450
会被mu BB的能量上移，并且向下旋转的电子是
going to get shifted up in energy by mu
B B and the spin down electrons are

439
00:39:57,450 --> 00:40:01,800
会变得更便宜，所以您将能源消耗降低了
going to become less expensive so you're
going to get shifted down in energy by

440
00:40:01,800 --> 00:40:08,040
 mu BB实际上也许我应该在这里删除一些内容，以便一些旋转
mu B B actually maybe I should erased
some things here so some of the spin ups

441
00:40:08,040 --> 00:40:12,480
将要变得翻身成为spin downs以降低
are going to get going to want to turn
over to become spin downs to lower the

442
00:40:12,480 --> 00:40:20,190
能量，让我们来画一下发生了什么，所以一旦添加
energy so let's let's draw that what
happens do this so once we add so this

443
00:40:20,190 --> 00:40:24,720
是因为B在这张图片中等于零B在这里等于零，让我们尝试绘制
is for B equals zero in this picture B
equals zero over here let's try to draw

444
00:40:24,720 --> 00:40:31,530
 B在这里不等于零，所以这里我们有能量这里我们有g用于自旋
B not equal to zero over here so here we
have energy here we have g for the spin

445
00:40:31,530 --> 00:40:41,520
 E的上升，这里我们有E的下降的G，发生的是
ups of E and here we have G for the spin
downs of E and what happens is that

446
00:40:41,520 --> 00:40:47,910
这些家伙的能量像这样被移动了，这个距离是亩B倍B 
these guys got shifted up in energy like
this this distance here is mu B times B

447
00:40:47,910 --> 00:40:57,090
这些家伙被这部如此多的电影所震撼了，好吧，然后我们
and these guys got shifted down this by
this much movie be okay and then we fill

448
00:40:57,090 --> 00:41:04,590
它们都由EF决定，这里是EF，这是EF，因此您会看到一些
them both up to EF here this is EF this
is EF so you see that that some of the

449
00:41:04,590 --> 00:41:13,010
那些旋转起来的家伙在这里清空，并在这里填充这些状态
ones that were spin up these guys here
emptied out and fill these states here

450
00:41:13,010 --> 00:41:18,390
清楚这是怎么发生的，所以这里的这些状态被推高了
is that clear how that happened
so these states here got pushed up in

451
00:41:18,390 --> 00:41:24,650
费米表面以上的能量，所以它们清空了，而这些状态在这里
energy above the Fermi surface so they
emptied out whereas these states here

452
00:41:24,650 --> 00:41:29,010
精力充沛，所以他们充满了好吗
got pulled down in energy so they filled
up okay

453
00:41:29,010 --> 00:41:36,300
是不是很清楚我知道这让我有点困惑
is that clear I know this this gets a
little bit confusing bear with me

454
00:41:36,300 --> 00:41:42,410
因此，如果我们要为自旋密度写一个表达式，我们可以写
so if we want to write an expression for
the spin up density we can write

455
00:41:42,410 --> 00:41:53,610
从零到EF减去mu BB的整数，因为实际上我们只想
integral from zero to EF minus mu B B
because actually we only want to

456
00:41:53,610 --> 00:41:59,010
整合到此处，因为在仅查找此区域时，此处没有此区域
integrate up to here because in finding
only this region not this region here

457
00:41:59,010 --> 00:42:02,610
因为一旦我们加了磁场，这里就只有这个区域
because once we add magnetic field it's
only this region here it's only the

458
00:42:02,610 --> 00:42:08,820
这些人已经被清空了
smaller region here that's filled up
these guys have now emptied de of G of e

459
00:42:08,820 --> 00:42:15,570
我猜超过两个，而下降的次数是从零到EF的整数倍
I guess over two whereas the number of
spin downs is integral from zero to EF

460
00:42:15,570 --> 00:42:23,550
再加上两个以上的g bb de g，因为我们要一直整合到
plus mu b b de g of e over two because
we want to integrate all the way up to

461
00:42:23,550 --> 00:42:27,290
在这里，既然那些人的精力都下降了
here since those have been pulled down
in energy is that clear

462
00:42:27,290 --> 00:42:33,630
好了，所以我们快到了，所以现在我要计算的是n 
alright so we're almost there so now
what I want is I want to calculate n

463
00:42:33,630 --> 00:42:44,790
 down减去n up，它是EF减去mu B到EF加mu BB的整数倍
down minus n up which is integral de
from EF minus mu B be to EF plus mu B B

464
00:42:44,790 --> 00:42:51,990
 EE的G超过2，那么我们就可以了
of G of EE over 2 and then we can just
since this is over over only a little

465
00:42:51,990 --> 00:42:55,950
一小部分能量，我们可以使用矩形规则来计算
small sliver of energy we can use the
rectangular rule to calculate that

466
00:42:55,950 --> 00:43:06,630
积分，我们得到1/2 G的EF x-mu BB um如果需要的话，取消二
integral and we get 1/2 G of EF x - mu
BB um cancel the twos if we want ok

467
00:43:06,630 --> 00:43:14,119
让我们写出EF的G乘以mu B是B然后
let's write out G of EF times mu B is B
and then

468
00:43:14,119 --> 00:43:19,609
最后，我们要寻找的磁化强度是
finally the magnetization which is what
we're looking for the magnetization is

469
00:43:19,609 --> 00:43:25,859
 Mubi每个电子的磁化强度为1，磁矩为1 
Mubi each electron has a magnetization
of one it has a magnetic moment of one

470
00:43:25,859 --> 00:43:29,670
 Bohr Magneton，然后我们得到了旋转下降的密度减去密度
Bohr Magneton and then we have the
density of spin downs minus the density

471
00:43:29,670 --> 00:43:32,729
旋转会给您磁化强度，如果您想知道
of spin ups will give you the
magnetization and if you want to know

472
00:43:32,729 --> 00:43:37,440
为什么以这种方式出现标志，为什么它跌宕-起伏不是起伏-跌倒是因为
why the sign comes out this way why it's
downs - ups not ups - downs it's because

473
00:43:37,440 --> 00:43:40,319
电子上的电荷为负，表示电子自旋
the charge on the electron is negative
meaning the spin of the electron

474
00:43:40,319 --> 00:43:45,259
实际上与它的磁化相反，这很令人困惑
actually points opposite its
magnetization which is pretty confusing

475
00:43:45,259 --> 00:43:49,890
但这是我们必须处理的，因为电子的符号是
but that's what we have to deal with
because the sign of the electron is is

476
00:43:49,890 --> 00:43:53,969
否定，所以我们要插上电源-n向下插
negative
okay so we'll plug in and up - n down so

477
00:43:53,969 --> 00:44:02,969
我们将得到G ef mu b乘以磁场的平方，然后将其称为
we'll get G e f mu b squared times the
magnetic field and you would call the

478
00:44:02,969 --> 00:44:09,690
在磁场而非磁场下磁化率chi的定义，因此我们确定
definition of susceptibility chi over mu
not magnetic field so we identify the

479
00:44:09,690 --> 00:44:15,180
保利功率磁化率是μB平方μn 
susceptibility the Pauli power magnetic
susceptibility is mu B squared mu naught

480
00:44:15,180 --> 00:44:23,369
 G TF，这又是我们的最终结果，我们可以将其与
G a TF and that's our final result again
we can take this result compared it to

481
00:44:23,369 --> 00:44:30,809
理论胜于Chi的实验，并且可以做各种不同的事情
the theory Theory over Chi experiment
and do it for various different things

482
00:44:30,809 --> 00:44:39,359
锂的比例约为2.5钠的比例为1.8钾的比例为1.6 
for lithium that ratio is about 2.5 for
sodium is 1.8 for potassium is 1.6 which

483
00:44:39,359 --> 00:44:44,099
如果我们正在考虑经典粒子，那么这是一个很好的协议
is pretty good agreement now if we were
thinking about classical particles

484
00:44:44,099 --> 00:44:51,630
古典单原子气体然后实际上当您施加磁场时
classical monatomic gas then in fact
when you apply to magnetic field nothing

485
00:44:51,630 --> 00:44:54,809
会阻止他们全部翻身他们会直接进入你的身边
would stop them all from flipping over
they would just go right into there you

486
00:44:54,809 --> 00:44:59,099
知道翻转状态，因为那将是较低的能量，并且是唯一
know the the flip down state because
that would be lower energy and the only

487
00:44:59,099 --> 00:45:02,609
之所以不能全部翻身是因为费米的统计费米
reason they don't all flip over is
because of Fermi statistics Fermi

488
00:45:02,609 --> 00:45:05,640
统计数据阻止它们全部翻转，因为某些状态
statistics prevents them all from
flipping over because some of the states

489
00:45:05,640 --> 00:45:08,759
已经充满了，因此您具有有限的磁化率，而经典
already filled and so you get a finite
susceptibility whereas the classical

490
00:45:08,759 --> 00:45:13,289
计算将预测出无限的敏感性，所以这是
calculation would would predict an
infinite susceptibility so this is

491
00:45:13,289 --> 00:45:16,589
非常好，我们在两到三倍的范围内就收到了结果
pretty good we're getting results there
you know within a factor of two or three

492
00:45:16,589 --> 00:45:22,109
实际的实验，所以这或多或少是我们不得不说的
of the actual experiments so this is a
more or less all we have to say about

493
00:45:22,109 --> 00:45:27,520
 Sommerfeld理论，让我们总结几件成功的事
Sommerfeld theory so let's summarize a
couple things successes

494
00:45:27,520 --> 00:45:35,930
自由电子理论理论的成功，这意味着德鲁德再加上
successes of free electron free electron
theory theory which means Drude up plus

495
00:45:35,930 --> 00:45:48,350
 Sommerfeld Garuda和Sommerfeld嗯，好吧，我们取得了一些成功
Sommerfeld Garuda plus Sommerfeld um
well okay a couple successes we got the

496
00:45:48,350 --> 00:45:52,730
正确的热容量我们得到了良好的导热性
heat capacity right we got the
conductivity good the thermal

497
00:45:52,730 --> 00:45:56,750
电导率好这两者之比都很好帕尔贴系数在
conductivity good the ratio of those two
are good the Peltier coefficient is in

498
00:45:56,750 --> 00:45:59,720
正确的球场易感性，实际上还有许多其他
the right ballpark the susceptibility
and there's actually many many other

499
00:45:59,720 --> 00:46:04,580
你可以计算出你会得到正确的东西，所以自由电子
things that you can calculate that
you'll get right so the free electron

500
00:46:04,580 --> 00:46:12,050
图片实际上还不错，但是仍然存在一些问题
picture is actually pretty good however
there are still some problems that we're

501
00:46:12,050 --> 00:46:16,720
必须要处理的是，正如我在本讲座前面提到的
going to have to deal with one is that
as I mentioned earlier in this lecture

502
00:46:16,720 --> 00:46:25,190
 lambda似乎太大太大了，这是一个大问题，因为
lambda seems too big too big that's a
big problem having a mean free path of a

503
00:46:25,190 --> 00:46:29,540
毫米，这似乎完全不合理，因为我们使用了
millimeter it just seems completely
unreasonable we used that the density of

504
00:46:29,540 --> 00:46:33,950
电子应该是每个原子一个电子，我们有一些直觉，为什么我们
electrons should be one electron per
atom and we had some intuition why we

505
00:46:33,950 --> 00:46:37,790
应该这样做，因为好吧，你知道有一堆电子
should do that because well you know
there's a bunch of the electrons are

506
00:46:37,790 --> 00:46:41,450
束缚在核心轨道上，也许他们不算他们没有绕着他们跑
bound in core orbitals and maybe they
didn't count they don't run around they

507
00:46:41,450 --> 00:46:45,080
只是保持固定不动，但还有其他一些你知道有效的原子
just stay stay fixed but there's some
other atoms you know that works

508
00:46:45,080 --> 00:46:48,410
钠和钾都很好，但是碳呢
perfectly well for sodium and for
potassium but what about for carbon

509
00:46:48,410 --> 00:46:53,090
碳使钻石在价态轨道上具有四个电子，如果
carbon makes Diamond it has four
electrons in its valence orbitals if

510
00:46:53,090 --> 00:46:56,540
在最外层的外壳中有四个电子，实际上它是一个绝缘体
four electrons in its outermost shell
and in fact it's an insulator there are

511
00:46:56,540 --> 00:47:00,170
没有电子自由运行，为什么不在那里发生什么，为什么我们
no electrons running around free so why
not what's going on there why do we

512
00:47:00,170 --> 00:47:03,920
每个原子计数一个电子为什么你根本不计算一些电子
count one electron per atom why do you
not count some electrons at all so what

513
00:47:03,920 --> 00:47:08,960
那些其他电子发生了什么，霍尔效应符号的符号呢
happened to those other electrons what
about the sign of the Hall effect sign

514
00:47:08,960 --> 00:47:13,940
 Rh我们的霍尔的霍尔系数总是应该具有相同的符号
of Rh our Hall the hall coefficient it's
always supposed to have the same sign in

515
00:47:13,940 --> 00:47:17,330
在索默费尔德理论中，犹大理论也总是被认为具有相同的符号，我们
Judah theory also in Sommerfeld theories
always supposed to the same sign and we

516
00:47:17,330 --> 00:47:20,540
实验测量了我们没有测量的东西，但是它被测量了
measured experimentally what we didn't
measure but it was measured

517
00:47:20,540 --> 00:47:25,460
从实验上看，实际上迹象出了错，有时又是另一回事
experimentally that in fact the sign
comes out wrong sometimes another thing

518
00:47:25,460 --> 00:47:32,300
我没提到光学特性之前就很有趣
that's kind of interesting I didn't
mention before optical properties are

519
00:47:32,300 --> 00:47:38,070
不同或者也许为什么不同为什么不同
different or maybe why different why
different

520
00:47:38,070 --> 00:47:42,720
我的意思是，不同的金属看起来不同，所以您知道黄金
what I mean by this is that different
metals look different so you know gold

521
00:47:42,720 --> 00:47:45,660
看起来有点像金色的银色看起来有点像银色的铜
looks kind of goldish silver looks kind
of silverish copper look kind of

522
00:47:45,660 --> 00:47:49,650
铜色，因此您知道这个名字，但锡看起来有点像您所知道的
copperish hence the name you know but
tin looks kind of you know lighter and

523
00:47:49,650 --> 00:47:52,950
酵素看起来更暗，它们看起来不同，它们具有不同的光学效果
leaven looks kind of darker they look
differently they have different optical

524
00:47:52,950 --> 00:47:56,880
它们反射不同颜色的特性为什么在自由电子理论中
properties they reflect different colors
why is that in the free electron Theory

525
00:47:56,880 --> 00:48:01,380
他们都应该看起来完全一样我们没有提到的另一件事是
they should all look exactly the same
another thing we didn't address is

526
00:48:01,380 --> 00:48:09,800
磁性磁性，例如铁， 
magnetism magnetism um
particular things like iron are

527
00:48:09,800 --> 00:48:15,570
铁磁性来自铁的名称，您可能会磁化
ferromagnetic coming from the name for
iron that you could have magnetization

528
00:48:15,570 --> 00:48:21,290
即使B等于零，也不等于零，您可能在
not equal to zero even when even when B
equals zero you probably studied this in

529
00:48:21,290 --> 00:48:26,640
你的电磁过程为什么自由电子图永远不会得到
your electromagnetism course why is that
free electron picture would never get

530
00:48:26,640 --> 00:48:38,430
最后，库仑相互作用如何？ 
that finally what about Coulomb
interactions Coulomb interactions when

531
00:48:38,430 --> 00:48:42,000
我们考虑到电子在固体中运转，你知道他们
we think about the electrons running
around in a solid you know that they

532
00:48:42,000 --> 00:48:45,390
有巨大的费米能量，你知道八万开尔文，但是如果你考虑一下
have a huge Fermi energy you know eighty
thousand Kelvin but if you think about

533
00:48:45,390 --> 00:48:48,780
库仑相互作用的能量规模也一样大
the the energy scale of the Coulomb
interaction it's just as big the

534
00:48:48,780 --> 00:48:52,560
电子与原子核的相互作用也接近八十
interaction of the electron with the
nucleus nuclei close by also eighty

535
00:48:52,560 --> 00:48:55,830
千开尔文也有大量的电子与其他相互作用
thousand Kelvin also a huge number the
interaction of the electron with other

536
00:48:55,830 --> 00:48:59,490
通过它运行的电子也有八万开尔文，我们把它扔掉了
electrons that are running by it also
eighty thousand Kelvin we threw it out

537
00:48:59,490 --> 00:49:03,780
完全我们只是将这些电子当作自由气体对待
completely we just treated these
electrons as if there are a free gas no

538
00:49:03,780 --> 00:49:07,980
相互作用，我们只将它们具有费米统计量的事实视为
interactions whatsoever we only treated
the fact they have Fermi statistics to a

539
00:49:07,980 --> 00:49:12,210
在很大程度上，所有这些问题都来自我们同一件事
large extent all of these problems come
from the same thing that we're

540
00:49:12,210 --> 00:49:16,110
一遍又一遍地忽略同一件事，我们将不得不
neglecting the same thing over and over
and over again and we're going to have

541
00:49:16,110 --> 00:49:20,370
更认真地对待一个特定的项目，而那个特定的项目是
to take more seriously one particular
item and that one particular item is

542
00:49:20,370 --> 00:49:25,140
材料具有微观结构详细的微观
that materials have microscopic
structures detailed microscopic

543
00:49:25,140 --> 00:49:28,710
结构原子通常以特定方式以特定方式粘在一起
structures atoms are stuck together in a
particular way frequently a particular

544
00:49:28,710 --> 00:49:32,370
周期性的方式，完全改变了属性，这就是我们
periodic way and that completely changes
that properties and that's what we're

545
00:49:32,370 --> 00:49:35,700
剩下的大部分时间都需要处理
going to have to deal with for much of
the remainder of the term all of these

546
00:49:35,700 --> 00:49:41,670
一旦我们诚实地处理了这一事实，大部分事情都会得到解决。 
things to a large extent will be sorted
out once we deal honestly with the fact

547
00:49:41,670 --> 00:49:45,690
原子以某种特定方式排列首先，我们是
that the atoms are arranged in some
particular way the first thing we're

548
00:49:45,690 --> 00:49:48,420
要做的是，我们必须了解为什么这是
going to have to do is we're going to
have to understand why it is that the

549
00:49:48,420 --> 00:49:51,960
原子实际上粘在一起开始，所以开始下一个
atoms actually stick together to begin
with and so starting the next

550
00:49:51,960 --> 00:49:56,430
所以我们将讨论一些化学和化学键
so we're going to discuss a bit of
chemistry and chemical bonding just a

551
00:49:56,430 --> 00:50:02,250
真的很快，多少人有化学反应哦，这很好
really quick thing how many people had a
level chemistry oh that's pretty good

552
00:50:02,250 --> 00:50:07,530
有多少人不好，你是幸运者，因为你不得不
how many did not okay you're the lucky
ones because you're gonna have to

553
00:50:07,530 --> 00:50:10,320
取消学习很多您可能不知道的事情， 
unlearn a lot of the things that you
know maybe not like it's always good to

554
00:50:10,320 --> 00:50:16,110
学习学习甚至化学的东西，但是您可能不得不学一些
learn learn things even chemistry but
but you may have to unlearn some of the

555
00:50:16,110 --> 00:50:18,570
您之前学到的东西，因为我们将从一个
things that you learned previously
because we're going to look at it from a

556
00:50:18,570 --> 00:50:24,090
更多的物理观点，我将在星期四，星期四，星期四见
more physics perspective and I will see
you Thursday Thursday Thursday

557
00:50:24,090 --> 00:50:28,579
好，星期四[鼓掌] 
okay Thursday
[Applause]