08 Microscopic View of Electrons in Solids in One Dimension Tight Binding Chain.srt

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好的，欢迎回来，这是浓缩物质课程的第八讲
okay welcome back this is now the eighth
lecture of the condensed matter course

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量子力学最奇妙的事情之一是粒子是
one of the more wonderful things about
quantum mechanics is that particles are

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波，波或粒子，这意味着我们在学习时所学到的东西
waves and waves or particles and that
means that things that we learn when we

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研究固体中的振动波通常可以应用于其他类型的波
study vibrational waves in solids can
often be applied to other types of waves

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在诸如电子波甚至电磁波之类的固体中，所以今天我们
in solids such as electron waves or even
electromagnetic waves so today we're

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00:00:25,669 --> 00:00:29,539
将要研究固体中的电子波，这堂课有点
going to be studying electron waves in
solids and this lecture is a little bit

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不合适，因为在课程的后期，我们将花费几天
out of place because later on in the
course we're going to spend several days

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研究固体中的电子带结构和一些电子波
studying electron band structure
electron waves in solids and some amount

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深度，但我在此插入本讲座的原因是为了指出重点
of depth but the reason I'm inserting
this lecture here is to make the point

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我们是否真的在研究同一件事，无论我们是否在研究振动
that we're really studying the same
thing whether we're studying vibrational

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或我们正在研究电子电子波
waves or we're studying electron
electron waves it's really very very

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类似地，我们还将看到我们今天所做的许多计算是
similar we're also going to see that
much of the calculation we do today is

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类似于我们研究共价键时的行为
similar to what we did when we study the
covalent bond which was a wave just

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在两个原子之间，所以我们今天要看的图片是
between two atoms so the picture we're
going to look at today is a

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一维一维紧密绑定链，这将是非常
one-dimensional one-dimensional tight
binding chain and it's going to be very

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类似于我们研究的一维振动链
analogous to the one-dimensional
vibrational chains that we looked at in

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最后一对讲座也类似于共价键，所以我们要
the last couple lectures also similar to
the covalent bond so we're going to

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想象像这样的链中有一束核，胶水给它们一个
imagine having a bunch of nuclei in a
chain like this and the glue give them a

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晶格常数相同原子核之间的距离，我们将添加
lattice constant a distance between
identical nuclei and we're going to add

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一个电子到这个原子核链，我们将看到当我们发生时会发生什么
one electron to this chain of nuclei and
we're going to see what happens as we

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让电子在不同原子核之间来回跳跃
allow the electron to hop back and forth
between the different nuclei so of

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当然，我们必须从哈密顿量开始，因为通常的P平方
course we have to start with a
Hamiltonian as the usual P squared over

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 2m任期加上它将与R的所有不同核V相互作用
2m term plus it will have an interaction
with all of the different nuclei V of R

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减去R sub J，其中R sub J是原子核的位置
minus R sub J where R sub J is position
position of nucleus

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 J，就像我研究共价键时所做的那样，我们将简称
J and as I did when we studied the
covalent bond we're going to abbreviate

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为了方便起见，这些术语称为k4动能， 
these terms for convenience this term
will be called k4 kinetic energy and

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这些项现在称为V sub J，用于与j DH核相互作用
these terms will be called V sub J for
interaction with the j DH nucleus now

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类似于我们研究共价键时所做的事情
similar to what we did when we studied
the covalent bond it's useful to think

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首先是关于电子，它仅与单个原子核相互作用，而不与任何原子核相互作用。 
first about an electron only interacting
with a single nuclei not with any of the

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其他原子核，所以我们写Hoop，我们写k加V sub M作为我们的
other nuclei so we'll write H oops
we'll write k plus V sub M as our

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哈密​​顿量，这意味着电子具有其动能，并且
Hamiltonian and that means that the
electron has its kinetic energy and it's

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仅与移情的leus交互，当它与之互动时，我们将为其赋予原子性
interacting with the empathic leus only
and we'll give it a atomic when it

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仅与原子核M相互作用，所以哈密顿量的本征态
interacts with nucleus M only so the
eigen state of the Hamiltonian which has

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与amnok leus进行互动时，我们称其为ket m， 
an interacting with the amnok leus we'll
call that the ket m and that will put

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蒽CLIA上的电子-好像所有其他核都不是
the electron on the anthony CLIA s-- as
if all of the other nuclei were not

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那里有一个标签，如果电子坐在这里就像是它
there at all so a label this cat one if
the electron is sitting here as if it

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并且不与任何其他原子核相互作用我们也称其为
and not interacting with any of the
other nuclei we'll call this one too

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这个一三等等好吧，到目前为止，好吧好吧，就像我们做的那样
this one three and so forth okay happy
so far okay all right now as we did when

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我们研究了共价键，我们将做出一个错误的假设
we studied the covalent bond we're going
to make a bad assumption bad assumption

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这就是说这些猫和M是正交的，如果
which is that these cats and an M are
orthonormal this is not too bad if the

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原子核彼此分开，因为一个电子坐在这里
nuclei are far apart from each other
because a electron sitting over here in

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那里的电子坐姿几乎彼此正交
electronic sitting way over there are
pretty much orthogonal to each other

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但是当原子核彼此靠近时，它们就不再正交了， 
but when nuclei get close together then
they're not orthogonal anymore and the

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我们做出这个近似的原因，即使那些不好的近似是为了
reason we make this approximation even
those are bad approximation is for

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简单性如果我们做到这一点，很多计算就变得容易得多
simplicity a lot of the calculation just
gets a lot easier if we make this

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假设，而做得更好，您不会学到很多东西
assumption and you don't learn a whole
lot more from doing more properly it's

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要做起来并不难，这本书上有一个练习
not that much harder to do it properly
there's an exercise on the book that

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引导您完成操作，并且您知道可以通过它进行操作，但是您会
walks you through it and you know you
can go through it if you want but you'll

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从这个简化的近似中获得大多数有趣的物理学
get most of the interesting physics out
of just this simplified approximation

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好吧，一旦我们做出了这个错误的假设，我们就可以写
okay so once we
made this bad assumption we can write

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向下试波函数试波函数与我们所做的非常相似
down our trial wavefunction trial
wavefunction very similar to what we did

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 sie等于n Phi nna的和的共价键
with the covalent bond which will have
the form sie equals sum over n Phi n n a

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原子轨道的线性组合，这是我们使用的等效词
linear combination of atomic orbitals
which is a word we used equivalent to

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类型绑定线性组合
type binding linear combination of

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轨道或lcao的原子轨道
atomic orbitals of orbitals or lcao

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我们正在使每个原子轨道上都有一堆电子
we're making we have a bunch of atomic
orbitals electrons sitting on each

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核，我们将使它们与系数v线性组合
nucleus and we're gonna make a linear
combination of them with coefficients v

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人们不喜欢这种近似的原因之一是因为您可以
sub n the reason people love this type
of approximation is because you can make

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通过向右侧添加更多内容，它变得越来越准确
it more and more accurate by just adding
more things to the right hand side with

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变量参数Phi在它们的前面，例如，您可以使用
variational parameters Phi in front of
them so for example you can have an

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电子以激发态α位于位点n上，所以可能是1s 2s 2p，所以
electron sitting on site n in excited
state alpha so this could be 1s 2s 2p so

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等等，我们可以使基础状态越来越大
forth and so on and we can just make our
basis state bigger and bigger basis set

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越来越大，并赋予所有这些系数， 
bigger and bigger and bigger and give
all of these coefficients and as we make

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基础越来越大，我们得到的越来越多
the basis set bigger and bigger and
bigger and bigger we get a more and more

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真正的波函数的精确近似，好吧，这有点
accurate approximation of the true
wavefunction ok this is sort of a

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变分方法，因此我们可以通过找到最佳方法来解决基态
variational approach so we can solve for
the ground state by finding the best

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系数，那么一旦我们有了基态，我们就可以求解第一个
coefficients then once we have the
ground state we can solve for the first

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通过找到最佳系数和最低能量来激发状态
excited state by finding the best
coefficients the lowest energy

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正交于我们刚发现的第一件事的系数
coefficients subject to being orthogonal
to the first thing we were just found

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依此类推，依此类推，所以现在我们要做的就是写下一个
and so forth and so on so now what we
have to do is we have to write down a

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这些系数的内部方程越短，方程又越短
shorter inner equation for these
coefficients the shorter equation again

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您要在作业中解决的问题将采用H的形式
something that you will solve for in
your homework will be of the form H and

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 M Phi M等于e Phi n，它看起来像一个短方程
M Phi M equals e Phi n it you know it
looks like a shorter equation

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像薛定inger方程那样的庸医，可能是它的缩短方程
quacks like a Schrodinger equation it
probably is a shortened equation it's

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不是真正的较短方程，因为真正的较短方程具有一个完整的
not the real shorter equation because
the real shorter equation has a full

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坐在这里的哈密顿量反而我们只有一个矩阵和HM，您可以
Hamiltonian sitting here here instead we
just have a matrix and H M and you can

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认为这是真正的较短方程在基础上的投影
think of this as being the projection of
the true shorter equation onto the basis

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由这些ket组成，并且我们与之合作，这使得
set made up of these ket's and that we
that we working with okay that makes

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感觉它看起来像一个较短的方程，它将像一个较短的方程
sense it looks like a shorter equation
it's going to act like a shorter

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等式，您也可以推导出它，所以在这里给出较短的等式
equation and you'll derive it also okay
so given our shorter equation over here

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我们现在必须计算这些矩阵元素
we have to now calculate these matrix
elements that go into our a shorter

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等式，让我们做，采用我们的哈密顿量并将其除法很有用
equation so let's do that it's useful to
take our Hamiltonian and divide it up

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分成几部分，因此我们首先将删除与
into pieces so first we'll take we'll
remove the empty interaction with the

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与所有其他原子核相互作用产生的空洞，因此这将是J 
empty leus from the interaction with all
the other nuclei so this will be J not

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等于MV sub J之所以这样做，是因为我们可以在H上写H 
equal to M V sub J the reason we do this
is because then we can write H on the

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 ket M等于K加cat M上的VM加上与所有其他元素的交互
ket M equals K plus VM on the cat M plus
the interaction with all the other

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骆驼上的核V sub J好的，现在我们将ket m定义为本征态
nuclei V sub J on camel okay now we
defined the ket m to be the eigenstate

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 K + VM的数据，所以这里的内容只是ket M上的原子
of K plus VM so this thing here is just
e atomic on ket M oh good

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对此感到满意，然后我们可以将内部产品关闭
happy with that so then we can take the
inner product close up the inner product

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在这里有ket n，我们得到e原子Delta和M加和，求和于J 
with the ket n over here and we get e
atomic Delta and M plus and sum over J

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不等于MV sub JM，这是个有趣的名词，所以这个名词
not equal to M V sub J M and this is the
interesting term here so this term here

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只是告诉我们，无论电子位于哪个原子核上
just tells us that no matter what site
the electron is sitting on which nucleus

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坐在它上面的电子具有能量e原子，这就是所有
the electron sitting sitting on it has
energy e atomic and this is all the

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与所有其他原子核的相互作用现在有一个
interaction with
all of the other nuclei now there's a

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这个词可能会发生的几件事一种可能性是N 
couple of things that could happen with
this term one possibility is that N

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等于M，在这种情况下，这就是告诉你有一些
equals M in which case this is what this
is telling you is that there's some

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由于与所有原子核相互作用，位于核M上的能量发生了变化
change in its energy sitting on nucleus
M due to its interaction with all of the

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其他原子不是M好吧，这将给我们带来这种东西和能量
other atoms not M okay so this would be
will give this thing and energy we'll

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称之为V naught等于和某些VGA不II J不等于M，我猜这些
call it V nought equals and some VGA not
II J not equal to M and I guess these

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可能是一团糟，这是与所有原子核相互作用的原因
can be a mess like that and this is so
this is interacting with all the nuclei

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不包括M，这是对能量的期望，所以它只是在改变
not including M and it's an expectation
of its energy so it's just shifting its

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该特定网站上的能量并不是特别有趣
energy on that particular site that's
not particularly interesting the more

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有趣的是发生了什么事并且不等于M，所以在这种情况下
interesting thing is what happens is and
not equal to M so in this case you have

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这个术语也许是我应该先称呼这个术语，这就是我们
this term is maybe I should call this
term something first this is what we

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之前直接打过电话，我想当我们谈到
call direct before and I guess I call it
D cross when we talked about the

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末端的共价键不等于M项，这就是我们之前所说的跳跃
covalent bond for the end not equal to M
term this is what we call hopping before

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我们之所以将其称为“跳跃”是因为，我们将赋予它价值
and the reason we called it hopping was
because and we'll give it the value

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00:10:01,360 --> 00:10:07,990
减去t减去T之所以称为跳变，是因为您认为
minus t minus T the reason we call it
hopping is if you think in the

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与时间相关的薛定inger方程类型的电子吸收方式
time-dependent schrodinger equation
type of way you can take on electron

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坐在站点m上，并最终到达站点n上，因此该对角项在
sitting on site m and have it end up on
site n so this off diagonal term in the

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哈密​​顿量允许电子从一个位置移动到另一个位置，因此我们称之为
Hamiltonian allows an electron to move
from one site to another hence we call

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它现在跳了一个近似值，实际上是一个相当不错的近似值
it hopping now an approximation which is
actually a fairly good approximation is

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 n减去m大于1的跳变为0 
that n minus m greater than 1 hopping is
0

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跳变等于0或大约为0，其原因是因为
hopping equals 0 or is approximately 0
and the reason for that is because it's

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如果想一想，电子很难一步一步跳到很远
very hard for an electron to hop very
far in one step if you think about it

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一秒钟，当我们采用这个矩阵时，我们真正计算的是
for a second what we're really
calculating when we take this matrix

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当我们明确写出矩阵元素时，它是某种
element is some sort of when we write
out the matrix element explicitly it's

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像这样的东西
something like this

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正确，这就是矩阵元素看起来像是胸罩，ket和交互作用
right this is what matrix elements look
like a bra a ket and an interaction now

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00:11:06,819 --> 00:11:13,209
如果phi n如果N和M相距遥远，那么这些波函数会随着
if phi n if N and M are far apart these
wave functions decay very quickly as you

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远离原子核，所以在空间上没有点
go away from the nucleus so there will
be no point in space where both this is

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很大，如果两个原子核相距很远，那么就很大，这就是为什么
large and this is large if the two
nuclei are very far apart so that's why

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我们可以假设这个矩阵元素将为0，除非N 
we can we can assume that this this
matrix element is going to be 0 unless N

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00:11:27,459 --> 00:11:35,170
和M本质上是邻居，好吧，所以我们现在拥有的是
and M are essentially neighbors okay
good so what we have now is we have this

125
00:11:35,170 --> 00:11:43,949
最后，我们可以在J上写n个和，不等于MV sub JM 
a in the end we have that we can write n
sum over J not equal to M V sub J M

126
00:11:43,949 --> 00:11:53,740
等于直接项V，如果n等于M，则不等于n，如果N等于M，则称其为负T 
equals the direct term V not if n equals
M we'll call it minus T if N equals M

127
00:11:53,740 --> 00:11:59,970
正负1，因此我们只能跳一个站点，否则跳零
plus or minus 1 so we can hop one site
only and zero otherwise

128
00:12:01,110 --> 00:12:06,749
好吧，好人对此感到满意
okay good people happy fairly happy with
that

129
00:12:06,749 --> 00:12:13,179
好是有人点头有人说有人说是谢谢，我发给你另一个
good yes someone nod someone say someone
say yes thank you I send you another

130
00:12:13,179 --> 00:12:18,819
巧克力，但是我昨天给你一个，很好，好，我们可以拿走
chocolate but I gave you one yesterday
so okay good so we can take our

131
00:12:18,819 --> 00:12:28,059
哈密​​顿量并将其重写为H，因为它是一个大矩阵H nm等于它是一个
Hamiltonian and rewrite it as H as a big
matrix H n m equals what is it it's a

132
00:12:28,059 --> 00:12:36,759
原子加上与所有其他原子核（不包括位点本身）的相互作用
atomic plus the interaction with all the
other nuclei not including site itself

133
00:12:36,759 --> 00:12:40,029
如果您坐在一个网站上，与其他网站互动， 
if you're sitting on one site
interacting with all the other ones not

134
00:12:40,029 --> 00:12:45,699
包括你自己，还有另一个名词，Delta n M加一个
including yourself plus there's going to
be another term which Delta n M plus one

135
00:12:45,699 --> 00:12:52,689
加Delta n逗号M减1，所以这里有这个附加项
plus Delta n comma M minus one so
there's this additional term here which

136
00:12:52,689 --> 00:12:58,899
允许您向左跳一步或向右跳一步
allows you to hop one step to the left
or one step to the right with an

137
00:12:58,899 --> 00:13:08,230
振幅很好，所以如果我们说我们是一个很大的矩阵
amplitude a T okay good so this is a
great big matrix if we have let's say we

138
00:13:08,230 --> 00:13:14,690
以nn nu开头，则H为N×n矩阵
have n n nu
to begin with then H is an N by n matrix

139
00:13:14,690 --> 00:13:21,589
和Aiden乘n哈密顿矩阵，我们需要找到其特征值
and Aiden n by n Hamiltonian matrix and
we need to find its eigenvalues so how

140
00:13:21,589 --> 00:13:25,490
如果n为a是a，我们这样做看起来像一个复杂的问题吗？ 
do we do that that looks like a
complicated problem if n is a is a

141
00:13:25,490 --> 00:13:30,500
很好，我们可以解决与我们所做的非常相似的事情
pretty large number well again we can
solve this very similar to what we did

142
00:13:30,500 --> 00:13:37,279
对于振动链，我们使用非点，这是点上的英文单词
for the vibrational chains we use a non
dots which is an English word on dots

143
00:13:37,279 --> 00:13:46,310
而我们使用他的文件的圆点是到IK的圆点上的平面波
and the on dots we use his file is a
plane wave on dots to the I K and a like

144
00:13:46,310 --> 00:13:53,360
现在，对此首先有一些评论，您可能会期待一个e 
this now a couple comments about this
first of all you may be expecting an e

145
00:13:53,360 --> 00:13:57,110
从我们对振动链所做的事情到I Omega T的原因是
to the I Omega T from what we did with
the vibrational chain the reason there's

146
00:13:57,110 --> 00:14:01,100
 I Omega T的原因在于我们正在解决与时间无关的薛定inger 
no e to the I Omega T is because we're
solving the time-independent schrodinger

147
00:14:01,100 --> 00:14:05,389
方程不是时间相关的薛定inger方程
equation not the time-dependent
schrodinger equation if we were to solve

148
00:14:05,389 --> 00:14:08,329
在时间相关的薛定inger方程中，欧米伽T等于e 
in the time dependent Schrodinger
equation there being e to the I Omega T

149
00:14:08,329 --> 00:14:11,690
好的，这就是为什么它在这种情况下不存在的原因， 
as well okay that's why it's not there
in this case it's just simpler in

150
00:14:11,690 --> 00:14:14,810
量子力学要与时间无关的本征态一起工作
quantum mechanics to work with time
independent eigen States than it is to

151
00:14:14,810 --> 00:14:19,310
使用与时间相关的波动函数，因此这是第一件事
work with time-dependent wave functions
so that's that's the first thing second

152
00:14:19,310 --> 00:14:23,000
事情是，如果您谨慎点，您可能放下
thing is that you probably if you're
careful you put down a square root of

153
00:14:23,000 --> 00:14:26,600
楼下的大写n，因此这是归一化的波动函数
capital n downstairs so that this is a
normalized wave function the

154
00:14:26,600 --> 00:14:30,139
规范化对我们来说并不重要，但严格来说应该
normalization isn't going to matter much
for us but strictly speaking it should

155
00:14:30,139 --> 00:14:36,589
可能在那里，第三点要注意的是，如果
probably be there and the third thing to
note is that this wave is the same if

156
00:14:36,589 --> 00:14:42,560
您在a上将k移至k加2pi是我们上次发现的
you shift k to k plus 2pi over a is
something we discovered last time the

157
00:14:42,560 --> 00:14:45,560
重要的不是动量而是水晶动量
thing that's important is not the
momentum but the crystal momentum if

158
00:14:45,560 --> 00:14:51,459
你把K移了2 pi，我又回到了同一波
you're shifted K by 2 pi over a I get
back exactly the same the same wave ok

159
00:14:51,459 --> 00:14:58,180
因此，让我们将点插入汉密尔顿式的其中一个步骤，我们得到
so let's take our on dots plug it into
that Hamiltonian one step here we get

160
00:14:58,180 --> 00:15:08,889
 ε等于负IK的零，e等于负IK n的负T加1 a 
epsilon naught e to the minus I K and a
minus T e to the minus I K n plus 1 a

161
00:15:08,889 --> 00:15:20,360
 ee等于负IK，负1 a等于ee等于负IK，因此
plus e e to the minus I K and minus 1 a
equals e e to the minus I K and a so

162
00:15:20,360 --> 00:15:24,290
只需将on点插入Schrodinger方程
that's just plug in the on dots into the
Schrodinger equation

163
00:15:24,290 --> 00:15:30,410
使用汉密尔顿式的形式好吧，这就是我没告诉过的
using that form of the Hamiltonian okay
so it's this is the oops I didn't tell

164
00:15:30,410 --> 00:15:34,100
你这是什么我没告诉你什么我不对这件事感到抱歉
you what this I didn't tell you what II
not is sorry about that this thing here

165
00:15:34,100 --> 00:15:43,880
我叫他不是，所以这是现场能量，它使您可以跳到左跳
I called he not so this is the energy on
site this allows you to hop to left hop

166
00:15:43,880 --> 00:15:50,690
在右边，这是另一面的特征能量
to the right and this is the eigenenergy
on the other side people happy okay good

167
00:15:50,690 --> 00:15:57,139
谢谢你，好，那么你就可以计算出一些因素。 
thank you good so then you just calc
cancel out a bunch of factors a bunch of

168
00:15:57,139 --> 00:16:06,620
指数因数，您得到e为零减去2t余弦K a，所以让我们
exponential factors and you get e is e
naught minus 2t cosine K a so let's

169
00:16:06,620 --> 00:16:20,449
实际绘制去，所以这里的PI超过这里的负PI超过这里
actually plot that go so here's PI over
a here's minus PI over a here's this is

170
00:16:20,449 --> 00:16:27,110
能量在这个轴上消失了，然后我们会有一个很好的余弦
energy on this axis energy this is e
naught and then we'll have a nice cosine

171
00:16:27,110 --> 00:16:32,209
看起来像这样的表格看起来像这样，如果您想
form that looks kind like this looks
kind of like this and if you wanted to

172
00:16:32,209 --> 00:16:37,190
您可以定期进一步进行下去，但我们真正感兴趣的是
you can continue it out periodically
further but what we're really interested

173
00:16:37,190 --> 00:16:45,769
进入的是区域1区域中从此处到此处的冲泡中的波浪，因为一旦我们
in is the waves within the brew on zone
1 zone from here to here because once we

174
00:16:45,769 --> 00:16:48,980
走到布里渊区之外，我们只是在复制相同的波
go outside of the Brillouin zone we're
just reproducing the same waves over and

175
00:16:48,980 --> 00:16:53,930
再来一次，如果我们想要不同的波浪，我们可以在
over again and if we want different
waves we have we all the ways within the

176
00:16:53,930 --> 00:16:56,630
*区彼此不同，但是一旦我们走出布朗斯自己
bran zone are different from each other
but once we go outside of the Brauns own

177
00:16:56,630 --> 00:17:03,079
我们开始重复我们已经认为可以的事情
we start repeating things that we've
already considered ok just a bit of

178
00:17:03,079 --> 00:17:06,790
 Naaman仓库非常有用，也许我把它放在这里
Naaman cloture which is which is fairly
useful maybe I put it over here an

179
00:17:06,790 --> 00:17:29,030
能带能带是指在每个K中一个情况的一个本征态。 
energy band energy band means one case
one eigenstate at each each K in the

180
00:17:29,030 --> 00:17:31,570
青铜自己
bronz own

181
00:17:35,890 --> 00:17:42,380
所以在这里我们画了一个能带，你会记得这看起来有点
so here we've drawn an energy band now
you'll remember this looks a little bit

182
00:17:42,380 --> 00:17:46,460
就像我们解决单原子谐波链时得到的一样
like what we got when we when we solve
the monatomic harmonic chain the

183
00:17:46,460 --> 00:17:50,930
每个原子在每个原子中都完全相同的单链振动
vibrations of a single chain of where
every atom is exactly the same in every

184
00:17:50,930 --> 00:17:54,770
春天与我上次讲解的完全相同
spring is exactly the same I last
lecture when we when we solve the

185
00:17:54,770 --> 00:17:58,370
双原子链或交替链，我们发现有两个分支
diatomic chain or the alternating chain
we found that there were two branches of

186
00:17:58,370 --> 00:18:04,210
激发在这个方向上每个K矢量都有两个正常模式
excitations there were two normal modes
at each K vector okay in this direction

187
00:18:04,210 --> 00:18:08,360
如果我们在这张图片中有类似的东西，其中有两个本征态
if we had something similar in this
picture where there were two eigenstates

188
00:18:08,360 --> 00:18:12,410
低能量的和高能量的，我们会说有两种能量
a low-energy one and a high-energy one
we would say that there are two energy

189
00:18:12,410 --> 00:18:16,850
低能带和高能带我们不使用声波
bands a low-energy one and a high-energy
band we do not use the words acoustic

190
00:18:16,850 --> 00:18:20,690
当我们谈论电子的原因是光学和光学时
and optical when we're talking about
electrons for reasons that will become

191
00:18:20,690 --> 00:18:27,950
一会儿就清楚了，但是将这种分散与我们将
clear in a moment but it is useful to
compare this this dispersion to what we

192
00:18:27,950 --> 00:18:38,690
有了振动振动了单原子链单原子我们有了欧米茄
got for vibrations vibrations the
monatomic chain monatomic we had Omega

193
00:18:38,690 --> 00:18:50,270
平方等于我猜这是两个Kappa / M减去两个Kappa / M余弦ka 
squared equals I guess it was two Kappa
/ M minus two Kappa / M cosine ka I

194
00:18:50,270 --> 00:18:55,820
相信这等同于您会看到两种色散看起来
believe it was equivalent to that you'll
see that the two dispersions look

195
00:18:55,820 --> 00:19:00,920
极为相似，但有显着差异，显着差异是
awfully similar but there is a notable
difference and the notable difference is

196
00:19:00,920 --> 00:19:05,000
对于较短的方程式，我们在左侧获得e， 
that with the shorter equation we got e
on the left-hand side and with the

197
00:19:05,000 --> 00:19:09,050
振动链，我们的欧米茄平方在左侧
vibrational chain we had omega squared
on the left-hand side the origin of this

198
00:19:09,050 --> 00:19:12,860
不同之处在于，如果考虑时间相关性，则较短的方程式
difference is if the shorter equation if
you think about the time-dependent

199
00:19:12,860 --> 00:19:17,240
薛定inger方程具有一阶导数，而牛顿方程F 
schrodinger equation it has one time
derivative whereas newton's equations F

200
00:19:17,240 --> 00:19:21,050
等于MA，加速度是两个时间导数，所以这就是为什么
equals MA the acceleration is two time
derivatives so that's why we're getting

201
00:19:21,050 --> 00:19:25,730
左侧有两个频率，而另一侧只有一个能量
two frequencies over on the left-hand
side but one only one energy on the

202
00:19:25,730 --> 00:19:28,340
当我们在思考薛定inger的时候
right-hand side when we're thinking
about Schrodinger and that actually

203
00:19:28,340 --> 00:19:32,960
当我们拥有
makes a rather important distinction
between the two when we had the

204
00:19:32,960 --> 00:19:38,020
在低能量的振动链中，线性模式降为零
vibrational chain at low energy we got
linear modes coming in down to zero

205
00:19:38,020 --> 00:19:44,700
此图片中零波矢量处的频率
frequency at zero wave vector in this
picture here

206
00:19:44,700 --> 00:19:51,400
如果在零波附近扩展，在电子情况下在电子中
in the in the electrons in the case of
electrons if you expand near zero wave

207
00:19:51,400 --> 00:19:58,920
向量不是负2t然后是小波向量K a的低替代值
vector it's not minus 2t and then low
substitute in for small wave vector K a

208
00:19:58,920 --> 00:20:08,100
像这样平方两个，我们看到这里得到2t，我们得到的是恒定的
squared over two like this we see that
we get 2t here what we get is constant

209
00:20:08,100 --> 00:20:16,540
我对加K的平方乘以T不感兴趣
which I'm not interested in plus K a
squared times T its quadratic at low

210
00:20:16,540 --> 00:20:21,760
余弦所带来的能量，这就是为什么我们不使用这个词
energy as you would expect from a cosine
and that's why we do not use the word

211
00:20:21,760 --> 00:20:25,990
这些低能量模式的声学振动特性
acoustic for these low energy modes for
vibrations acoustic modes by definition

212
00:20:25,990 --> 00:20:30,880
声音模式具有线性分散，频率应与
sound modes have linear dispersion that
the frequency should be proportional to

213
00:20:30,880 --> 00:20:36,160
波矢量是声音的定义，所以这里是二次方，所以我们不
wave vector that's the definition of
sound so here is quadratic so we don't

214
00:20:36,160 --> 00:20:41,020
称之为声学，如果有更高的能量，我们也不会称之为光学的
call it acoustic and we don't call it
optical either if there's higher energy

215
00:20:41,020 --> 00:20:47,050
品牌分支，但是这张照片显示的是能量的二次方波动
brands branches but this this picture of
the energy being quadratic in wave

216
00:20:47,050 --> 00:20:53,410
在其他情况下，vector应该对您相当熟悉，例如，如果您
vector should be fairly familiar to you
from other contexts for example if you

217
00:20:53,410 --> 00:20:59,050
有自由电子，有自由电子，有外太空中的电子飞来飞去或
have a free electron free electrons an
electron in outer space flying around or

218
00:20:59,050 --> 00:21:09,190
它的能量是恒定的，加上h bar平方K平方超过2m K 
something its energy is some constant
plus h bar squared K squared over 2m K

219
00:21:09,190 --> 00:21:16,180
平方超过2m，在MK中也是平方，您可以选择C 
squared over 2m right it's also
quadratic in M K you might choose C to

220
00:21:16,180 --> 00:21:19,960
如果您愿意，则为0；如果您愿意，则可以选择为MC平方
be 0 if you wanted to or you could
choose it to be MC squared if you're

221
00:21:19,960 --> 00:21:25,090
在思考其剩余质量能量的同时，重要的是
thinking about its rest mass energy as
well the important thing is that it's

222
00:21:25,090 --> 00:21:30,970
它是二次方的，并且波矢量与此处的电子相同，因此
it's quadratic in and wave vector the
same as this electron over here so it

223
00:21:30,970 --> 00:21:36,250
也许对我们有用，以自由电子的方式思考一秒钟
might be useful for us to sort of think
in terms of free electrons for a second

224
00:21:36,250 --> 00:21:44,610
所以我们要做的是定义一个有效质量有效质量M星
and so what we do is we define an
effective mass effective mass M star

225
00:21:44,610 --> 00:21:55,360
使得H bar在2 m星上平方等于T的平方乘以
such that H bar squared over 2 m star
equals a squared times T with this

226
00:21:55,360 --> 00:22:00,580
然后定义我们的能带和我们电子中的能带
definition then for our energy band
and for energy band for electrons in our

227
00:22:00,580 --> 00:22:08,080
能带能带我们仍然有一个常数等于一个常数。 
energy band energy band we still have
energy equals a constant same constant

228
00:22:08,080 --> 00:22:15,940
再加上h平方，K平方超过2m星，好吧，它看起来像免费的
plus h bar squared K squared over 2m
star okay it looks just like free

229
00:22:15,940 --> 00:22:21,640
电子，它仍然是K的二次方，至少是小K，但现在是
electrons it's still quadratic in K at
least it's small K but now it's an

230
00:22:21,640 --> 00:22:26,050
有效质量M star而不是电子M的实际物理质量
effective mass M star rather than the
real physical mass of the electron M

231
00:22:26,050 --> 00:22:31,780
好吧，现在您知道应该认真考虑一下
okay now this you know should really
think about this for a second the

232
00:22:31,780 --> 00:22:35,380
我们在这个紧密的绑定链中到达这里的有效质量
effective mass that we're getting here
in this tight binding chain that we're

233
00:22:35,380 --> 00:22:40,240
解算与电子的实际质量无关
solving has nothing to do with the
actual mass of the electron it actually

234
00:22:40,240 --> 00:22:45,130
与原子核之间的跳跃有关，质量得到更大的跳跃
has to do with the hopping between
nuclei for greater hopping the mass gets

235
00:22:45,130 --> 00:22:52,720
较小，跳跃越小，质量就越大，我认为因此
smaller for smaller hopping the mass
gets larger I think and so it has it's

236
00:22:52,720 --> 00:22:58,270
完全与电子的实际物理质量无关
totally unrelated to the to the actual
physical mass of the electron you will

237
00:22:58,270 --> 00:23:04,330
查找您知道的样品系统实际物理系统金属或其他地方
find sample you know systems actual
physical systems metals or whatnot where

238
00:23:04,330 --> 00:23:09,580
质量比电子或电子的物理质量小一百倍
the mass is a hundred times less than
the physical mass of the electron or a

239
00:23:09,580 --> 00:23:13,390
比电子的物理质量大一百或一千倍
hundred or a thousand times more than
the physical mass of the electron

240
00:23:13,390 --> 00:23:17,350
尽管在实际材料中具有有效质量的物质并不罕见
although it's not uncommon to have
effective masses in real materials which

241
00:23:17,350 --> 00:23:22,030
在物理电子的实际质量的数量级上
are on the order of the actual mass of
the physical electron the other thing to

242
00:23:22,030 --> 00:23:26,620
请记住，我们在能带中讨论的K不是
keep in mind here is that the K we're
talking about in the energy band is not

243
00:23:26,620 --> 00:23:35,530
动量，但它是晶体动量晶体动量或或h-bar K是
momentum but it's crystal momentum
crystal momentum or or h-bar K is

244
00:23:35,530 --> 00:23:41,200
如果您将H形杆插入其中，则表示动量很大，这意味着您仅知道我们在
crystal momentum if you put the h-bar in
which means only that you know we're

245
00:23:41,200 --> 00:23:48,010
仅定义我们的动量以晶体波矢量为模
only defining our momentum modulo the
crystal wave vector the reciprocal

246
00:23:48,010 --> 00:23:52,330
如果我在I上将所有内容都移两个pi，则晶格波矢量在两个pi上
lattice wave vector two pi over a if I
shifted everything by two pi over a I

247
00:23:52,330 --> 00:23:57,160
回到相同的物理波，所以它与自由略有不同
get back the same physical wave so it's
slightly different from the free

248
00:23:57,160 --> 00:24:02,050
我们熟悉的最后一件事就是电子波
electron waves that were that we're
familiar with the last thing that I want

249
00:24:02,050 --> 00:24:07,040
这里要强调的是本征态
to emphasize here are the eigenstates
eigenstates

250
00:24:07,040 --> 00:24:16,610
我们的波浪我们的飞机波浪只是波浪现在为什么这很有趣
our waves our plane waves are just waves
now why is that interesting it means

251
00:24:16,610 --> 00:24:21,920
我可以拿一个电子，如果我把它放在某种状态下，那有点像
that I could take an electron and if I
put it in some state here it's sort of a

252
00:24:21,920 --> 00:24:26,150
向左移动，向右移动，这是向右移动，向右移动
left a right moving wave it's a right
moving wave in that right moving wave

253
00:24:26,150 --> 00:24:31,970
延伸到整个系统，这对我们的浪潮是至关重要的
extends clear across the system it's an
e to the i k aien our form of the wave

254
00:24:31,970 --> 00:24:35,900
我从面板顶部滚动下来，但这是一个平面波
which I get scrolled off the top of the
board but it's a plane wave that goes

255
00:24:35,900 --> 00:24:41,000
在整个系统中清晰可见，这也许令人惊讶，为什么
clear across the system and that might
be surprising for a second why well

256
00:24:41,000 --> 00:24:44,240
记得当我们研究索末菲理论时，就有这个问题
remember when we when we studied
Sommerfeld theory there was this issue

257
00:24:44,240 --> 00:24:50,060
固体中电子的散射长度似乎过长
that the scattering lengths of electrons
in solids seemed unreasonably long you

258
00:24:50,060 --> 00:24:53,660
所有这些原子核都带有正电荷，它们遍布各处
have all these nuclei with positive
charges and they're all over the place

259
00:24:53,660 --> 00:24:57,590
到每几埃，您就会遇到带正电荷的原子核
to every few angstroms you run into a
nucleus with a positive charge and yet

260
00:24:57,590 --> 00:25:00,620
电子的散射长度可以是一百埃
the electron has a scattering length
that can be a hundred angstroms a

261
00:25:00,620 --> 00:25:05,150
现在有一个千埃或一百万埃长的模型
thousand angstroms or a million
angstroms long now here we have a model

262
00:25:05,150 --> 00:25:09,200
我们有很多原子核周期性地排列在一起
where we have lots of nuclei lined up
periodically one after the other after

263
00:25:09,200 --> 00:25:12,470
另一个接一个，电子从一个跳到另一个
the other after the other and the
electron hops from one to the next to

264
00:25:12,470 --> 00:25:16,610
下一波，使波在整个系统中清晰可见，在
the next next and makes a wave that goes
clear across the system no scattering at

265
00:25:16,610 --> 00:25:20,540
所有您将观察到的这是一个非常好的波形包
all you would observe this as a
perfectly good wave packet going

266
00:25:20,540 --> 00:25:25,670
整个系统中都没有回散一点点，所以这是我们的第一个
entirely across the system without back
scattering one bit so that's our first

267
00:25:25,670 --> 00:25:29,240
令人惊讶的结果，我们将非常频繁地返回到该结果
surprising result and we're going to
come back to that result very frequently

268
00:25:29,240 --> 00:25:33,400
后来，这个术语大家都对此感到满意
later on in the term okay is everyone
fairly happy with that

269
00:25:33,400 --> 00:25:39,830
是的，还有很多我们尚未考虑的其他事情
yes well there's lots of other things
that we haven't accounted for us for

270
00:25:39,830 --> 00:25:44,030
例如我们可以不谈论它可能会扩散到其他电子中
example it can we haven't talked about
it could scatter into other electrons so

271
00:25:44,030 --> 00:25:46,970
实际上，这是我们不会在整个学期讨论的内容
that's actually in fact that's something
we're not going to discuss all term

272
00:25:46,970 --> 00:25:50,510
因为您可以忽略对其他电子的散射的原因是
because the reason you can ignore
scattering against other electrons is

273
00:25:50,510 --> 00:25:53,870
实际上非常微妙，明年您甚至可能不会学到
actually extremely subtle and you
probably won't even learn it next year

274
00:25:53,870 --> 00:25:59,720
直到1980年代或更晚，人们才对它有所了解
it wasn't understood properly until
probably the 1980s or even later and it

275
00:25:59,720 --> 00:26:04,010
在它之下直到1950年代甚至一点都不了解
was under it wasn't understood at all
not even one little bit until 1950s so

276
00:26:04,010 --> 00:26:06,650
这是一个非常棘手的问题，但是您还可以分散其他内容
that's a really tough question but
there's other things you can scatter

277
00:26:06,650 --> 00:26:10,670
有杂质，所以晶体并不完美，里面还有其他垃圾， 
there's impurities so crystals aren't
perfect there's other junk in it and

278
00:26:10,670 --> 00:26:16,070
有振动，所以你可以让电子移动，它们可以撞击
there's vibrations so you can have an
electron moving along and they can hit a

279
00:26:16,070 --> 00:26:19,850
振动声子并从声子上散射出去
vibrational a phonon and scatter off of
a phonon as well so at finite

280
00:26:19,850 --> 00:26:23,070
温度，电子声子的散射经常发生
temperature there's
electron phonon scattering is frequently

281
00:26:23,070 --> 00:26:29,610
限制过程好不好的好问题，我再给你巧克力
the limiting process okay good good
question I give you another chocolate

282
00:26:29,610 --> 00:26:35,340
但是您已经可以了，所以我们要做一个以前做过的练习
but you already have one okay so we're
going to do an exercise we did before

283
00:26:35,340 --> 00:26:43,610
计算计数状态时，我们在计算正常模式时是这样做的
counting counting states we did this for
when we counted normal modes in our

284
00:26:43,610 --> 00:26:52,440
振动链，所以我们将拥有n个原子核，这意味着
vibrational chain so will we have n
nuclei nuclei we have so that means the

285
00:26:52,440 --> 00:26:57,480
我们系统的长度是a的n倍，如果系统的长度是a的n倍
length of our system is n times a and if
the length of the system is n times a

286
00:26:57,480 --> 00:27:03,990
我们大声的情况必须是L乘以P的两个PI的形式，其中PP是
the case we are loud must be of the form
two PI over L times P where P P is an

287
00:27:03,990 --> 00:27:09,240
整数P是大家熟悉的整数
integer P is an integer you guys
familiar with that that notation integer

288
00:27:09,240 --> 00:27:20,600
是的，好的P是整数，所以不同k的数量不同k是
yeah okay good P is integer so the
number of different k's different k's is

289
00:27:20,600 --> 00:27:25,860
等于k的范围不同，k的范围是两个pi 
equal to the range of k's that are
different which is two pi over a the

290
00:27:25,860 --> 00:27:31,110
从这里到这里拥有的棕熊除以相邻KS之间的间距
Bruins own from here to here divided by
the spacing between adjacent KS which is

291
00:27:31,110 --> 00:27:37,950
 L上的两个PI，a上的L，n不足为奇
two PI over L which is L over a which is
n not surprising the number of different

292
00:27:37,950 --> 00:27:41,910
在青铜器中允许的洞穴数量等于
caves were allowed in the bronz own is
equal to the number of unit cells in the

293
00:27:41,910 --> 00:27:48,120
在这种情况下，整个系统很容易以不同的方式理解
entire system in this case it's easy to
understand that in a different way we

294
00:27:48,120 --> 00:27:55,080
从一堆猫开始，其中有n个和n个， 
started with a bunch of cats m there
were n of them and n of these and

295
00:27:55,080 --> 00:28:01,919
这些轨道位于原子核上，一个原子核两个原子核三个
orbitals these were sitting on on nuclei
in nucleus one nucleus two nucleus three

296
00:28:01,919 --> 00:28:07,020
依此类推，然后对角化我们的哈密顿量，得到本征态K 
and so forth and then we diagonalized
our Hamiltonian and we get eigenstates K

297
00:28:07,020 --> 00:28:13,860
毫不奇怪，他们中的n个和本征态都很大
and they're unsurprisingly there n of
them and eigenstates we had an m and big

298
00:28:13,860 --> 00:28:18,600
通过大n矩阵对角化，我们得到了n个特征值和本征态
and by big n matrix and we diagonalized
it we got n eigenvalues and eigenstates

299
00:28:18,600 --> 00:28:25,350
好吧，所以您输入n个状态就得出n个状态，这是另一种命名法
okay so you put in n states you get out
n states one other piece of nomenclature

300
00:28:25,350 --> 00:28:32,309
这在这里相当有用这是我想的能量规模
which is fairly useful here which is
this energy scale here which I guess for

301
00:28:32,309 --> 00:28:36,000
我们把它滚动到最上面
our
well I scrolled it off the top it's

302
00:28:36,000 --> 00:28:47,100
其实40知道在哪里，所以它是余弦的2倍，余弦2t 
actually 40 see where was it yeah so
it's a 2 times the cosine the cosine 2t

303
00:28:47,100 --> 00:28:53,190
余弦乘以该范围，T余弦从正1到负1，所以
times the cosine is the the range and T
cosine goes from plus 1 to minus 1 so

304
00:28:53,190 --> 00:28:59,360
从Emacs的总范围，我们称它为Emacs，在这里，我们称其为e min 
the total range from Emacs let's call it
Emacs here let's call this e min here a

305
00:28:59,360 --> 00:29:17,790
 min在这里，因此Emacs Emax减去e min被称为带宽宽度，其中
min here so Emacs Emax minus e min is
known as the band width width which in

306
00:29:17,790 --> 00:29:26,520
这种情况是40好，现在我们该怎么办我们带宽是多少取决于它
this case is 40 okay now what do we what
do we what is the bandwidth depend on it

307
00:29:26,520 --> 00:29:31,320
取决于您跳的次数越多，跳的越大
depends on how much hopping you have the
more hopping you have the bigger the

308
00:29:31,320 --> 00:29:34,679
带宽，所以让我们实际画一些图片
band width
so let's actually draw a little picture

309
00:29:34,679 --> 00:29:40,650
这里的带宽实际上取决于T取决于什么
here what is the bandwidth actually
depend on what is what is T depend on

310
00:29:40,650 --> 00:29:44,820
好吧，我们拥有的东西我们可以改变的是
well the thing we had at our disposal
the thing we could change was the

311
00:29:44,820 --> 00:29:50,910
晶格常数a原子核之间的距离为
lattice constant a the distance between
the nuclei as the distance between the

312
00:29:50,910 --> 00:29:55,500
原子核上升，跳变自然下降，越难跳越
nuclei goes up the hopping naturally
goes down it's harder to hop over a

313
00:29:55,500 --> 00:30:02,210
距离越远，T就会上升
longer distance so T goes up goes up
going this way

314
00:30:02,210 --> 00:30:06,890
好人对此感到满意，然后精力就会这样
good people happy with that okay and
then energy is going to be this way and

315
00:30:06,890 --> 00:30:12,450
然后我们可以画一幅这样的图片，我们从一堆
then we can draw a picture that kind of
looks like this we start with a bunch of

316
00:30:12,450 --> 00:30:20,040
所有原子都具有本征态且能量为e的原子
atoms all having eigenstates with energy
e atomic when the atoms are very far

317
00:30:20,040 --> 00:30:24,480
它们都具有相同的能量电子可以坐在这里
apart they all have the same the same
energy the electron can sit here can sit

318
00:30:24,480 --> 00:30:29,250
在这里，它可以坐在任何一个原子上，并且它的能量总是e原子， 
here it can sit on any one of the atoms
and its energy is always e atomic and

319
00:30:29,250 --> 00:30:38,370
有n个本征态，我想它现在可以坐在N个原子中的任何一个上
there are n eigenstates and I guess it
can sit on any one of the N atoms now as

320
00:30:38,370 --> 00:30:44,190
我们将原子靠近在一起，而我们的原子核靠近在一起
we bring the atoms closer together as we
nuclei closer together T is going to

321
00:30:44,190 --> 00:30:51,030
增加，我们将获得一支乐队，如果它从e min到Emacs 
increase and we're going to get a band
and if it goes from e min to Emacs here

322
00:30:51,030 --> 00:30:56,610
并且在MN和Emacs之间会有这个本征态
and there will be eigenstates within
this range between MN and Emacs so you

323
00:30:56,610 --> 00:31:01,620
在MN和Emacs之间选择任何能量，那里会有一些本征态
pick any energy between MN and Emacs and
there will be some eigenstates there

324
00:31:01,620 --> 00:31:07,140
好吧，所以这里和这里之间的任何能量总会有一些K有一个
okay so any energy between here and here
there will always be some K which has an

325
00:31:07,140 --> 00:31:12,120
在这种能量下本征态实际上应该看起来有点
eigenstate at that at that energy now
this should actually look a little bit

326
00:31:12,120 --> 00:31:19,500
从研究共价键时开始熟悉
familiar from from when we studied the
covalent bond when we studied the

327
00:31:19,500 --> 00:31:25,050
共价键当电子在一个原子上时，我们有两个带能量的原子
covalent bond we had two atoms with
energy when the electron set on one

328
00:31:25,050 --> 00:31:28,530
原子核中的能量不是，当坐在另一个原子核上时是
nucleus this energy was not and when sat
on the other nucleus energy was anat

329
00:31:28,530 --> 00:31:33,570
然后我们将它们放在一起，我们得到了反键轨道反键， 
then we brought them together and we got
an antibonding orbital anti bond and we

330
00:31:33,570 --> 00:31:40,530
在这里得到了一个比原来低一高的键合轨道键
got a bonding orbital bond down here one
lower and one higher than the original

331
00:31:40,530 --> 00:31:45,900
能量，这正是我们在这里拥有的
energies and that's exactly what we have
over here we have a whole bunch of

332
00:31:45,900 --> 00:31:49,770
我们以相同的能量将所有轨道合并在一起，我们允许
orbitals all with the same energy we
bring them together we allow the

333
00:31:49,770 --> 00:31:54,090
电子来回跳动，一些能量下降，而一些
electron to hop back and forth and some
of the energies go down and some of the

334
00:31:54,090 --> 00:31:59,520
能量上升，并散布成一个带，这确实有意义
energies go up and it spreads out into
into a band good does that make sense

335
00:31:59,520 --> 00:32:06,030
是的，希望如此，以便传播到乐队中
yes hopefully
so the spreading into a band is coming

336
00:32:06,030 --> 00:32:15,990
从电子来回跳来跳去，所以现在让我们想象一下
from the hopping of the electron back
and forth so right so let's imagine now

337
00:32:15,990 --> 00:32:20,490
我们不仅有一个我们要考虑的电子，而且我们有
that we have not just a single electron
that we want to consider but we have

338
00:32:20,490 --> 00:32:30,480
我们想要考虑许多电子，因为我们不可能有K个状态K 
many electrons that we want to consider
since we had n possible K States K

339
00:32:30,480 --> 00:32:42,050
状态，但每个K状态每个K可以有两个旋转，可以有向上旋转或向下旋转
States but each K state each K can have
two spins can have spin up or spin down

340
00:32:42,050 --> 00:32:57,929
这意味着总共有两个n个电子可以适合我，我可以适合乐队，所以很好
that means there's two n total electrons
can fit I can fit in band okay good so

341
00:32:57,929 --> 00:33:00,450
我们填满所有的K State，您可以用任一旋转填满它们
we fill up all the K State's you can
fill them up with either spin

342
00:33:00,450 --> 00:33:05,310
或自旋向下旋转，因此我们可以在该频带中总共容纳两个n个电子
or spin-down so we can have a total of
two n electrons fitting in that band

343
00:33:05,310 --> 00:33:14,280
所以让我们首先考虑一个，我们将把它放在这里
so let's first consider a we're going to
put this maybe here mono valent mono

344
00:33:14,280 --> 00:33:26,160
每个原子核具有一个电子的价原子，因此总共可以得到
valent atom which has one electron per
nucleus so that will give us a total of

345
00:33:26,160 --> 00:33:40,130
总共n个电子，总共n个电子，并且一半填充，一半填充带
n electrons total n electrons total and
that half fills and half filled band

346
00:33:41,150 --> 00:33:48,480
好吧，如果我们总共有n个电子或每个原子有1个电子，我们将充满
okay so if we have n electrons total or
one electron per atom we will fill up

347
00:33:48,480 --> 00:33:56,340
乐队到这里的一半，现在已经充满了旋转和
the band halfway up to here and this is
filled now with both spin-up and

348
00:33:56,340 --> 00:34:03,720
自旋降落的电子好了现在这张照片
spin-down
electrons alright now this picture of a

349
00:34:03,720 --> 00:34:14,370
半满的带半满的带首先它具有一些有趣的特性
half filled band half filled and it has
some interesting properties the first

350
00:34:14,370 --> 00:34:26,340
东西1有费米表面费米表面um那是什么
thing 1 it has a Fermi surface has a
Fermi surface surface um what's that

351
00:34:26,340 --> 00:34:30,659
费米表面费米表面在这里是填充状态相遇的地方
Fermi surface the Fermi surface is this
point here where the filled States meet

352
00:34:30,659 --> 00:34:34,559
空国家，这里是填充国与空国家交汇的地方

353
00:34:34,560 --> 00:34:38,340
状态，所以它们只是两个，因为它是一维的，实际上只是两个
States so they're just two since it's
one dimensional it's actually just two

354
00:34:38,340 --> 00:34:44,219
费米指出您在其中具有最高能量填充状态的两点
Fermi points two points where where you
have the highest energy filled state in

355
00:34:44,219 --> 00:34:50,429
附近的空状态，因为它具有费米表面，所以有可能
the empty state nearby because since it
has a Fermi surface it's possible to

356
00:34:50,429 --> 00:34:54,689
通过从费米正下方获取一些电子来进行低能激发

357
00:34:54,690 --> 00:35:00,810
表面并将其激发到费米表面正上方，这意味着
surface and exciting it to just above
the Fermi surface okay and that means

358
00:35:00,810 --> 00:35:05,640
就像我们在计算中得出的那样，热容量将与T成正比
that the heat capacity is going to be
proportional to T like we calculated in

359
00:35:05,640 --> 00:35:09,900
 Sommerfeld理论，因为您可以很好地进行许多低能激发
the Sommerfeld theory because you can
make as many low energy excitations well

360
00:35:09,900 --> 00:35:13,830
您可以进行许多低能量激发，而可能没有问题
you can make lots of low energy
excitations with no problem maybe less

361
00:35:13,830 --> 00:35:18,750
的-是这是一种金属
of the
- is that this is a metal it conducts

362
00:35:18,750 --> 00:35:24,120
用电，这可能很难想象，但让我们考虑一下
electricity so that's maybe a little bit
harder to to imagine but let's think

363
00:35:24,120 --> 00:35:29,760
关于它的一秒钟在这里，我们所拥有的，我们已经在这里，我们有权利
about it for a second here what we have
is we have over here we have right

364
00:35:29,760 --> 00:35:34,980
移动的电子也许我会给它们贴上标签，因为这些家伙在这里有
moving electrons maybe I'll label them
since this one these guys over here have

365
00:35:34,980 --> 00:35:41,340
正KH横条K在右边，所以这些都是右移右移
positive K H Bar K is to the right so
these are right movers right movers over

366
00:35:41,340 --> 00:35:48,420
在图的这一边，在这里，我们还剩下左动子
here on this side of the diagram and
over here we have left movers left

367
00:35:48,420 --> 00:35:57,600
推动者在这一方面如此积极与消极动量，如果我们
movers over on this side so positive
versus negative momentum and if we if we

368
00:35:57,600 --> 00:36:01,890
施加很少的能量消耗的电场，我们可以采取一些
apply an electric field with very little
energy cost we could take some of the

369
00:36:01,890 --> 00:36:06,900
从这里过来的电子并将它们移动到这里，我们可能会过度填充
electrons from over here and moving them
to over here we could over populate the

370
00:36:06,900 --> 00:36:10,470
右边稍微增加一点，并在下面填充一个左边
right hand side by a little bit and
under populate the left hand side by a

371
00:36:10,470 --> 00:36:14,610
只需花费很少的精力即可完成操作，只需移动费米
little bit it cost you only very little
energy to do so just shift the Fermi

372
00:36:14,610 --> 00:36:22,950
只是一点点的表面，然后就有净电流，所以金属可以移动
surface just a little bit and then you
have a net current so metal can shift

373
00:36:22,950 --> 00:36:40,140
费米表面费米表面实际上以一种方式获得电流
Fermi surface Fermi surface and get
current in fact the way the way it

374
00:36:40,140 --> 00:36:43,680
实际发生的是您应该认真考虑一下
actually happens is that you should
really think about it you apply an

375
00:36:43,680 --> 00:36:48,960
电场和每个电子态在一个方向上加速一点点
electric field and each electron state
accelerates a little bit down in one

376
00:36:48,960 --> 00:36:53,310
方向，因此每个电子都会稍微改变其动量，直到这些家伙
direction so each electron changes his
momentum a little bit until these guys

377
00:36:53,310 --> 00:36:58,170
人口过剩，这些人人口不足，那么你就拥有了
get overpopulated and these guys get
underpopulated and then you have the net

378
00:36:58,170 --> 00:37:04,080
当前，这的确是一种金属，并且确实非常频繁地
current so this is indeed a metal and
indeed it's very very frequently the

379
00:37:04,080 --> 00:37:11,970
单价原子是金属的情况下，我的意思通常是简单的图片
case that mono valent atoms are metals
often I mean a simple picture should

380
00:37:11,970 --> 00:37:20,730
总是真实的，但通常是一价单价物质
always be true but often mono valent
mono valent materials

381
00:37:20,730 --> 00:37:34,690
我们的金属现在进一步让我们回到这里看这张照片的情况
our metals now further let's come back
over here to the case to this picture

382
00:37:34,690 --> 00:37:40,570
想象一下，当我们充满乐队之后会发生什么，所以我们将
and imagine what happens when we have
fill the band okay so we're going to

383
00:37:40,570 --> 00:37:45,010
拿这个乐队，我们要填充一半，现在我们在这里填充这些状态， 
take this band we're going to half fill
it now we filled these states here and

384
00:37:45,010 --> 00:37:50,380
我们将这些状态留在此处为空，所以这是一个半满的带，现在是
we've left these states up here empty so
this is a half filled band now this is

385
00:37:50,380 --> 00:37:58,570
与共价键非常相似，它的净能量较低
very similar to the covalent bond that
there is it's a lower net energy when

386
00:37:58,570 --> 00:38:04,180
电子可以定位在两个原子核之间，因为它们正在填充
the electrons can be localized between
the two nuclei because they're filling

387
00:38:04,180 --> 00:38:07,570
只要是键合轨道，只要您不必填充反键
only the bonding orbital and as long as
you don't have to fill the anti-bonding

388
00:38:07,570 --> 00:38:12,520
轨道，所以在这里当您开始让电子来回跳动时
orbital so here when you start letting
the electrons hop back and forth they

389
00:38:12,520 --> 00:38:16,600
与他们开始时的能量相比，可以降低他们的能量
can lower their energy compared to this
energy that they started with this

390
00:38:16,600 --> 00:38:21,460
他们开始的原子能，所以
atomic energy they started with so
there's a tractive force between the

391
00:38:21,460 --> 00:38:24,790
原子核试图使它们彼此靠近，从而使跳跃更加频繁
nuclei trying to get them closer
together trying to make the hopping much

392
00:38:24,790 --> 00:38:33,370
更大，这就是形成金属之间的金属键
greater so that is what's forming the
metallic bond between forms metallic

393
00:38:33,370 --> 00:38:42,670
键与原子核之间的金属键非常相似
bond metallic bond between the nuclei
this is very very similar to the

394
00:38:42,670 --> 00:38:47,140
通过让电子离域来使它们共价键
covalent bond by letting the the
electrons delocalized by letting them

395
00:38:47,140 --> 00:38:51,670
通过散布其波函数来降低其动能
reduce their kinetic energy by spreading
out their wave function it forms a

396
00:38:51,670 --> 00:38:55,990
现在将原子核保持在一起的键合力
bonding force that holds the nuclei
together now

397
00:38:55,990 --> 00:38:59,890
您知道在这张非常简单的图片中，我们遇到的问题与
you know in this very simple picture we
have the same problem that we had when

398
00:38:59,890 --> 00:39:03,730
我们研究了共价键，通过这张图您可以期望
we studied the covalent bond that by
this picture you would expect that the

399
00:39:03,730 --> 00:39:06,880
随着细胞核越来越近，细胞核将变得越来越幸福
nuclei would be happier and happier and
happier as they got closer and closer

400
00:39:06,880 --> 00:39:10,930
在一起，一切都会下降到零距离，那不是
together and everything would go down to
sort of zero distance and that's not

401
00:39:10,930 --> 00:39:15,520
正确，因为我们的某些近似值开始分解我们的其中一个
true because some of our approximations
start to break down one of our

402
00:39:15,520 --> 00:39:19,330
开始分解的近似值是我们假设正交
approximations that this starts to break
down is that we assumed orthogonal

403
00:39:19,330 --> 00:39:22,930
简化假设的轨道
orbitals that simplifying assumption
that bad assumption that we started with

404
00:39:22,930 --> 00:39:26,680
另一个要分解的东西另一个分解的东西是我们
another so that's going to break down
another thing that breaks down is we

405
00:39:26,680 --> 00:39:30,220
忘记了或多或少的核相互作用
forgot about the nuclear nuclear
interaction which is more or less

406
00:39:30,220 --> 00:39:34,840
类似于共价键，这种情况或多或少被直接
similar to the covalent bond it case is
more or less canceled by the direct

407
00:39:34,840 --> 00:39:38,320
相互作用，但是当原子核开始变得非常接近时， 
interaction but when the nuclei start to
get very close to each other that

408
00:39:38,320 --> 00:39:42,070
取消不再有用，原子核开始收获全部，因此它们不再
cancellation is no good anymore and the
nuclei start to reap all so they're not

409
00:39:42,070 --> 00:39:45,730
将会无限接近，但这大致是导致
going to get infinitely close anymore
but roughly this is what causes the

410
00:39:45,730 --> 00:39:51,490
结合在金属中，这允许电子散布到许多
bonding in metals it's allowing the
electrons to spread out over many over

411
00:39:51,490 --> 00:40:00,250
许多原子并降低其动能，好开心，好，所以我们可以
many atoms and lower their kinetic
energy okay happy okay good so we can

412
00:40:00,250 --> 00:40:07,120
现在考虑一个更普遍的二价原子二价原子的情况
now consider a more general case of
divalent atoms divalent atoms I guess

413
00:40:07,120 --> 00:40:10,720
更一般地说，这只是二价原子的不同情况，例如
not more general this is just a
different case divalent atoms like maybe

414
00:40:10,720 --> 00:40:19,150
氦氦有两个电子在这种情况下，我们有两个电子和两个
helium helium has two electrons in this
case we have two and electrons and two

415
00:40:19,150 --> 00:40:30,340
和本征态我们可以填充，所以我们得到了一个完全填充的地图填充带
and eigenstates we can fill so we get an
entirely filled map filled band well

416
00:40:30,340 --> 00:40:35,620
好吧会发生什么，然后让我们回到这张照片，现在
okay what happens then let's go back to
this picture so now with a divalent

417
00:40:35,620 --> 00:40:41,410
材料，我们完全充满了这个乐队，我们完全像这样充满
material we completely fill this band
and we completely fill like this

418
00:40:41,410 --> 00:40:47,830
一切都充满了，在这种情况下，充满的乐队非常无聊
everything is filled in this case the
filled band has is very boring the

419
00:40:47,830 --> 00:40:57,460
填充带是惰性的填充带可能是惰性的，如果您认为为什么惰性好
filled band is inert filled band may be
inert why is it inert well if you think

420
00:40:57,460 --> 00:41:01,930
大约可以吸收罐，填充带是否有可能吸收任何能量， 
about can absorb can is it possible for
the filled band to absorb any energy the

421
00:41:01,930 --> 00:41:05,020
答案是不，不能吸收任何能量，因为所有状态都已充满
answer is no can't absorb any energy
because all the states already filled

422
00:41:05,020 --> 00:41:08,680
你不能将电子从一种状态来回转移到另一种状态
you can't transfer an electron back and
forth from one state to another to in

423
00:41:08,680 --> 00:41:11,680
为了提供能量或带走能量，所有状态都充满了
order to give an energy or take energy
away all the states are filled it's a

424
00:41:11,680 --> 00:41:16,150
独特的状态，没有什么可以移动的，它可以承载电流吗
unique state there's nothing that can be
moved around can it carry current it

425
00:41:16,150 --> 00:41:19,660
无法承载电流，因为您无法更改左动子的数量
can't carry current because you can't
change the number of left movers versus

426
00:41:19,660 --> 00:41:27,600
正确的搬运工他们都被装满了，所以没有热容量没有热容量
right movers they're all filled so it
has no heat capacity no heat capacity

427
00:41:29,580 --> 00:41:42,380
没有电流，没有电流，也没有金属结合
carries no current carries no current
and it also has no metallic bonding

428
00:41:42,380 --> 00:41:50,880
这又与我们在滑雪时的情况非常相似
and this is again very similar to what
we had in the case of the of the ski

429
00:41:50,880 --> 00:41:55,530
当我们考虑两个氦原子之间可能的键合
when we considered possible bonding
between the helium atom between two

430
00:41:55,530 --> 00:41:58,410
氦原子（如果必须填充键合轨道和反键） 
helium atoms if you have to fill the
bonding orbital and the anti-bonding

431
00:41:58,410 --> 00:42:04,380
轨道，您不会获得任何能量，因此，如果我们有一个充满的频段， 
orbital you don't gain any energy so
here if we have a filled band you had

432
00:42:04,380 --> 00:42:07,770
两个您填充较低的能量状态，但您必须填充较高的能量状态
two you fill the lower energy states but
you had to fill the higher energy states

433
00:42:07,770 --> 00:42:16,440
也是如此，所以跳楼再也不会给你带来任何好处
too so it doesn't gain you anything to
have the the hopping go up anymore okay

434
00:42:16,440 --> 00:42:27,329
那么你的状况怎么样呢，例如在振动变化的情况下
so how are you doing okay good so as in
the case of the vibrational change that

435
00:42:27,329 --> 00:42:32,640
我们本周早些时候研究过的情况还可能是
we we studied earlier this week you can
also have a situation where not every

436
00:42:32,640 --> 00:42:37,440
原子或您正在考虑的每个轨道都不相同，所以现在让我们
atom or not every orbital you're
considering is the same so let's now

437
00:42:37,440 --> 00:42:44,640
考虑一种情况，其中有两个轨道，每个单位两个不同的轨道
consider a case where there are two
orbitals two different orbitals per unit

438
00:42:44,640 --> 00:42:51,510
细胞并且可以杀死它可以以两种不同的方式或几种不同的方式发生
cell and this can kill can can occur in
two different ways or several different

439
00:42:51,510 --> 00:42:55,650
一种可能的方式是在晶胞中有两个不同的原子
ways one possible way is that you have
two different atoms in the unit cell and

440
00:42:55,650 --> 00:42:58,980
他们有不同类型的轨道，其中有氯原子中的钠
they have different types of orbitals in
them sort of a sodium in a chlorine atom

441
00:42:58,980 --> 00:43:03,599
或像这样的情况在单元格中另一种可能的情况是
or something like that in a unit cell
another possible case is you have one

442
00:43:03,599 --> 00:43:08,069
原子，但该原子上有两个不同的轨道，例如S轨道和ap轨道或
atom but two different orbitals on that
atom like an S orbital and ap orbital or

443
00:43:08,069 --> 00:43:13,680
你想考虑的一个和两个轨道
one s orbital and the 2's orbital that
you want to consider so without actually

444
00:43:13,680 --> 00:43:21,180
详细解决此问题可以告诉您什么频谱
solving this problem in in detail can
show you what the what the spectrum

445
00:43:21,180 --> 00:43:24,240
看起来像散布看起来很像
looks like what the dispersion looks
like it's very similar to the case of

446
00:43:24,240 --> 00:43:31,170
振动链，所以这里的K是这里的PI，这里是负的PI是这里的PI 
the vibrational chain so here's K here's
PI over a here's minus PI over a here's

447
00:43:31,170 --> 00:43:40,920
 e，您会得到一个类似于我们的声学模式的低能带
e and you get a low energy band
analogous to the acoustic mode that we

448
00:43:40,920 --> 00:43:45,750
进入振动链，您会获得类似于
got in the vibrational chain and you get
a high energy band analogous to the

449
00:43:45,750 --> 00:43:48,720
我们在振动链中拥有的光学模式
optical mode that we had in the
vibrational chain okay that's supposed

450
00:43:48,720 --> 00:43:53,950
看起来很对称道歉，所以现在我们有两个
to look symmetric apologize about that
so now we have two

451
00:43:53,950 --> 00:43:58,900
能量带较低的能量带和较高的能量带我们也可以
energy bands the lower energy band and a
higher energy band we could also if we

452
00:43:58,900 --> 00:44:07,089
想要这是缩小区域方案缩小区域方案，我们还可以
want this is the reduced zone scheme
reduced zone scheme and we could also

453
00:44:07,089 --> 00:44:13,980
画在扩展区方案中，我实际上将在这里进行
draw it in the extended zone scheme
which I which I'll do over here actually

454
00:44:17,099 --> 00:44:27,849
扩展区域方案区域方案我们可以画出相同的东西
extended zone scheme zone scheme we
could draw the same thing which would

455
00:44:27,849 --> 00:44:41,290
看起来像这样好吧II，这里的PI等于这里的负PI，这里的两个PI 
look kind of like this okay II here's PI
over a here's minus PI over a here's two

456
00:44:41,290 --> 00:44:47,440
在-2上的PI在a上的PI，扩展区域方案的思想是
PI over a here's -2 PI over a and the
idea of the extended zone scheme is to

457
00:44:47,440 --> 00:44:58,810
分散乐队到结核菌素区域，所以这是第一个区域，首先是BZ 
spread out the bands into tuberin zones
so this is the first zone first BZ here

458
00:44:58,810 --> 00:45:09,280
第二个在这里，第二个BZ在这里，所以我所做的就是
and the second is out here and out here
second BZ out here okay so all I did was

459
00:45:09,280 --> 00:45:14,740
我拿走了这件作品，并把它移到了这里，移到了两个pi的上方。 
I took this piece and moved it over here
by two pi over a and this piece and

460
00:45:14,740 --> 00:45:18,849
由两个PI移到此处，因此每个可能的K只有一个
moved over here by two PI over a such
that each possible K there's only one

461
00:45:18,849 --> 00:45:25,660
一种激励，无论哪种情况，您都应该在
one excitation and in either case your
this is supposed to line up right at the

462
00:45:25,660 --> 00:45:29,980
棕色区域边界此间隙进入此处该间隙进入此处
brown zone boundary this gap comes in
here this gap comes in here at the

463
00:45:29,980 --> 00:45:35,140
青铜自己的边界请注意，区域边界处有一个间隙
bronze own boundary notice that there's
a gap right at the zone boundary and

464
00:45:35,140 --> 00:45:38,349
如果您将它们放在一起并斜视一下，这不是偶然的
it's not coincidental that if you sort
of put this together and squint your

465
00:45:38,349 --> 00:45:43,720
眼睛看起来像一个整体抛物线，那么自由电子可能具有
eyes this looks like one overall
parabola then a free electron might have

466
00:45:43,720 --> 00:45:49,170
但是您已经在区域边界处打开了空白，我们将解释为什么
but you've opened up gaps at the zone
boundary and we'll explain why that is

467
00:45:49,170 --> 00:45:55,390
在学期后期，您可能会想到的一件事是
later on in the term now one thing that
you might consider here is suppose

468
00:45:55,390 --> 00:46:04,059
假设晶胞具有三个电子
suppose the unit cell unit cell has
three electrons

469
00:46:04,059 --> 00:46:11,979
里面有三个电子那么会发生什么然后你填满整个
has three electrons in it well then what
happens then you fill up the entire

470
00:46:11,979 --> 00:46:17,259
带有自旋和自旋电子的低能带，你有一半的
lower band with both spin-up and
spin-down electrons and you have half of

471
00:46:17,259 --> 00:46:23,499
上面的乐队被填满，这里的一半乐队被填满
the upper band filled up to here and
half of the upper band here is filled

472
00:46:23,499 --> 00:46:27,219
填满了上部频段的一半，所以您填满了下部频段，而​​它填充了一半
filled up half of the upper band so you
filled the lower band and it filled half

473
00:46:27,219 --> 00:46:31,599
您会注意到我们已经完全填满了较低频段
of the upper band well you'll notice
that we have entirely fill lower band

474
00:46:31,599 --> 00:46:40,359
较低的频带是惰性的较低的频带是惰性的，我们只需要担心
the lower band is inert lower band is
inert and we only need to worry if we're

475
00:46:40,359 --> 00:46:45,160
考虑运输特性或没有特定热量的热特性
thinking about transport properties or
heat properties that no specific heat

476
00:46:45,160 --> 00:46:48,640
热容量之类的，我们只需要担心上频带
heat capacity something like that we
only have to worry about the upper band

477
00:46:48,640 --> 00:46:51,579
因为较低的频段至少是惰性的，除非您给它一个
because the lower band is completely
inert at least unless you give it an

478
00:46:51,579 --> 00:46:55,209
激发下层频段到上层频段的巨大能量
enormous energy to excite things out of
the lower band and up to the upper band

479
00:46:55,209 --> 00:47:00,819
能量很高，因为这些间隙通常很大，您可以忽略
very high energy because these gaps are
typically very large you can just ignore

480
00:47:00,819 --> 00:47:05,289
整个较低的频段，只担心较高频段的电子
the lower band altogether and only worry
about the electrons in the upper band

481
00:47:05,289 --> 00:47:08,469
但这实际上是这些问题之一的答案之一
but this is actually this is actually
the answer to one of those questions one

482
00:47:08,469 --> 00:47:14,469
我们在本学期早期写下的这些难题中
of these puzzles that we wrote down
earlier in the term and the puzzle from

483
00:47:14,469 --> 00:47:18,160
这个学期的早期是为什么您有时不必担心
earlier in the term was why is it that
you sometimes don't have to worry about

484
00:47:18,160 --> 00:47:23,259
核心电子如果电子在核心轨道中，为什么可以丢
core electrons if electrons are in core
orbitals why is it you can just throw

485
00:47:23,259 --> 00:47:26,679
当你数电子时，它们就消失了。你知道钠只计算一个
them away when you're counting electrons
you know for sodium you only count one

486
00:47:26,679 --> 00:47:31,660
尝试时，每个原子每个单位的电子，而不是每个原子11个电子
electron per unit per per atom and not
11 electrons per atom when you're trying

487
00:47:31,660 --> 00:47:35,890
计算金属密度的原因是因为基本上
to calculate the metallic density and
the reason is because basically those

488
00:47:35,890 --> 00:47:40,299
其他电子完全充满了能带，能带变得惰性， 
other electrons are completely filling
bands and the band's become inert and

489
00:47:40,299 --> 00:47:43,929
只是你知道不是像热容这样有趣的物理学的一部分
just you know aren't part of the
interesting physics like heat capacity

490
00:47:43,929 --> 00:47:52,209
然后在这里进行一些其他评论，所以我们得到的是
and and conduction a couple more
comments here so what we've had is that

491
00:47:52,209 --> 00:47:58,029
有这些重要原理是半满的带子是金属
have these sort of important principles
is it half filled band band is a metal

492
00:47:58,029 --> 00:48:07,450
通常是金属，通常是金属，这是一个电子
usually a metal usually metal
and and this is for one electron for

493
00:48:07,450 --> 00:48:12,310
晶胞或实际上每个晶胞有任何奇数个电子，因此它们将是
unit cell or actually any odd number of
electrons per unit cell so they'd be a

494
00:48:12,310 --> 00:48:19,150
每个晶胞电子的奇数个奇数，而偶数个电子
odd odd number of electrons per unit
cell whereas an even number of electrons

495
00:48:19,150 --> 00:48:26,260
每个单位电池的偶数个单位电池的电子数
per unit cell even number of electrons
per unit cell per unit cell you might

496
00:48:26,260 --> 00:48:34,900
认为这是迈克·马累（Mike Male）的权利，可能是绝缘体，因为您会
think that this is Mike male right might
be be an insulator because you would

497
00:48:34,900 --> 00:48:41,740
完全充满乐队，然后你只是一个惰性的情况，但它变成
completely fill bands and then you have
just an inert situation however it turns

498
00:48:41,740 --> 00:48:46,600
在很多情况下，每个电子有偶数个电子
out that there are many many cases where
you have an even number of electrons per

499
00:48:46,600 --> 00:48:51,550
晶胞，但它是金属，可能发生的原因是如果您
unit cell and yet it's a metal and the
reason that that can occur is if you

500
00:48:51,550 --> 00:48:55,870
具有如下所示的波段结构，所以如果您更复杂
have a band structure that looks as
follows so if you're more complicated

501
00:48:55,870 --> 00:49:02,620
像这样的带结构，所以这里是K，这里是PI，这是这里的负号
band structure that looks like this so
here's a K here's PI over a here's minus

502
00:49:02,620 --> 00:49:08,890
 PI在上，这是这样的较低频段，然后想象我们有一个
PI over a and here's the lower band like
this and then imagine that we have an

503
00:49:08,890 --> 00:49:15,580
高端频段可以做到这一点，所以您可以分散更复杂的模型
upper band that does this okay so it's
dispersion more complicated models you

504
00:49:15,580 --> 00:49:19,240
可以有一个看起来像这样的色散，特别是上频段滑移
can have a dispersion that looks like
this in particular the upper band slips

505
00:49:19,240 --> 00:49:24,220
低于较低带的下端的顶部，所以如果我每个单元有两个电子
below the top of the lower of the lower
band so if I have two electrons per unit

506
00:49:24,220 --> 00:49:33,550
实际上，您可以充满整个较低的频段，但能量较低
cell in fact you could fill the entire
lower band but it's lower energy to

507
00:49:33,550 --> 00:49:39,790
部分填充该波段并部分填充该波段，那会
partially fill this band and partially
fill this band and that would that would

508
00:49:39,790 --> 00:49:43,630
每个晶胞组成两个电子，而不是完全填充较低的
make up your two electrons per unit cell
instead of entirely filling the lower

509
00:49:43,630 --> 00:49:47,530
乐队，而在这些情况下将这些国家留空
band and leaving these States empty in
that case you have two partially filled

510
00:49:47,530 --> 00:49:50,950
乐队，而不是一个完全填充的乐队，因为你有两个部分
bands instead of one completely filled
bands and since you have two partially

511
00:49:50,950 --> 00:49:54,730
填充的频段，那么您在这里具有低能量激发和低能量
filled bands you then have low energy
excitations here and low energy

512
00:49:54,730 --> 00:49:58,840
激发在这里，所以这个东西实际上变成了金属，所以有
excitations here and so this thing
actually becomes a metal so there are

513
00:49:58,840 --> 00:50:03,370
很多情况下，例如钙，钙甚至是金属
lots of cases of that things like
calcium has calcium is a metal even

514
00:50:03,370 --> 00:50:06,460
尽管它每个晶胞有两个电子，而这正是这种物理学
though it has two electrons per unit
cell and it's exactly this physics

515
00:50:06,460 --> 00:50:10,150
这两个频段实际上正在穿越能量，所以
that's occurring the two bands are
actually crossing an energy so instead

516
00:50:10,150 --> 00:50:14,890
完全填满一个乐队，你有两个部分填满的乐队，好吧，我想
of completely filling one band you have
two partially filled bands okay I think

517
00:50:14,890 --> 00:50:18,780
也许我最好停在那里，我你明天猜猜
maybe I better stop there and I
you I guess tomorrow

518
00:50:18,780 --> 00:50:24,359
 [掌声] 
[Applause]