06 Microscopic View of Vibrations in Solids in One Dimension I The Monatomic Harmonic Chain.srt

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这是凝聚态课程的第六讲
this is a the sixth lecture of the
condensed matter course

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在上一讲中，我们讨论了离子键和共价键的键合以及
in last lecture we discussed bonding
ionic bonding and covalent bonding and

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还有其他三种类型的绑定，它们在我们要列出的列表中
there were three other types of bonding
they were on a list that we were going

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最终覆盖范德华键合金属键合和氢
to cover eventually van der Waals
bonding metallic bonding and hydrogen

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绑定，由于种种原因，我将把这些推迟到以后
bonding and for various reasons I'm
going to push those off until later

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我们将尝试将它们放到适合的位置，而我们要做的是
we'll try to pick them up where they fit
in instead what we're gonna do is we're

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要回去重新考虑原子的运动，我们要简化你的生活
gonna go back and reconsider the motion
of atoms we're gonna simplify your life

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今天很多，我们只考虑一个原子的运动
an awful lot today and we're only going
to consider the motion of atoms in one

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维度，所以今天我们生活在一个一维的世界中
dimension so for the purpose of today we
live in a one-dimensional world we're

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我们将非常简单地总结一下我们所了解的有关粘接的所有信息
gonna summarize everything we know about
bonding very simply we're going to

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想象我们有两个原子相距一段距离X， 
imagine we have two atoms some distance
apart what's called a distance X and

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 X的势能V代表
there will be some potential V of X
which represents the force between the

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我们上次讨论的X的两个原子和X的电位V可能看起来
two atoms and the potential V of X as we
discussed last time it probably looks

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 X的V之类的东西，所以它具有吸引人的结合
something like this V of X something
like that so it has a attractive bonding

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力，然后如果原子距离太近，势能会突然上升
force and then if the atoms get too
close the potential shoots off to

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无限好，所以我们通常要做的就是扩展
infinity okay so what we're usually
going to do is we're going to expand

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围绕这个最小值的底部以二次方的方式将其近似为
around the bottom of this minimum in a
quadratic way and approximate it as a

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抛物线，所以我们将X的V表示为V等于V等于V等于底V 
parabola so we'll write V of X is some V
naught V naught is the bottom V naught

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在这里加上它的距离
here
plus it's called this distance here from

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这里到这里，我们称X平衡为
here to here let's call that X
equilibrium that's the the bottom of the

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换句话说，原子最想坐的距离
well in other words the distance at
which the atoms would most like to sit

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能量的最小值，因此想在此处给出一个二次项X减去X 
the minimum of the energy so would like
to give a quadratic term here X minus X

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平衡平方加点点点好，所以如果两个
equilibrium squared plus dot dot dot
okay so the energy is minimum if the two

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原子被这个平衡距离分开，然后是某种
atoms are separated by this equilibrium
distance and then it's some sort of

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抛物线，如果原子更远或更近
parabola if you if the atoms are either
farther apart or closer together okay

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00:02:06,840 --> 00:02:12,420
现在我们在执行此操作时应谨慎一些，因为偶尔
now we should be a little bit cautious
in doing this because occasionally by

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近似抛物线，你把婴儿扔出去
approximating something
parabola you throw the baby out with the

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沐浴水，尤其是如果您担心热膨胀， 
bathwater and in particular if you're
worried about thermal expansion it's

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保持点对点术语非常重要，事实上， 
very important to keep the dot-dot-dot
terms the fact that in fact the

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潜力不是抛物线，所以让我们考虑一下
potential is not a parabola so let's
think about that for a second

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如果原子处于低温状态，热膨胀如何发生？ 
how does thermal expansion happen well
if you're at low temperature the atoms

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您有点想像潜在阱底部的粒子，但是
you sort of think of a like a particle
in the bottom of potential well but

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我们真正在谈论的是两个原子之间的距离
we're really talking about is the
distance between the two the two atoms

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但是您可以考虑一下来回摆动的距离
but you can sort of think about that
distance oscillating back and forth just

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就像是井底的粒子，所以它来回摆动
like it was a particle in the bottom of
well so it oscillates back and forth the

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原子变得越来越远，越来越近，它们来回振荡
atoms get farther apart and closer
together they oscillate back and forth

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并且平均距离几乎保持在X平衡，但是如果我们给
and pretty much the average distance
stays at X equilibrium but if we give

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原子的能量更高，温度比原子高
the atoms some higher amount of energy
here a higher temperature then the atoms

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可以摆动到这里，但一直到这里都可以，因为
can oscillate into here but out all the
way to here okay because the the

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内部的电势比外部的电势陡
potential is steeper on the inside than
it is on the outside

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通常，原子将能够使其在此处达到此x max和此X min X 
generally the atoms will be able to make
it far this x max here and this X min X

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最小值将与X平衡值不同，特别是X Max Plus X 
min will differ a different amount from
X equilibrium in particular X Max Plus X

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在2上的最小值将大于在T等于零时的X平衡，因此
min over two will be greater than X
equilibrium at T equals zero so the

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原子开始振荡时彼此之间的平均距离
average distance that the atoms are from
each other when they start oscillating

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将会开始增加，这是由于以下事实： 
will start to increase and this comes
from the fact that when the when the

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原子振荡，它们可以推动一点点，但潜力确实是
atoms oscillate they can push in a
little bit but the potential is really

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陡峭的，所以他们不能推那么多，但是当他们振荡出
steep so they can't push in that much
but then when they oscillate out the

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势较弱，所以他们可以走得更远，所以这就是
potential is softer so they can go much
farther distance out so this is what

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为您提供热膨胀的潜在形式
gives you thermal expansion the
particular form of the potential

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原子彼此远离并变得非常非常柔软的功能
function that's softer as the atoms go
away from each other and gets very very

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当原子彼此靠近时陡峭，但是只要我们不
steep when the atoms get close to each
other okay but as long as we're not

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考虑热膨胀之类的事情，可以截断我们的
considering things like thermal
expansion it's okay to just truncate our

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二次方的势，我们有一个纯胡克定律类型的弹簧
potential at quadratic order and we have
a pure Hookes law type spring between

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到目前为止，我们的原子还不错，每个人都满意，好吗？ 
our atoms so far so good
everyone happy with that okay so what

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我们将在本讲座的其余部分中讲到， 
we're going to do with the rest of the
lecture is we're going to take a very

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原子振动的简单模型实际上是
simple model of atomic vibration
actually extremely simple model of

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原子振动，称为单原子单原子谐波链
atomic vibration which is known as the
monatomic monatomic harmonic chain

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谐波链可能是我们要使用的最重要的模型
harmonic chain which is potentially the
most important model we're going to

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全年学习不仅因为它碰巧出现在期末考试中
study all year not only because it
happens to show up on the final exams

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如此频繁的单原子意味着只有一种类型的原子谐波
very frequently so monatomic means that
there is only one type of atom harmonic

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意味着它将是原子和链之间的简单弹簧
means that it's going to be just simple
Springs between the atoms and chains

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意味着我们会有很多这样的原子，原因很简单
mean we're gonna have a lot of these
atoms and the reason it's a very simple

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模型，但之所以如此重要，是因为它引入了很多想法
model but the reason it's so important
is because it introduces a lot of ideas

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在整个学期中都会反复出现
that will come back over and over again
throughout the term so this is what it

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看起来我们有一堆原子，我们又生活在一个维度中
looks like we have a bunch of atoms and
again we're living in one dimension the

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排成一排的每个原子都有一些质量M，它们全部相同，然后
lined up in a row each atom has some
mass M all of them identical and then

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有一个弹簧常数，所有弹簧常数之间完全相同
there's a spring constant all the spring
constants identical identical between

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两个原子Kappa Kappa Kappa和Kappa的弹簧常数来自
the two atoms Kappa Kappa Kappa and the
Kappa the spring constant comes from the

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键电位的扩展我们的谐波扩展
expansion of the of the bonding
potential the harmonic expansion that we

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没关系用完两者之间的平衡距离
used up above okay
the equilibrium distance between the two

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原子X平衡在这里我们称其为
atoms X equilibrium here let's call it a
for the purpose of

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通常在这里这个距离a的论点称为晶格
argument generally over here this
distance a is known as a lattice

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常数，通常我们会经常使用晶格常数一词
constant and generally we're gonna use
the word lattice constant frequently

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术语“晶格常数”的后面一般是指
later in the term lattice constant
general generally means distance between

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在相同的原子之间，在这种情况下，我们所有的原子都是相同的，所以
between identical atoms and in this case
all of our atoms are identical so it's

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只是这里原子之间的距离让我们在这里定义一些位置
just the distance between the atoms here
let's define some positions here so

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我们称这个为一个x也许这个为X 2 X 3 
let's call this one x one maybe this one
will be X 2 X 3

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一般说X sub n是原子n的位置，让我们让X和
that so generally say that X sub n is
position of atom n and let's let X and

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上标0是原子an的平衡位置，因此
the superscript 0 be the equilibrium
position position of atom an so that's

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你想象让链条静止下来，然后测量位置
you imagine letting the the chain come
to rest and you measure the positions

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它们之间的距离都是a，所以X和0就是
they're all spaced by a distance a so X
and 0 is just we can let it be if we let

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如果原子从位置0开始则为0，那么我们可以将X和0设置为n乘以a 
the 0 if atoms start at position 0 then
we can just set X and 0 to be n times a

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每个原子与下一个原子之间的距离为a，而实际上我们是
each atom separated by a distance a from
the next and the quantity we're actually

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感兴趣的是与平衡位置的偏差，我们称之为
interested in is the deviation from the
equilibrium position which we'll call

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 Delta X，所以Delta X sub n是X sub n减去xn naught，我们要做的是
Delta X so Delta X sub n is X sub n
minus xn naught and what we're going to

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尝试做的是，我们将尝试找出该链条的振动
try to do is we're going to try to
figure out the vibrations of this chain

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首先使用完全经典的物理学，我们将担心量子
using completely classical physics to
begin with and we'll worry about quantum

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物理之后，所以您可能已经完成了耦合
physics later
so in you've probably done coupled

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第一年可能会有春天和质量问题，这很漂亮
spring and mass problems probably in
your first year and it's pretty

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你应该做的很简单，就是你应该写下来
straightforward what you're supposed to
do is you're supposed to write down

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牛顿方程适用于所有质量，所以很容易我们有了F 
Newton's equations for all of the masses
so that's pretty easy we just have F

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等于MA，所以原子n上的f是质量乘以加速度Delta X的两倍
equals MA so f on the atom n is mass
times the acceleration Delta X double

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点，那么第n个质量上的X上的力是多少
dot and so what is the force on on X on
on the nth mass well okay it has a force

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从原子到它的右边，所以它是Kappa Hookes定律Delta X n加1减去Delta 
from the atom to its right so it's Kappa
Hookes law Delta X n plus 1 minus Delta

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 X n，然后它在左边的Kappa Delta X上受到原子的作用力为负
X n and then it has a force from the
atom on its left Kappa Delta X and minus

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 1减去Delta X，我想我们可以通过写简化一下
1 minus Delta X and I guess we can
simplify that a little bit by writing is

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 Kappa Delta X和加1加Delta X和减1减2 Delta X和
Kappa Delta X and plus 1 plus Delta X
and minus 1 minus 2 Delta X and

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我们要解决的是我们要解决普通模式
and what we'd like to solve for is we
would like to solve for the normal modes

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想要此链正常模式的正常模式，提醒您意味着所有
want-want normal modes of this chain
normal mode to remind you means all

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原子以相同的频率假设在相同的频率，如果您还记得
atoms postulate at a common frequency at
common frequency and if you remember how

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您在第一年的课程中做到了这一点，最终将成为
you did this in in your first-year
courses it would ended up being an

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具有质量数维矩阵的特征值问题
eigenvalue problem with sort of a matrix
the dimension of the number of masses

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如果我们现在拥有的话，我们可能会有无数的群众要处理
you have now we might have an infinite
number of masses here to deal with if we

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有一个很长的链，所以它看起来像一个无限大的特征值问题
have a very long chain so it looks like
an infinitely large eigenvalue problem

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听起来可能有点吓人，但事实证明
and that might be sound a little bit
frightening but it turns out that

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解决这样的问题真的很容易，我们将再次使用
solving problems like this is really
easy and we're gonna do again we use the

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一年又一年，都是同样的把戏，把戏是猜测
same kind of trick over and over this
year and the trick is to guess the

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回答，幸运的是所有的猜测
answer
and fortunately the guesses are all the

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一样，所以很容易猜到猜测是您使用了点波
same so it's easy to guess the guess is
you use what is known as a wave on dots

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点上是德语单词，意思是猜测，所以我们要
on dots is a german word that means
something like guess so we're gonna

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猜想解决方案是波形，有人会说德语
guess that the solutions are waveforms
someone who speaks german is probably

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00:10:11,050 --> 00:10:15,940
说不，那不是什么意思，就是你说的是的，我有
saying no it's that's not what it means
is that what you're saying yeah I have

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不知道这是什么经验法则或什么意思，我可以接受
no idea what does it mean rule of thumb
or something it means what okay I stand

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纠正物理我们经常很好地谢谢你
corrected
thank you well in physics we frequently

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我们用它来表示猜测
we use it to mean to mean guess

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所以我不知道它到底来自哪里，所以诀窍是我们要
so I don't know where it came from all
right so so the trick is we're going to

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猜想振荡的形式是简单的波，所以
guess that that the the the form of the
of the oscillations are simple waves so

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我们将为波形写下Delta X是I Omega t的常数ae 
we'll write down a for a wave form Delta
X is some constant a e to the I Omega t

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00:10:55,490 --> 00:11:09,130
减去IK乘以X，而不是K，这里是波矢，而Omega是频率
minus I K times X and not K here is the
wave vector and Omega is the frequency

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现在这可能有点令人困惑，因为我们写下来的是
now this might be a little bit confusing
because what we've written down is we've

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写下一些复杂的内容，据我们所知，这些职位感兴趣
written down something complex where as
we know the positions were interested in

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实际上是真实的，但这就像当您知道学习电路
are actually real but this is just like
when you you know study circuits in your

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第一年，您试图考虑的是振荡电流， 
first year you're trying to think about
currents that are oscillating and

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00:11:27,290 --> 00:11:31,060
而不是写正弦和余弦，而是写下一些复杂的东西
instead of writing sines and cosines you
write down some sort of complex

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00:11:31,060 --> 00:11:34,760
表达，而你真正的意思是你应该把真实
expression and what you really mean is
that you're supposed to take to the real

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在一天的结尾部分，这就是我们真正的意思是在这里结束
part at the end of the day and that's
what we really mean here is at the end

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00:11:37,820 --> 00:11:40,550
一天之内您应该参与其中，而我们这样做的原因是
of the day you should take the real part
and the reason we do this is because

128
00:11:40,550 --> 00:11:44,200
使用指数的总是比使用指数的容易
it's always easier to work with
Exponential's than it is to work with

129
00:11:44,200 --> 00:11:49,400
现在是正弦和余弦，因为我们将尽我们所能
sines and cosines now because we're
going to take the real part we could put

130
00:11:49,400 --> 00:11:55,790
在指数中减去整体负号后仍会得到相同的答案，因此我们可以
an overall minus sign up in the exponent
and still get the same answer so we can

131
00:11:55,790 --> 00:12:00,350
修复Omega始终大于或等于零
fix that Omega is always greater than or
equal to zero

132
00:12:00,350 --> 00:12:03,650
不失一般性，因为如果您只是更改符号
without any loss of generality because
if you just change the sign of

133
00:12:03,650 --> 00:12:06,140
一切和指数一旦您发挥了真正的作用，您最终就会得到
everything and the exponent once you
take the real part you end up getting

134
00:12:06,140 --> 00:12:10,730
结果相同，但我们必须跟踪K可以有一个正弦的事实
the same result but we must keep track
of the fact that K can have either sine

135
00:12:10,730 --> 00:12:18,950
好吧，要么是正弦波，要么是向左波，要么是向右波
okay either sine which corresponds to
either a left going wave or a right

136
00:12:18,950 --> 00:12:24,560
我想我们应该在这里替代另一件事
going wave the other thing I guess we
should probably substitute in here the

137
00:12:24,560 --> 00:12:29,840
 xn的值为零，这是n倍，所以我们要做的就是
value of xn zero which is n times a okay
so then all we have to do is we have to

138
00:12:29,840 --> 00:12:33,260
在Zots上接受这一浪潮，将其插入牛顿方程
take this wave on Zots
plug it into Newton's equations up there

139
00:12:33,260 --> 00:12:39,190
看看如果我们将其插入左侧，我们会很好
and see what we get all right well if we
plug it in on the left hand side we get

140
00:12:39,190 --> 00:12:45,570
减去M Omega平方乘以P至I Omega T 
minus M Omega squared times a P to the I
Omega T

141
00:12:45,570 --> 00:12:51,720
好的，衍生权使负的Omega平方下降，然后在
okay and a right to derivatives brings
down minus Omega squared and then on the

142
00:12:51,720 --> 00:13:00,030
右手边，我们得到Kappa，是的，Kappa到我有一个e 
right hand side we get Kappa a
yeah okay Kappa a there's an e to the I

143
00:13:00,030 --> 00:13:04,560
 Omega T是所有术语通用的，然后我们得到e减去IK 
Omega T which is common to all the terms
and then we get e to the minus I K and

144
00:13:04,560 --> 00:13:18,480
负IK n加上1 a加e n负1 a负2唯一的ikna好现在我们
plus 1 a plus e to the minus I K n minus
1 a minus 2 unique i k n a good now we

145
00:13:18,480 --> 00:13:22,290
可以从这个方程式中消除很多事情来简化我们的生活
can cancel out a whole bunch of things
from this equation to simplify our life

146
00:13:22,290 --> 00:13:32,280
所以我们得到负的MΩ平方等于I的Kappa e加上负的e 
so we get minus M Omega squared equals
Kappa e to the i ka plus e to the minus

147
00:13:32,280 --> 00:13:41,550
 IK减2，那么我们可以使用Trig Identity来确定这个东西实际上是2 
I K a minus 2 then we can use a trig
identity that this thing is actually 2

148
00:13:41,550 --> 00:13:47,370
倍余弦，实际上我也将它们移到另一边
times the cosine and I actually move
them to the other side as well we get

149
00:13:47,370 --> 00:13:55,730
欧米茄平方，它是Kappa / M 2减去2余弦ka 
omega squared
it's Kappa / M 2 minus 2 cosine ka

150
00:13:55,730 --> 00:14:04,020
另一个三角身份，我们可以将1减去余弦K a替换为正弦
another trig identity that we can
replace 1 minus cosine K a as sine

151
00:14:04,020 --> 00:14:11,010
超过2的ka的平方，这样一来我们总共有4个，而ya总共有4 K超过m 
squared of ka over 2 so that gives us a
total of 4 with ya total of 4 K over m

152
00:14:11,010 --> 00:14:17,820
将ka的正弦平方乘以2，然后我将取平方根
times sine squared of ka over 2 and then
I'll just take the square root of this

153
00:14:17,820 --> 00:14:23,940
整个方程式，我们得到的最终结果Ω是m上Kappa的2平方根
whole equation we get the final result
Omega is 2 square root of Kappa over m

154
00:14:23,940 --> 00:14:30,360
正弦K的绝对值超过2，我们已经解决了
absolute value of sine K over 2 and
there we have it we've we solved for the

155
00:14:30,360 --> 00:14:36,500
给定波矢，链的正常模式的频率现在可以了
frequency of the normal modes of our
chain given the wave vector okay now

156
00:14:36,500 --> 00:14:47,079
可能值得绘制答案去，所以看起来像这样，我们在这里
probably worth plotting the answer
go so was it looked like so here we have

157
00:14:47,079 --> 00:14:53,750
垂直的K的欧米茄，然后像这样水平放置K，我们将
Omega of K vertically then we have K
horizontally okay like this and we'll

158
00:14:53,750 --> 00:14:58,089
提出一些要点，让我们对此点进行PI，也许这是
put some points on let's make this point
PI over a maybe this point over here is

159
00:14:58,089 --> 00:15:04,399
在a上减去PI，我们得到正弦类型外观的正弦绝对值
minus PI over a and we have this sine
absolute value of sine kind of looks

160
00:15:04,399 --> 00:15:10,069
像这样的正弦绝对值看起来像这样，好吧， 
like this absolute value of sine kind of
looks like this that okay well there's

161
00:15:10,069 --> 00:15:18,920
应该是相同的高度，这条曲线的高度是平方
supposed to be the same height and the
height of this curve here is to square

162
00:15:18,920 --> 00:15:26,180
 K的根超过m，并且在PI超过a时达到峰值，而在a超过PI时达到峰值，因此
root of K over m and it has its peak at
PI over a and at minus PI over a so this

163
00:15:26,180 --> 00:15:35,110
曲线称为弥散曲线弥散曲线
curve is known as a dispersion curve
dispersion dispersion

164
00:15:36,290 --> 00:15:41,860
只是表示欧米茄是K的函数
it just means Omega as a function of K

165
00:15:41,980 --> 00:15:48,500
现在在这张图片的某个地方，我们应该期望会有声波
now somewhere in this picture we should
expect that there should be sound waves

166
00:15:48,500 --> 00:15:53,389
想着谈论一些固体的振荡，我的意思是我们有这个
with think talking about oscillations of
some solid I mean we have this we have

167
00:15:53,389 --> 00:15:56,660
我们的固体图片，就是原子与弹簧粘在一起， 
our picture of a solid which is just
atoms stuck together with Springs and

168
00:15:56,660 --> 00:16:01,449
某处应该有一些像声波的东西
somewhere in there there should be
something that looks like sound waves

169
00:16:01,449 --> 00:16:05,029
声波在哪里好声波是很长的波长
where are the sound waves well sound
wave is a very long wavelength

170
00:16:05,029 --> 00:16:07,699
现象声音的定义是应该是长波长
phenomenon the definition of sound is
that it should be a long wavelength

171
00:16:07,699 --> 00:16:14,389
振荡，所以我们真的应该在这里小K处寻找
oscillation so we should really be
looking down here at small K to look for

172
00:16:14,389 --> 00:16:20,329
声波只是作为您所知道的助记符
sound waves just to as a you know sort
of mnemonic the sound that you hear has

173
00:16:20,329 --> 00:16:24,350
波长介于一厘米到几米之间
wavelength somewhere between like a
centimeter and several meters that's

174
00:16:24,350 --> 00:16:28,730
您可以听到的声音取决于您耳朵的音质，而
what you can you can hear depending on
how good your ears are whereas the

175
00:16:28,730 --> 00:16:35,899
我们在上面讨论这个pi的原子之间的间距是一个o如此
spacing between atoms that we're talking
about this pi over a a here one o is so

176
00:16:35,899 --> 00:16:40,970
在波矢量上与此pi关联的波长约为
this the wavelength associated with this
pi over a wave vector is on the order of

177
00:16:40,970 --> 00:16:45,380
出现的原子之间的距离还可以，所以波上有2 pi 
the distance between atoms okay with a
appears here so 2 pi over the wave

178
00:16:45,380 --> 00:16:49,399
向量会给你一个原子间距的数字，所以
vector will give you a number on the
order of the spacing between atoms so

179
00:16:49,399 --> 00:16:53,620
这张照片的大部分是非常非常非常小的波长
most of this picture is very very very
small wavelength compared

180
00:16:53,620 --> 00:16:57,580
我们通常认为他的声音是针对非常小的波长的
what we usually think of his sound sound
is for very very small wavelengths right

181
00:16:57,580 --> 00:17:04,119
靠近K的中间等于零，所以我们实际上尝试找出
down near the middle near K equals zero
so let's actually try to figure out what

182
00:17:04,119 --> 00:17:07,269
声音将按照我们弄清楚声音在做什么的方式进行

183
00:17:07,270 --> 00:17:13,329
到K非常接近于零我们扩展了这个东西我们扩展了符号然后我们最终
to K very close to zero we expand this
thing we expand the sign and we end up

184
00:17:13,329 --> 00:17:17,680
得到什么我们会好起来的1/2取消2，我们得到的平方根
getting what do we get well the 1/2
cancels the 2 and we get square root of

185
00:17:17,680 --> 00:17:24,309
 Kappa超过m乘以绝对K的倍数，您会记得
Kappa over m times a times absolute K
and you'll remember the definition of

186
00:17:24,309 --> 00:17:28,630
声速是频率应该是声速乘以绝对值
the sound velocity is that frequency
should be sound velocity times absolute

187
00:17:28,630 --> 00:17:34,750
 K，因此我们得出的声速是m乘以a的Kappa的平方根
K and so we've derived sound velocity is
square root of Kappa over m times a

188
00:17:34,750 --> 00:17:40,720
如果您上学期参加了流体课程，那么我们现在的声速
there we have our sound velocity now if
you took the fluids course last term

189
00:17:40,720 --> 00:17:43,960
您会记住，声音也可以用诸如
you'll remember that sound can also be
defined in terms of things like

190
00:17:43,960 --> 00:17:48,910
流体的可压缩性，我们应该能够获得相同的速度
compressibility of a fluid and we should
be able to get the same velocity of the

191
00:17:48,910 --> 00:17:54,270
从公式中得出的声音是您得出的上一项是1以上的平方根
sound from the formula that you derive
last term which is square root of 1 over

192
00:17:54,270 --> 00:18:07,929
密度质量密度这里的质量密度乘以β的可压缩性还可以
density mass density here mass density
times the beta the compressibility okay

193
00:18:07,929 --> 00:18:12,100
那么我们的链条的质量密度是多少，这很容易
so what's the mass density of our chain
that's easy it's just one mass per

194
00:18:12,100 --> 00:18:15,190
距离a，首先要压缩的是什么
distance a and what's the
compressibility well first we have to

195
00:18:15,190 --> 00:18:19,470
弄清楚可压缩性的定义是V减去-1 
figure out what the definition of
compressibility is it's minus 1 over V

196
00:18:19,470 --> 00:18:25,929
体积和压力，但我们处于一维，因此被负号代替
the volume and pressure but we're in 1
dimension so that gets replaced by minus

197
00:18:25,929 --> 00:18:31,210
 1超过长度D的长度D的力，因此，如果您考虑其中的一小部分
1 over length D length D force so if you
think about one little piece of the

198
00:18:31,210 --> 00:18:37,660
链的长度为1，长度为a，则您从胡克斯（Hookes）延伸森林长度
chain the length is 1 over a length is a
and you length the forest from Hookes

199
00:18:37,660 --> 00:18:43,210
定律是Kappa的负1，所以可压缩性是Kappa的1 
law is minus 1 over Kappa so the
compressibility is just 1 over Kappa a

200
00:18:43,210 --> 00:18:50,290
然后如果我们将这两个表达式插入那好吧
and then if we plug in these two
expressions into that well okay well do

201
00:18:50,290 --> 00:18:55,420
然后在此以这种可压缩性使用质量密度为1 
it here this is then using this mass
density in this compressibility it's 1

202
00:18:55,420 --> 00:19:03,820
超过a的质量，然后所有Kappa都超过Kappa a，这使我们
over mass over a and then all Kappa oops
one over Kappa a and that gives us in

203
00:19:03,820 --> 00:19:07,000
事实与我们在这里得到的结果相同
fact the same result that we had over
here

204
00:19:07,000 --> 00:19:12,550
 kappa超过m倍a，因此与我们从a得到的表达式一致
kappa over m times a so it agrees with
the expression we would get from a

205
00:19:12,550 --> 00:19:19,870
声波应该没有问题的流体力学图片
hydrodynamic picture of what sound waves
should be okay so everything all sorts

206
00:19:19,870 --> 00:19:25,900
似乎工作得很好，但是在这种情况下，它的作用是
seems to work pretty well but that is
way down here in this regime with a

207
00:19:25,900 --> 00:19:31,390
在较高的k处色散是线性的，它不再是线性的
where the dispersion is linear
at higher k it's not linear anymore it

208
00:19:31,390 --> 00:19:35,260
开始弯曲，现在回想起德拜在说德拜是
starts to curve now think back to what
it was that Debye was saying Debye was

209
00:19:35,260 --> 00:19:40,870
说他只是假设他的声音模式的频率是
saying he was going to just assume that
the the frequency of his sound modes was

210
00:19:40,870 --> 00:19:46,180
总是在K线性，然后他以某个最大频率切断频谱
always linear in K and then he cut off
the spectrum at some maximum frequency

211
00:19:46,180 --> 00:19:51,670
好吧，它开始是线性的，它不会保持线性曲线，但是确实
well it starts out linear it doesn't
stay linear curves but indeed it does

212
00:19:51,670 --> 00:19:56,110
最大频率高于某个频率，就没有声音模式了
have a maximum frequency above some
frequency there's no sound modes left we

213
00:19:56,110 --> 00:20:00,100
找到了系统的所有正常模式，这就是所有这些，而且它们具有
found all of the normal modes of the
system this is all of them and they have

214
00:20:00,100 --> 00:20:05,410
只有能量，它们的频率只有零到这个频率之间，所以
only energy they only have frequencies
between zero and this frequency here so

215
00:20:05,410 --> 00:20:10,870
在这方面，德拜是对的，确实存在最大声波
Debye was right in that respect that
there really is a maximum sound wave

216
00:20:10,870 --> 00:20:16,870
频率不是很好的称为声音，而是您的振动频率
frequency it's not well mudan called
sound but your vibrational frequency of

217
00:20:16,870 --> 00:20:21,130
在这一点上，值得在此更仔细地研究一下
this chain it's worth looking a little
bit more closely at this point here

218
00:20:21,130 --> 00:20:26,670
这是最高频率激励的最高频率正常模式
which is the highest frequency
excitation highest frequency normal mode

219
00:20:26,670 --> 00:20:33,850
所以发生在K等于PI超过a，我们将如何看待更多
so that occurs at K equals PI over a and
how are we gonna look at that more

220
00:20:33,850 --> 00:20:41,980
很好，让我们写下波形对我来说是什么
closely well let's let's write down what
the wave form would be it's a e to the I

221
00:20:41,980 --> 00:20:54,730
 Ω减去I PI超过I倍，否则将成为IΩT倍
Omega t minus I PI over a times a n or
it would be a into the I Omega T times

222
00:20:54,730 --> 00:21:01,960
负1到N表示每个交替的偶数与奇数质量都在移动
minus 1 to the N which means that every
alternate even versus odd mass is moving

223
00:21:01,960 --> 00:21:04,960
相反的方向，这是您可能获得的最高频率
in the opposite direction and that's the
highest frequency you can possibly get

224
00:21:04,960 --> 00:21:07,870
好吧，现在值得看一部电影
okay
at this point it's worth seeing a movie

225
00:21:07,870 --> 00:21:16,020
所以，让我，哦，在这里，我们今天看看它是否可以正常工作
so let me uh-oh here we go
see if it works today beautiful okay

226
00:21:16,020 --> 00:21:20,850
这是我们可以从我的网站下载的程序，它是由Mike编写的
so this is a program we can download
from my website it was written by Mike

227
00:21:20,850 --> 00:21:25,890
 Glazer可以在Windows上运行，可以在Linux上的Wine下运行，我认为可能
Glazer it works on Windows it works
under wine on Linux I think it probably

228
00:21:25,890 --> 00:21:29,309
在Mac上可以使用，但我不确定
works on on Mac but I'm not sure about
that

229
00:21:29,309 --> 00:21:35,250
因此，如果它的目的是向您展示其中之一的振荡
so the the its if the purpose of it is
to show you what oscillations of one of

230
00:21:35,250 --> 00:21:39,059
这些链实际上看起来很详细，因此在这里我们单击单原子链
these chains actually looks like in
detail so here we click monatomic chain

231
00:21:39,059 --> 00:21:43,040
我们将点击纵向，这意味着天哪，我改变了速度
and we're gonna click longitudinal which
means oops my gosh change the speed

232
00:21:43,040 --> 00:21:47,580
我们去了，所以您也可以更改振荡速度以使其更容易
there we go so you can change the speed
of oscillation as well to make it easier

233
00:21:47,580 --> 00:21:50,130
看看我们在这个角落有什么
to see so what we have here in this
corner

234
00:21:50,130 --> 00:21:54,690
在这里，您具有色散曲线，可以从
over here you have this dispersion curve
and you can change K back and forth from

235
00:21:54,690 --> 00:22:00,600
你知道高K工具好吧，所以让我们从一个非常低K吧开始吧， 
you know high K tool okay so let's start
with a very fairly low K okay make the

236
00:22:00,600 --> 00:22:04,140
速度相当快，所以您可以看到这里发生的事情，还可以看到
speed fairly fast so you can see what's
going on here okay and you can see at

237
00:22:04,140 --> 00:22:09,030
相当低的K来吧，你可以看到这是一个很长的波长
fairly low K come on yeah you can see
that this is sort of a long wavelength

238
00:22:09,030 --> 00:22:13,080
每个人向左摆动然后向右摆动以及向后摆动的振荡
oscillation where everyone sloshes left
then sloshes to the right and back and

239
00:22:13,080 --> 00:22:18,929
随着K的增加，它来回振荡得更快， 
forth and back and forth now as we
increase K it oscillates faster and

240
00:22:18,929 --> 00:22:23,070
更快，波长变得越来越小，直到最终
faster and the wavelength gets gets
smaller and smaller and until eventually

241
00:22:23,070 --> 00:22:26,070
它的振荡速度真的非常快，我不得不放慢速度
it's oscillating really really fast and
I'm gonna have to slow down the speed so

242
00:22:26,070 --> 00:22:30,120
您实际上可以了解到在这一点上进行PI时发生了什么
you can actually see what's going on
when you get to PI over a this point

243
00:22:30,120 --> 00:22:35,490
在这里，您可以看到其他所有质量都朝相反的方向前进， 
here you can see that every other mass
is going in the opposite direction and

244
00:22:35,490 --> 00:22:38,940
如果您考虑了一会儿，您将意识到这是一个很好的选择
if you think about it for a while you'll
realize that this is there's a good

245
00:22:38,940 --> 00:22:43,440
这是您可能获得的最高频率的原因
reason why this is the highest frequency
that you can possibly get

246
00:22:43,440 --> 00:22:47,280
你不能做得比其他所有人都反对它的邻居更好
that you can't do any better than having
every other mass opposing its neighbor

247
00:22:47,280 --> 00:22:51,150
如果您想获得很高的压缩率或高频，这就是
that if you want to get really high
compression or high frequency this is

248
00:22:51,150 --> 00:22:58,530
您可能会遇到的最高频率，所以我将其留给你们
the highest frequency you could possibly
get okay so I'll leave that for you guys

249
00:22:58,530 --> 00:23:01,320
玩，我建议把它弄乱
to play with
I recommend messing around with it a

250
00:23:01,320 --> 00:23:07,200
一点点，呃，我们将在将Beca再次推向所有人时遇到问题
little bit uh-oh we're gonna have
problems turning Beca that back on all

251
00:23:07,200 --> 00:23:11,990
好吧，所以最重要的是
right anyway
good so the most important thing that we

252
00:23:11,990 --> 00:23:21,500
我们在此计算中得出的是，色散在K中是周期性的
that we deduced in this calculation is
that the dispersion is periodic in K

253
00:23:21,500 --> 00:23:26,780
它与一个符号成比例，您会看到，您知道色散
it was proportional to a sign and you'll
see that you know the the dispersion

254
00:23:26,780 --> 00:23:31,820
只是一遍又一遍地不断地像自己一样前进
just keeps going exactly like itself
over and over we sort of a periodicity

255
00:23:31,820 --> 00:23:35,510
你知道从这里到这里，你只是不断地替换它
you know from here to here you just keep
replacing over and over it looks exactly

256
00:23:35,510 --> 00:23:40,160
同样，这不是巧合，有一个普遍的法律，如果你有
the same this is not a coincidence
there's a general law that if you have

257
00:23:40,160 --> 00:23:48,830
在现实空间中是周期性的，在现实空间中，例如具有
something that's periodic in real space
real space like a chain which has a

258
00:23:48,830 --> 00:23:54,260
如果您采用了我们的整个链，并且您转移了，则周期性周期Delta x等于a 
periodic periodicity Delta x equals a if
you take our whole chain and you shifted

259
00:23:54,260 --> 00:23:58,640
如果有东西，您将再次返回完全相同的链
over by a you get back exactly the same
chain again if you have uh something

260
00:23:58,640 --> 00:24:01,730
在现实空间中是周期性的，我也应该说现实空间也是已知的
that's periodic in real space I should
also say that real space is also known

261
00:24:01,730 --> 00:24:11,120
作为直接空间直接空间意味着您将具有分散
as direct space direct space that will
imply that you have a dispersion which

262
00:24:11,120 --> 00:24:24,710
是周期的，并且k空间中的K也称为具有周期的倒数空间
is periodic and K in k space also known
as reciprocal space with a periodicity

263
00:24:24,710 --> 00:24:32,930
 Delta K的a等于a的2pi，因此名称倒数，因为这里a 
of Delta K equals 2pi over a and hence
the name reciprocal because here the a

264
00:24:32,930 --> 00:24:38,570
在楼上，这里的a在楼下，现在这是一些
is upstairs and here the a is is
downstairs now the periodic here's some

265
00:24:38,570 --> 00:24:42,770
一些更多有用的词，这里的周期单位是已知的
some more words that are going to be
useful the periodic unit here is known

266
00:24:42,770 --> 00:24:50,780
作为棕熊区1区，也许我会写一些定义
as the Bruins zone 1 zone maybe I'll
write that definition a little bit is

267
00:24:50,780 --> 00:25:02,090
这相当重要，k空间中k空间中的周期单位等于棕熊
this rather important the periodic unit
in k space in k space equals the Bruins

268
00:25:02,090 --> 00:25:09,440
我自己，我不仅向讲德语的人道歉，而且向
own and I apologize
not only to German speakers but to

269
00:25:09,440 --> 00:25:15,470
现在讲法语的人说，右侧Louie Dubrow上的这个扫帚是
French speakers now that the this word
broom on the right Louie Dubrow on is a

270
00:25:15,470 --> 00:25:18,680
说英语的人很难正确发音的单词
word that English speakers have a
terribly hard time pronouncing correctly

271
00:25:18,680 --> 00:25:22,100
可能会讲法语的人都会畏缩于
and probably anyone who speaks French
will cringe at how

272
00:25:22,100 --> 00:25:25,460
我屠杀它，但我不是每个讲英语的人中唯一的一个
I butcher it but I'm not the only one
every English speaker

273
00:25:25,460 --> 00:25:30,679
屠夫这个名字我真的很抱歉我在法国差点失败。 
butchers this name I really apologize
about that I almost failed French in

274
00:25:30,679 --> 00:25:34,640
高中，所以我从来没有做得更好，但我认为这是
high school so I've never gotten any
better at it but I think it's something

275
00:25:34,640 --> 00:25:41,240
就像布里恩·旺（Brie Whan）一样，这是我现在最好的猜测
like brie whan that's my best guess
about how it's supposed to be okay now

276
00:25:41,240 --> 00:25:45,350
为什么这是一个非常重要的原则，我们在
why is it this is a very important
principle that we have periodicity in

277
00:25:45,350 --> 00:25:49,480
 k空间为什么我们在k空间具有周期性
k-space why is it we have periodicity in
k-space

278
00:25:49,480 --> 00:25:57,380
好，让我们找出我们的波形Delta xn是I Omega t的ae 
well let's find out we have our wave
form Delta xn is a e to the I Omega t

279
00:25:57,380 --> 00:26:06,110
减去IK na，然后取K并将其偏移2 pi 
minus I K na and then let's take K and
shift it by 2 pi over a shift it by this

280
00:26:06,110 --> 00:26:11,299
从这里到这里的整个长度，我们将取一个特定的K并将其移动2 
entire length from here to here we'll
take some particular K and move it by 2

281
00:26:11,299 --> 00:26:16,760
 pi在a上，看看为什么我们能很好地返回同一件事，让我们将e转换为
pi over a and see why we get back the
same thing well let's do that shift e to

282
00:26:16,760 --> 00:26:24,799
我的I Omega T，然后我们得到负IK + 2 PI乘以na 
the I Omega T and then we get minus I K
plus 2 PI over a times na and that

283
00:26:24,799 --> 00:26:35,179
等于a等于I Omega T减去IK，乘以e等于负I 2 pi N和
equals a e to the I Omega T minus I K
and a times e to the minus I 2 pi N and

284
00:26:35,179 --> 00:26:41,480
这个东西是1，所以实际上我们要返回完全相同的波形
this thing here is 1 so in fact we're
getting back exactly the same waveform

285
00:26:41,480 --> 00:26:46,100
我们开始时色散曲线看起来与此相同的原因
that we started with the reason the
dispersion curve looks the same at this

286
00:26:46,100 --> 00:26:50,360
在这里，这是因为他们正在准确地描述
point here and this point here is
because they're describing exactly the

287
00:26:50,360 --> 00:26:56,210
相同的波，通过将K移2 pi可以使波保持不变，这是一个
same wave the wave is unchanged by
shifting K by 2 pi over a ok this is an

288
00:26:56,210 --> 00:27:00,980
非常重要的原则，我们会一遍又一遍地重复
extremely important principle that we'll
come back over and over and over now

289
00:27:00,980 --> 00:27:06,740
这可能会让您感到不安，从嗯开始吧
this might be something that you you'll
find disturbing to begin with uh-huh ok

290
00:27:06,740 --> 00:27:11,840
现在我们在这里有问题，好吧，现在看看我能不能告诉你，糟糕
now we have a problem here ok now see if
I can show you this oops come back ok

291
00:27:11,840 --> 00:27:20,679
让我向您展示其他东西，看看是否最终恢复正常，是的，很好，所以
let me show you something else see if it
comes back up eventually yes ok good so

292
00:27:20,679 --> 00:27:27,380
您可能会问，在一个波长范围内，K是否等于k加2pi 
you might ask well if K is the same as k
plus 2pi over a what's the wave length

293
00:27:27,380 --> 00:27:32,540
对，我的意思是我们通常认为波长是2pi超过2pi 
right I mean we usually think of the
wave length as being 2pi over a 2pi over

294
00:27:32,540 --> 00:27:38,540
 k等于波长2pi超过k等于2pi，所以如果K相同，则知道它是
k equals a wave length 2pi over k equals
so if K is the same you know which it's

295
00:27:38,540 --> 00:27:43,820
在这里和这里在物理上是同一波波长是正确的
physically the same wave here and here
which one's the wavelength well the way

296
00:27:43,820 --> 00:27:52,280
可以从这张照片中了解到，所以这里绘制了Delta X DX 
to understand that is from this picture
here so Delta X DX here is plotted on

297
00:27:52,280 --> 00:27:57,350
垂直轴只是为了使其更清楚，所以我在这里画了两个波
the vertical axis just to make it more
clear so I've plotted two waves here one

298
00:27:57,350 --> 00:28:03,830
其中实线位于K或实际上该虚线为K且
of them the solid line is at K or
actually this dash line is K and the and

299
00:28:03,830 --> 00:28:12,260
实线在ok处在k加2 pi处，两条曲线在该位置一致
the solid line is at k plus 2 pi over a
ok the two curves agree at the position

300
00:28:12,260 --> 00:28:16,790
群众，我们只是在谈论群众的流离失所， 
of the masses and we're only talking
about the displacement of the masses the

301
00:28:16,790 --> 00:28:21,020
波形在质量位置之间不表示任何东西
wave doesn't mean anything in between
the position of the masses the wave form

302
00:28:21,020 --> 00:28:25,340
是用来描述群众在做什么，你不能问
is meant to describe what the masses are
doing you can't ask what is the value of

303
00:28:25,340 --> 00:28:29,330
群众之间的波浪没有价值，意味着波浪
the wave in between the masses it
doesn't have a value the wave is meant

304
00:28:29,330 --> 00:28:32,930
描述海浪在做什么，群众在做什么
to describe what the waves are doing
what the what the masses are doing so it

305
00:28:32,930 --> 00:28:37,850
您是否在实体中延伸曲线实际上并没有任何意义
doesn't actually mean anything whether
you extend the the curve in the solid

306
00:28:37,850 --> 00:28:42,050
两个质量之间的直线或虚线，但仅此而已
line way or the dashed line way in
between the two masses but only all that

307
00:28:42,050 --> 00:28:46,760
重要的是群众是多少，你知道个别群众有多少流离失所
matters is what the mass is you know how
much the individual masses are displaced

308
00:28:46,760 --> 00:28:51,710
这是一种被称为锯齿的现象，如果您想到了这一点，您将
this is a phenomenon known as aliasing
and if you get this in your head you'll

309
00:28:51,710 --> 00:28:56,150
很好地理解了为什么在这一点上，而这一点实际上是
understand pretty well why it is at this
point and this point are actually

310
00:28:56,150 --> 00:29:01,700
即使他们有不同的洞穴，也可以描述同一波
describing the same wave even though
they have different caves okay good all

311
00:29:01,700 --> 00:29:07,760
好吧好吧，这是Naumann clay Chur多了一点
right all right so it's a couple a
little bit more Naumann clay Chur that's

312
00:29:07,760 --> 00:29:21,520
相当有用的实际空间晶格让我将其定义为所有点
fairly useful the real space lattice
lattice let me define it as all points

313
00:29:21,520 --> 00:29:26,530
等效点
points equivalent

314
00:29:27,460 --> 00:29:35,169
两个x等于零，所以在我们的图片中XN等于a的M倍
two x equals zero so in our picture
there that would be X N equals M times a

315
00:29:35,169 --> 00:29:41,350
因为您可以在整个春季之前移动每个点，然后回到
because you can shift each point by
entire spring and you get back to

316
00:29:41,350 --> 00:29:45,820
看起来完全一样的东西我们也有倒数的空间格
something that looks exactly the same we
also have the reciprocal space lattice

317
00:29:45,820 --> 00:29:55,870
倒数空间我们饮的空间晶格是k中的所有点
reciprocal space we sip space lattice
which is all points in k space points in

318
00:29:55,870 --> 00:30:06,100
等于k的k空间等于0，对于此特定模型，则为
k space equivalent to k equals 0 and for
this particular model it would then be

319
00:30:06,100 --> 00:30:14,140
我们称这些点G sub M等于2 pi M在所有这些点上
let's call those points G sub M equals 2
pi M over a all those points are

320
00:30:14,140 --> 00:30:22,179
等价于k等于0，在实际中，将2 pi的2 pi单位偏移了0 
equivalent to k equals 0 shifted by 2 pi
units of 2 pi over a now it's actually

321
00:30:22,179 --> 00:30:25,690
一些有趣的事情，将在今年晚些时候变得重要
something that is kind of interesting
and will be important later on this year

322
00:30:25,690 --> 00:30:32,289
 EBI GM xn在所有点上总是等于1，并且有直接
is the EBI GM xn is always equal to 1
for all of the points and there's direct

323
00:30:32,289 --> 00:30:34,750
晶格和倒易晶格中的所有点，您可以选择
lattice and all the points in the
reciprocal lattice you can choose any

324
00:30:34,750 --> 00:30:38,080
这些点的结合，实际上总会给你一个
combination of those points and will
always give you one in fact we're gonna

325
00:30:38,080 --> 00:30:41,860
当我们使用更高的尺寸时，我们将使用该属性作为一种方式
use that when we go to higher dimensions
we're gonna use that property as a way

326
00:30:41,860 --> 00:30:45,789
定义倒易格，所以把它放在你的脑后
to define the reciprocal lattice so sort
of put it in the back of your your head

327
00:30:45,789 --> 00:30:49,899
因此，在这一点上，实际上对我们而言非常有用的一件事是
so one thing that's actually really
useful for us to do at this point is to

328
00:30:49,899 --> 00:30:55,090
计算我们为正常波模求解的枚举波总数
count the total number of waves that we
enumerated we solved for normal modes of

329
00:30:55,090 --> 00:31:01,049
该系统中有多少个它们？有多少个正常模式
the system so how many are them of them
are there how many how many normal modes

330
00:31:01,049 --> 00:31:12,370
正常模式，我们知道如果我们要计算不同的正常模式，我们只会
normal modes well we know that if we're
counting different normal modes we only

331
00:31:12,370 --> 00:31:16,330
不得不看一下布鲁因斯区域，那么如果我们走到格兰德区域之外，我们可以
have to look within the Bruins zone then
if we go outside the grande zone we can

332
00:31:16,330 --> 00:31:20,049
总是我们描述的东西等同于
always we're describing something that's
equivalent to another point inside the

333
00:31:20,049 --> 00:31:24,070
棕褐色区域，我们在此处选择此点，例如，它等同于此点
Bruins zone we pick this point here for
example its equivalent to this point

334
00:31:24,070 --> 00:31:28,149
在这里，我们只需要描述*区域内的事物就可以了
here so we only have to describe things
within the bran zone we have to figure

335
00:31:28,149 --> 00:31:31,210
找出*区域内有多少种模式，所以我们将做平常的事情
out how many modes there are within the
bran zone so we'll do the usual thing

336
00:31:31,210 --> 00:31:40,070
使用周期性边界条件周期性边界，所以我们将
use periodic boundary conditions
periodic boundaries so we're putting the

337
00:31:40,070 --> 00:31:44,990
弹起一个大圆圈让我们让圆圈的长度为L 
springs in in a big circle
let's let the circle have a length L

338
00:31:44,990 --> 00:31:52,220
这是n，所以n在圆周长度L周围有n个质量，因此K必须为2 PI 
which is n so n masses around in a
circle length L so K then has to be 2 PI

339
00:31:52,220 --> 00:32:00,290
在L倍上，一些整数PP是整数的元素，通常是
over L times some integer P P is an
element of integers usual way that

340
00:32:00,290 --> 00:32:04,220
当我们将所有内容放入定期框中时发生，因此
happens when we put everything in a
periodic box so the total number of

341
00:32:04,220 --> 00:32:13,130
然后，模式数就是看2pi的k的总范围
modes number of modes is then the total
range of k's that were looking at 2pi

342
00:32:13,130 --> 00:32:17,150
在a上，这就是我们正在寻找的k的整个范围
over a so that's the entire range of k's
we're looking at

343
00:32:17,150 --> 00:32:24,920
除以模式2 PI到L的间距，所以我们要做的是
divided by the spacing between modes 2
PI over L so what we're doing is we're

344
00:32:24,920 --> 00:32:32,390
将自己的青铜器分成许多小块，间距为2 PI 
dividing up this bronze own into lots of
little pieces whose spacing is 2 PI over

345
00:32:32,390 --> 00:32:41,240
 L ok，所以在a或M上为L，因此我们拥有的正常模式总数
L ok and so that is L over a or M so the
total number of normal modes that we've

346
00:32:41,240 --> 00:32:45,830
解决的是n有n个正常模式，因为有n个质量开始
solved for is n there are n normal modes
because there were n masses to begin

347
00:32:45,830 --> 00:32:49,580
这并不奇怪，您在学习时可能会发现
with this should not be surprising you
probably found this when you studied

348
00:32:49,580 --> 00:32:53,990
质量和弹簧桩，如果您从n个质量开始，则不可避免地得到
mass and spring piles before if you
start with n masses you inevitably get

349
00:32:53,990 --> 00:32:59,090
和正常模式，这实际上正是Debye之前猜到的
and normal modes and this is actually
exactly what Debye had guessed before

350
00:32:59,090 --> 00:33:03,080
他想拥有与他的度数相同的模式数
that he's wanted to have just the same
number of modes as he had degrees of

351
00:33:03,080 --> 00:33:09,080
他系统的自由还可以，到目前为止，我们所做的一切都是完全
freedom in his system okay so so far
everything we've done is completely

352
00:33:09,080 --> 00:33:17,870
到目前为止，我们现在将继续讨论量子力学
classical so far so now we're going to
move on to quantum mechanics quantum so

353
00:33:17,870 --> 00:33:21,710
在量子力学中，有一个非常简单的规则，那就是如果您有一个
in quantum mechanics there's a very
simple rule which is that if you have a

354
00:33:21,710 --> 00:33:32,059
普通模式频率的经典普通模式频率欧米茄
normal mode a classical normal mode of
frequency a frequency Omega

355
00:33:32,059 --> 00:33:37,480
当你进入成为本征态的量子力学时
when you go to quantum mechanics that
becomes eigenstates

356
00:33:38,230 --> 00:33:51,770
能量为H bar V sub n等于H bar Omega n加1/2那么这是什么
with energy H bar V sub n equals H bar
Omega n plus 1/2 so what is this what is

357
00:33:51,770 --> 00:33:55,880
这n告诉我们这n告诉我们您选择一个特定的正常模式
this n telling us this n is telling us
you pick a particular normal mode that

358
00:33:55,880 --> 00:33:59,900
你在看，这个n基本上是在告诉你
you're looking at and this n is
basically telling you the amplitude of

359
00:33:59,900 --> 00:34:03,950
该模式，然后，如果您让它振荡很多，那么它就有很多能量
that mode then if you if you let it
oscillate a lot it has a lot of energy

360
00:34:03,950 --> 00:34:07,550
如果你允许的话，我会说一点，它只有一点能量，所以
if you let it I'll say a little bit it
has only a little bit of energy so it's

361
00:34:07,550 --> 00:34:11,840
区别在于最终，但关键是能量在
the difference bein an end but the key
thing is that the energy is quantized in

362
00:34:11,840 --> 00:34:21,409
 H bar Omega的整数单位，这是一个重要的定义， 
integer units of this H bar Omega now
this is an important definition that one

363
00:34:21,409 --> 00:34:33,379
量子量子振动的振动量子是声子。 

364
00:34:33,380 --> 00:34:42,280
声子是声子，它完全类似于光子
phonon is a phonon phonon this is
entirely analogous to a quantum of light

365
00:34:42,280 --> 00:34:56,050
作为光子，它的能量是K的h-barΩ，具体取决于哪个
being a photon and it's energy energy is
h-bar Omega of K depending on which

366
00:34:56,050 --> 00:35:03,620
 K实际上正在考虑，所以我现在不得不
which K were actually considering so I
now I have to I have to rant for a

367
00:35:03,620 --> 00:35:08,120
关于我正常的宠物烦恼的第二件事，所以我的怒气是，如果你打开很多
second about my normal pet peeve
so my rant is that if you open up a lot

368
00:35:08,120 --> 00:35:13,310
他们会告诉你，声子是振动的量子
of books they will tell you that a
phonon is a quantum of vibrational

369
00:35:13,310 --> 00:35:18,530
能量或光子是光能的量子，我认为这实际上是一个坏问题
energy or a photon is a quantum of light
energy and I think that's actually a bad

370
00:35:18,530 --> 00:35:23,420
定义我认为您不应该指定能量，因为光子的确是
definition I think you shouldn't specify
energy because it's true that a photon

371
00:35:23,420 --> 00:35:27,950
携带光，但也携带角动量
carries light but it also carries
angular momentum it carries momentum so

372
00:35:27,950 --> 00:35:33,530
你为什么要指定能量而不是仅仅而不是仅仅指定一个
why did you specify energy and not just
and not just specify this a quantum of

373
00:35:33,530 --> 00:35:36,260
与声子类似地发光这里的声子
light
similarly with phonon here a phonon

374
00:35:36,260 --> 00:35:40,610
携带能量，但也携带动量和其他东西，所以我
carries carries energy but it also
carries momentum and other things so I'm

375
00:35:40,610 --> 00:35:44,750
只是要说一个振动量子，这就是声子，所以要说一点
just gonna say a quantum of vibration
here's the phonon so to be a little bit

376
00:35:44,750 --> 00:35:50,630
通过声子更具体地说明我的意思，我们选择一个特定的
more specific about what I mean here by
by a phonon we pick a particular

377
00:35:50,630 --> 00:35:54,380
模式或特定的振荡频率是特定的波矢K 
mode or a particular oscillation
frequency is a particular wave vector K

378
00:35:54,380 --> 00:35:59,089
然后我们说如果它处于基态N等于零，我们说那里
and then we say if if it's in this
ground state N equals zero we say there

379
00:35:59,089 --> 00:36:05,240
如果您最多激发一个，就不会有声子填充该状态，我们说有一个
are no phonons filling that state if you
excite n up to one we say there's one

380
00:36:05,240 --> 00:36:10,400
在这种模式下的声子，如果我们最终激发到两个，我们说其中有两个声子
phonon in that mode if we excite end up
to two we say there's two phonons in

381
00:36:10,400 --> 00:36:15,410
该模式与您现在对光子和光所做的完全相似
that mode it's exactly similar to what
you do with photons and light now of

382
00:36:15,410 --> 00:36:22,339
当然这些声子必须是玻色子，因为你可以把声子是玻色子
course these phonons have to be bosons
because you can put phonon is a boson is

383
00:36:22,339 --> 00:36:30,670
玻色子，因为在给定的K模式下，您可以在给定模式下放置多个声子
a boson because you can put more than
one phonon in a given mode given K mode

384
00:36:30,670 --> 00:36:40,279
在特定的K模式下，其声子的能量为h-bar Omega K，因此
and its energy energy of the phonons in
a particular K mode is h-bar Omega K so

385
00:36:40,279 --> 00:36:44,599
这是特定K模式下声子的能量乘以玻色系数
this is the energy of the phonons in a
particular K mode times the Bose factor

386
00:36:44,599 --> 00:36:52,329
还有另一个迹象表明我们在这里谈论玻色子Bose factor 
there's another indication that we're
talking about bosons here Bose factor

387
00:36:52,329 --> 00:36:57,470
是因子加一半，所以这里的Bose因子告诉您预期
was factor plus one half so the the Bose
factor here tells you the expected

388
00:36:57,470 --> 00:37:03,710
在特定模式下，给定温度下给定声子数
number of phonons in the particular mode
had a given at a given temperature okay

389
00:37:03,710 --> 00:37:06,890
每个人都很舒服吗，它看起来应该像您所做的一样
is everyone comfortable this it should
look a lot like like what you did with

390
00:37:06,890 --> 00:37:13,400
光子较早是的，是的，很好，现在我们可以做什么
photons earlier yes yeah okay
good so now what we can do we can use

391
00:37:13,400 --> 00:37:17,420
这些信息实际上可以计算出量子力学的热容量
this information to actually calculate
the quantum mechanical heat capacity of

392
00:37:17,420 --> 00:37:22,430
我们的链现在我们的链是一个相当简化的链，这是一个简化的模型
our chain now our chain is a rather
simplified chain it's a simplified model

393
00:37:22,430 --> 00:37:26,450
它发生了什么，您知道它具有这些简化吗？ 
of what's going on it has you know it
has these a number of simplifications

394
00:37:26,450 --> 00:37:29,210
你知道泉水只在最近的邻居之间， 
that you know the springs are only
between nearest neighbors and they're

395
00:37:29,210 --> 00:37:33,589
完美的谐波弹簧，但经过这些简化，我们要做什么
perfectly harmonic Springs but given
those simplifications what we're going

396
00:37:33,589 --> 00:37:37,759
最后给出的是热容量的精确答案
to end up with is an exact answer for
the heat capacity no approximations at

397
00:37:37,759 --> 00:37:41,930
因此，对于热容量的确切答案是链中的总能量
all so the exact answer for the heat
capacity the total energy in the chain

398
00:37:41,930 --> 00:37:47,829
是所有模式的总和K等于a的2 PI的负PI 
is the sum over all of the modes K
equals minus PI over a 2 PI over a of

399
00:37:47,829 --> 00:37:58,430
 h-bar欧米茄模式Boze factor beta H bar欧米茄K加1/2的情况
h-bar Omega that mode Boze factor beta H
bar Omega K plus 1/2 where the case are

400
00:37:58,430 --> 00:38:07,910
现在通常以2 PI超过L um的步骤
taken in in steps of
steps of 2 PI over L um now the usual

401
00:38:07,910 --> 00:38:14,000
可以用很多次以前的方式用整数DK代替总和
way that we have before so many times we
can replace that sum with an integral DK

402
00:38:14,000 --> 00:38:19,940
超过系统长度的2 pi倍，我们只需从负值进行积分
over 2 pi times the length of the system
and we only have to integrate from minus

403
00:38:19,940 --> 00:38:24,410
 PI超过2 PI的PI，因为一旦不在Brauns自己的范围内，我们将被读取
PI over a 2 PI over a because once were
outside the Brauns own we're just read

404
00:38:24,410 --> 00:38:29,120
列举一些我们之前已经描述过的模式，以及
ascribing some of the modes that we have
already previously described and if we

405
00:38:29,120 --> 00:38:33,890
计算系统中的模式总数总数模式总数
count the total number of modes in the
system total number of modes total

406
00:38:33,890 --> 00:38:41,630
模式数量模式数量很好，我们可以写成L over 2 
number of modes number of modes it would
be well okay we can write it as L over 2

407
00:38:41,630 --> 00:38:49,040
 pi L由2个PI减去2个PI减去1个DK上的2 pi积分
pi L over 2 pi integral from minus PI
over a 2 PI over a DK of the number 1 so

408
00:38:49,040 --> 00:38:52,040
我们将要整合所有模式的总数
we're just gonna integrate over all the
number of modes count the total number

409
00:38:52,040 --> 00:38:58,340
会给我们一个L或系统中原子数的模式
of modes that will give us what L over a
or the number of atoms in the system

410
00:38:58,340 --> 00:39:02,900
这再次与Debye中断了整合的过程非常相似
which again exactly similar to what
Debye did he cut off his integration

411
00:39:02,900 --> 00:39:08,780
使得他在系统中恰好有n个模式，并且没有
over modes such that he would have
exactly n modes in the system and no

412
00:39:08,780 --> 00:39:18,350
不少，所以我们使用我们使用的Omega K 
more no less so we use we use our Omega
of K that we derived wherever it is well

413
00:39:18,350 --> 00:39:23,030
我会再写一遍，我们使用2平方根的Kappa 
off the top of the board I'll write it
again we use 2 square root of Kappa over

414
00:39:23,030 --> 00:39:32,450
 m超过2的ka的绝对值符号，如果将其插入到那个和中
m absolute value sign of ka over 2 and
if we plug that in to that sum which we

415
00:39:32,450 --> 00:39:36,200
可以变成积分，我们将获得确切的能量
can turn into an integral we will get
the exact amount of energy in that

416
00:39:36,200 --> 00:39:39,830
系统，他给一个温度，您可以区分它来获取热量
system and he give a temperature and you
can differentiate it to get the heat

417
00:39:39,830 --> 00:39:50,330
购买二手的容量，使ik等于B声音乘以K的绝对值， 
capacity to buy used oh make ik a equals
B sound times absolute value of K which

418
00:39:50,330 --> 00:39:55,070
在小K上非常吻合，但在大K上不同意，但这是一个
agrees pretty well at small K but
doesn't agree at large K but it was a

419
00:39:55,070 --> 00:40:00,070
可以很好地逼近爱因斯坦，也可以用同样的方式描述爱因斯坦
pretty good approximation Einstein you
can describe Einstein the same way

420
00:40:00,070 --> 00:40:08,900
爱因斯坦曾用过他所有的振荡器Omega K的相同频率Omega 0 
Einstein used all his oscillators Omega
K are at the same frequency Omega 0 that

421
00:40:08,900 --> 00:40:11,870
会给你爱因斯坦模型，基本上说有n个隔离子
will give you the Einstein model
basically saying there's n Isolators and

422
00:40:11,870 --> 00:40:15,320
它们无论如何都具有相同的频率
they all have the same same frequency at
any rate they

423
00:40:15,320 --> 00:40:19,010
使我们至少可以写下来，至少可以代数地写下来
enables us to write down at least at
least you can algebraically write it

424
00:40:19,010 --> 00:40:24,080
精确地表达我们的链量子的热容
down an exact expression for exactly the
heat capacity of our chain quantum

425
00:40:24,080 --> 00:40:28,610
从机械上讲，我们需要一个更重要的概念
mechanically there's one more rather
important concept that we need to

426
00:40:28,610 --> 00:40:36,140
今天介绍一下，这又回到了K和K plus是
introduce today which is again coming
back to this idea that K and K plus are

427
00:40:36,140 --> 00:40:43,760
在a上的倒数晶格两个pi M的成员代表同一波I 
a member of the reciprocal lattice two
pi M over a represent the same wave I

428
00:40:43,760 --> 00:40:49,130
意味着我们知道我们喜欢将这些声子视为新的事实
mean the fact that we you know we like
to think of these phonons as being new

429
00:40:49,130 --> 00:40:52,340
粒子以与我们认为光子相同的方式思考量子
particles the same way we think of
photons as being you know quantum

430
00:40:52,340 --> 00:40:55,310
力学很棒，因为您可以将事物视为粒子或波浪
mechanics is great because you can think
of things as particles or waves waves

431
00:40:55,310 --> 00:40:59,000
和颗粒，然后来回走，但是你喜欢
and particles and go back and forth
however however you like

432
00:40:59,000 --> 00:41:02,150
所以有时候我们喜欢把这些声子看作是波浪
so sometimes we like to think about
these phonons as being waves and

433
00:41:02,150 --> 00:41:05,270
有时我们喜欢将它们视为粒子和我们喜欢的粒子
sometimes we like to think about of them
as being particles and particles we like

434
00:41:05,270 --> 00:41:09,050
想一想它们是您知道某个正在移动的物体，它承载着一些
to think about them is you know some
object that's moving and it carries some

435
00:41:09,050 --> 00:41:14,390
动量，实际上有时候在某些模型中，这些声子可以
amount of momentum and in fact sometimes
in some models these phonons can you

436
00:41:14,390 --> 00:41:16,070
知道比这更复杂的模型
know more complicated models than this
one

437
00:41:16,070 --> 00:41:20,810
但是在你的声子中它们可以以它们的方式散布到事物中
but in your phonons can can scatter into
things in their way they can scatter

438
00:41:20,810 --> 00:41:23,960
进入其他声子，从其他电话中散射出来，可以从电子中散射出来
into other phonon scatter off of other
phones they can scatter off of electrons

439
00:41:23,960 --> 00:41:28,220
例如，它们可以散射光，我们将讨论其中的一些
they can scatter off of light for
example and we'll discuss some of those

440
00:41:28,220 --> 00:41:33,470
学期后期的事情，以及您可能认为会有一个
things later in the term and the thing
you might think that there would be a

441
00:41:33,470 --> 00:41:38,660
在散射过程中保持了动量，但是我们怎么可能
conserved momentum in the scattering
process but how is it possible we can

442
00:41:38,660 --> 00:41:43,820
如果K的定义不明确，则在散射过程中保持动量
conserve momentum in a scattering
process if K isn't even well-defined up

443
00:41:43,820 --> 00:41:49,580
对于这一变化，K和K加G代表同一波，那么
to this change that K and K plus G
represent the same wave so what one does

444
00:41:49,580 --> 00:41:57,950
是一个定义定义什么叫晶体动量晶体晶体重量
is one defines define what is known as
crystal momentum crystal crystal weight

445
00:41:57,950 --> 00:42:01,430
好吧，如果是K，这是通常的事情，如果是K，则它的波矢，如果是h-bar 
well okay if it's K it's the usual thing
if it's K its wave vector if it's h-bar

446
00:42:01,430 --> 00:42:09,770
 K它的动量为水晶动量，所以我将输入H条形的晶体动量
K its momentum crystal momentum so I'll
put in the H bars crystal momentum which

447
00:42:09,770 --> 00:42:24,010
被定义为H Bar K模2pi在现在的模上是什么意思模是一个词
is defined as H Bar K modulo 2pi over a
now what's modulo mean modulo is a word

448
00:42:24,010 --> 00:42:36,609
最多等于加法项的加法项，例如12 mod 
which means up to additive terms of
additive terms of so for example 12 mod

449
00:42:36,609 --> 00:42:44,770
 5等于7 mod 5 modulo 5，因为最多5 12和7的加法项是
5 equals 7 mod 5 modulo 5 because up to
an additive term of 5 12 and 7 are the

450
00:42:44,770 --> 00:42:50,440
同样的2 mod 5也一样，所以换句话说，我们就是在承认
same 2 mod 5 is also the same okay
so in other words we're we're confessing

451
00:42:50,440 --> 00:42:56,650
动量在2 pi之内不能守恒
that momentum is not conserved up to
this factor of 2 pi over a that you can

452
00:42:56,650 --> 00:43:02,710
取并给出您知道的2 pi的加法项的自由因子， 
take and give factors you know additive
terms of 2 pi over a freely and this

453
00:43:02,710 --> 00:43:05,920
可能会打扰您，因为您可能知道自己可能真的很喜欢这个东西
might disturb you because you might you
know you might really like this thing

454
00:43:05,920 --> 00:43:11,710
叫做动量守恒，但是我们真的需要回想一下为什么
called momentum conservation but we
really need to think back to why is it

455
00:43:11,710 --> 00:43:14,980
如果首先考虑对称性，我们首先要保持动量
that we wanted momentum conservation in
the first place if you took the symmetry

456
00:43:14,980 --> 00:43:19,619
在相对论课程上学期，您会记住动量守恒
in relativity course last term you'll
remember that momentum conservation

457
00:43:19,619 --> 00:43:24,070
来自空间的对称性，而来自它的空间对称性是
comes from a symmetry of space and the
symmetry of space that it came from was

458
00:43:24,070 --> 00:43:27,609
空间的平移对称性，因此您可以翻译，知道自己的
translational symmetry of space and so
you can translate you know your

459
00:43:27,609 --> 00:43:31,270
在太空中关注您想要的任何地方，所有物理定律仍然是
attention in space to anywhere you want
and all the laws of physics remain the

460
00:43:31,270 --> 00:43:36,040
在我们的模型中相同，我们不再拥有它，因为只允许
same in our model here we don't have
that anymore because we're only allowed

461
00:43:36,040 --> 00:43:42,460
我们的翻译小组翻译的内容比您知道的要小
to translate by a our translation group
is smaller than in you know in out in

462
00:43:42,460 --> 00:43:46,450
这里的空间真空我们只能通过和来翻译
the vacuum of space here we can we're
only allowed to translate by a and

463
00:43:46,450 --> 00:43:50,800
因此，我们得到的保护律不那么强， 
because of that we get a less strong
conservation law that instead of having

464
00:43:50,800 --> 00:43:55,510
动量完全守恒，我们只有动量守恒在2 pi以上
momentum completely conserved we only
have momentum conserved up to 2 pi over

465
00:43:55,510 --> 00:44:00,130
好吧，这就是这里发生的事情，它与这个想法非常相似
a ok so that's what's gone gone on here
it's very similar to this idea of

466
00:44:00,130 --> 00:44:03,970
诺斯里奇定理这不是完全东北定理，而是人们在说
northridge theorem it's not exactly
northeast theorem instead people talking

467
00:44:03,970 --> 00:44:08,260
大约是卑鄙的话，她从
about a mean orther she was amazing a
mathematician from the early part of the

468
00:44:08,260 --> 00:44:12,460
严格来说1900年代的Northridge定理仅适用于连续
1900s northridge theorem strictly
speaking only applies to continuous

469
00:44:12,460 --> 00:44:16,150
对称性，这是一个离散的对称性，但是非常相似，并且
symmetries and this is a discrete
symmetry but it's very very similar and

470
00:44:16,150 --> 00:44:21,490
我们获得的动量守恒能力不如您的原因
the reason we're getting less strong
momentum conservation than we had in you

471
00:44:21,490 --> 00:44:26,590
知道当您想到空间中的规则粒子是因为我们不
know in in when you think about regular
particles in space is because we don't

472
00:44:26,590 --> 00:44:29,350
具有相同的空间平移对称性，我们只有一个离散的
have the same translational symmetry of
space we only have a discrete

473
00:44:29,350 --> 00:44:36,400
平移对称好了，我们还有几分钟，所以我要去
translational symmetry ok we have a few
minutes left so I'm going to I'm going

474
00:44:36,400 --> 00:44:40,880
添加我答应过的主题
to add
subject I promised you I would address

475
00:44:40,880 --> 00:44:46,010
而且那个主题是氢键，所以这是一个完全不同的主题
and that subject is hydrogen-bonding so
this is a totally different subject I'm

476
00:44:46,010 --> 00:44:49,130
只是拿起它，因为它只会花我们五分钟的氢气
just picking this up because it's only
gonna take us five minutes hydrogen

477
00:44:49,130 --> 00:44:55,130
氢键的一般概念是氢是非常特殊的
bonds the general idea of a hydrogen
bond is that hydrogen is a very special

478
00:44:55,130 --> 00:45:00,170
元素，因为它基本上只是一个质子和一个电子，所以可能会提示
element because it's basically just a
proton and an electron so tip you might

479
00:45:00,170 --> 00:45:05,870
想像一下有一个大原子，所以这里有一个像氟的大原子，然后
imagine having a big atom like so here's
a big atom like fluorine and then it

480
00:45:05,870 --> 00:45:11,270
与氢原子或多或少地离子键合，但可能略微共价键合
bonds to hydrogen atom more or less
ionically but maybe slightly covalently

481
00:45:11,270 --> 00:45:15,500
所以氢原子坐在这里是质子，电子是更多
so the hydrogen atom is sitting out here
it's a proton and it's electron is more

482
00:45:15,500 --> 00:45:20,030
或更少地坐在这里，其电子已被吸入氟原子
or less sitting over here its electron
has been sucked to the the fluorine atom

483
00:45:20,030 --> 00:45:24,980
剩下的是光子，所以这是我们的氢核
and what's been left behind is a bare
proton so this is our hydrogen nucleus

484
00:45:24,980 --> 00:45:32,870
剩下的是光子质子，这是非常特别的，因为
and what's left is a bare proton proton
and that's extremely special because

485
00:45:32,870 --> 00:45:37,370
这是化学界唯一一次获得裸质子的时间
it's the only time in chemistry where
you ever get a bare proton it's not

486
00:45:37,370 --> 00:45:41,480
被周围的一些电子屏蔽，所以只要有化学键
screened by some electrons around it so
whenever you have chemical bonding

487
00:45:41,480 --> 00:45:44,480
通常一些电子在周围移动，但是总有一些壳
usually some of the electrons are moved
around but there's always some shell

488
00:45:44,480 --> 00:45:49,160
除了氢可以去除一个电子的情况外
left over except in the case of hydrogen
where the one electron can be removed

489
00:45:49,160 --> 00:45:52,460
从那里，你剩下一个光子，这就是为什么氢键在
from it and you're left with a bare
proton and that's why hydrogen bonds in

490
00:45:52,460 --> 00:45:55,730
非常特殊的方式，那么如果您有另一个原子发生，那将会发生什么
very special ways so what can happen
then is if you have another atom over

491
00:45:55,730 --> 00:46:00,470
这里说氧或另一个氟原子，这个裸质子将
here say oxygen or maybe another
fluorine atom this bare proton will then

492
00:46:00,470 --> 00:46:05,720
使这个氟原子非常强地极化并吸引它，这是
make a very strong polarization of this
fluorine atom and attract it and this is

493
00:46:05,720 --> 00:46:12,290
氢键H键，是氢与另一个原子的键合
the hydrogen bond H bond which is the
bonding of the hydrogen to another atom

494
00:46:12,290 --> 00:46:19,190
通过这种强大的偶极力，让我向您展示如何
via this strong dipole force so let me
show you a classic picture of this how

495
00:46:19,190 --> 00:46:24,560
这表明冰是典型的例子，其中您有一个冰
this shows up so ice is sort of the
classic example of that where you have a

496
00:46:24,560 --> 00:46:28,820
氢坐在氧气表面上，使您的水变成氧气
hydrogen sitting on the surface of
oxygen to make your h2o but the oxygen

497
00:46:28,820 --> 00:46:32,240
基本上已经将氢中的电子去除了
has essentially removed the electrons
from the hydrogen to a very large extent

498
00:46:32,240 --> 00:46:39,610
留下一个裸露的氢载体核，一个裸露的质子
leaving behind a bare hydrogen bearer
nucleus a bare proton that their proton

499
00:46:39,610 --> 00:46:44,750
使另一个电子极化，并吸引另一个电子
polarizes this other electron over here
and attract this other electron here on

500
00:46:44,750 --> 00:46:50,180
这种氧气并吸引氧气，为此形成非常弱的点状键
this oxygen and attracts the oxygen
making a very weak dotted bond to this

501
00:46:50,180 --> 00:46:55,059
大氧气坐在这里，这你
big
oxygen sitting over here and this you

502
00:46:55,059 --> 00:46:58,569
知道它是弱键，比共价键或弱键弱得多
know it's a weaker bond it's a much
weaker bond than a covalent bond or an

503
00:46:58,569 --> 00:47:02,109
离子键，但足以结冰
ionic bond
but it's strong enough to freeze ice you

504
00:47:02,109 --> 00:47:07,240
知道低于零，所以它是一个a，你知道这是一个足够强大的结合力，可以做一些
know below below zero so it's a you know
it's a strong enough bond to do some

505
00:47:07,240 --> 00:47:13,510
有趣的是，氢键进入的另一个经典案例是生物学
interesting things another classic case
where hydrogen bonds enter is in biology

506
00:47:13,510 --> 00:47:19,569
特别是DNA，如果您上学期是生物学部分， 
and particularly in DNA if you took the
biology segment last term you'll

507
00:47:19,569 --> 00:47:23,140
如果您现在感觉不好，就将其识别为DNA 
recognize this as being DNA if you
didn't well okay now you're seeing it

508
00:47:23,140 --> 00:47:29,829
这是DNA，您会看到DNA是由双链组成的，对
this is DNA and you'll see so DNA is
made up this double strand and right

509
00:47:29,829 --> 00:47:32,740
在双股线的中间向下，两股线拉在一起
down the middle of the double strand
where the two strands zipped together

510
00:47:32,740 --> 00:47:38,079
有一个氢键与氧键，这个键是氢键
there's a hydrogen bonded to an oxygen
this bond is the hydrogen bond the

511
00:47:38,079 --> 00:47:41,920
氢实际上是与氮结合的
hydrogen is really bonded to the
nitrogen over here and it has a weak

512
00:47:41,920 --> 00:47:45,760
在这里与氧键结合，您会发现存在类似的氢键
bond to the oxygen over here and you'll
see that there's a similar hydrogen bond

513
00:47:45,760 --> 00:47:49,990
从这里的氢到这里的氮同样地，氢键在这里
from a hydrogen here to a nitrogen here
similarly hydrogen bonds right down here

514
00:47:49,990 --> 00:47:53,799
氢键就在这里，所以氢键保持在一起
and hydrogen bonds down right here so
it's a hydrogen bond that holds together

515
00:47:53,799 --> 00:47:59,140
 DNA沿着拉链向下延伸，因此当您拉动DNA链时
the DNA right down down this zipping so
so when you pull the strands of DNA

516
00:47:59,140 --> 00:48:04,180
分开氢键，所以我认为这就是您所需要的
apart you're breaking hydrogen bonds so
I think that's all pretty much you need

517
00:48:04,180 --> 00:48:09,430
知道氢键，我想我们会完成的，我们会再次捡起
to know about hydrogen bonds and I guess
we will finish we'll pick up again

518
00:48:09,430 --> 00:48:15,449
明天[鼓掌] 
tomorrow
[Applause]

之间有角度，那是说如果有一个角度
yeah if there's an angle between pee
you're saying that if there's an angle

514
00:50:55,559 --> 00:50:59,309
在p1到p2之间​​会有余弦Thetas等等，类似的东西
between p1 to p2 there would be cosine
Thetas and so forth and things like that

515
00:50:59,309 --> 00:51:03,960
是的，但是在力量仍然具有吸引力的地方，实际上
yeah but that where the force will still
be attractive you can actually there's

516
00:51:03,960 --> 00:51:07,020
书中有一个练习可以帮助您完成练习，并且您可以
there's an exercise in the book that
works you through it and you can and you

517
00:51:07,020 --> 00:51:10,680
可以使所有角度正确，并且总是会出现有吸引力的好问题
can get all the angles right and it will
always come out attractive good question

518
00:51:10,680 --> 00:51:15,530
但不值巧克力棒，所以还是个好问题，谢谢
but not worth the chocolate bar so but
it's still a good question thank you

519
00:51:15,530 --> 00:51:22,799
无论如何，所以基本上你在一个原子上会有一点波动
anyway the so it's basically you get a
little bit of a fluctuation on one atom

520
00:51:22,799 --> 00:51:26,160
它引起另一个原子的波动，它们吸引，无论是哪个原子
it causes a fluctuation the other atom
and they attract and no matter which

521
00:51:26,160 --> 00:51:28,710
方向波动是响应波动在
direction the fluctuation is the
responding fluctuation is in the

522
00:51:28,710 --> 00:51:31,859
相反的方向，他们总是吸引，所以你仍然可以得到一个非零
opposite direction and they always
attract so you can still get a nonzero

523
00:51:31,859 --> 00:51:35,640
范德华力没有范德华力比共价离子弱得多
Van der Waals force no the van der Waals
force is much weaker than covalent ionic

524
00:51:35,640 --> 00:51:41,880
或氢键，但它仍然足够强到可以引起诸如
or hydrogen bond but it's still strong
enough to cause important effects such

525
00:51:41,880 --> 00:51:46,559
如您所知，它可以将低温下的氩气和
as you know you know it holds together
things like argon at low temperature and

526
00:51:46,559 --> 00:51:51,539
当氩气变成固体时，范德华斯更有趣
when argon becomes a solid a more
interesting case of where van der Waals

527
00:51:51,539 --> 00:51:56,190
如果你去过热带气候，部队就会和这个家伙在一起
forces show up is with this guy if
you've ever been to tropical climate

528
00:51:56,190 --> 00:52:00,150
这是壁虎的小蜥蜴，他们也出售汽车保险
this this is a gecko a little lizard
they also they sell car insurance in the

529
00:52:00,150 --> 00:52:03,869
美国我不知道为什么这是事实，这是美国的吉祥物
United States I don't I don't know why
that's true this is the mascot of the

530
00:52:03,869 --> 00:52:11,099
壁虎Geico壁虎公司无论如何壁虎是惊人的小动物
Gecko Geico gecko company anyway
geckos are amazing little creatures

531
00:52:11,099 --> 00:52:15,960
因为它们可以爬上几乎完全平坦的玻璃墙而没有
because they can crawl up almost
completely flat glass walls with no

532
00:52:15,960 --> 00:52:20,010
根本没有麻烦，实际上把它们粘在墙上的东西变成了
trouble at all and the thing that
actually sticks them to the wall turns

533
00:52:20,010 --> 00:52:23,490
成为范德华力，他们让你知道他们的脚很扁平， 
out to be Van der Waals forces they have
you know they have very flat foot and

534
00:52:23,490 --> 00:52:27,599
他们将脚踩在玻璃上，并且有足够的范德华力
they stick their foot onto the glass and
there's enough van der Waals force

535
00:52:27,599 --> 00:52:30,869
在他们的脚和玻璃之间，它们是相当轻的小动物
between their foot and the glass and
they're fairly light little creatures

536
00:52:30,869 --> 00:52:34,589
他们实际上可以走到墙上，因为那很好，我向我道歉
that they can actually walk up the wall
because of that all right I apologize I

537
00:52:34,589 --> 00:52:40,130
过去了，我星期三见
went over it I will see you on Wednesday
you