We're going to start at the bottom of the staircase, which is basic examples of spinors in physics.

我們將從樓梯的最底層開始，這是物理學中旋子的基本例子。

In the next four or five videos, I'm going to talk about two examples of how spinors come up in physics, the polarization of light waves and quantum spin states.

在接下來的四、五個視頻中，我將講述物理學中如何出現自旋體的兩個例子，即光波的偏振和量子自旋態。

We're going to find that both of these phenomena can be described by spinors, in the form of two-by-one columns with complex number entries.

我們會發現，這兩種現象都可以用旋子來描述，即帶有複數項的二乘一列的形式。

Afterward, we'll see how we can visualize these spinors as flagpoles on a sphere, called the Poincaré sphere or Bloch sphere.

之後，我們將看到如何把這些旋量可視化為一個球體上的旗杆，這個球體被稱為 "波昂卡萊球體 "或 "布洛赫球體"。

In this video, we're going to discuss all the different polarizations of light waves and their corresponding spinors, which are called Jones vectors in this specific case.

在本視頻中，我們將討論光波的所有不同偏振及其相應的旋光子，在這種特定情況下，旋光子被稱為瓊斯向量。

And in the next video, we will discuss how to rotate between these polarizations using SU2 matrices.

在下一個視頻中，我們將討論如何利用 SU2 矩陣在這些極化之間進行旋轉。

So I'm hoping you're familiar with electric and magnetic fields.

所以，我希望你們熟悉電場和磁場。

The electric field is a set of vectors everywhere in space that point in the direction that a positive charge will be pulled in due to electric forces.

電場是一組在空間各處指向正電荷在電場力作用下被拉向的方向的矢量。

The magnetic field is another set of vectors everywhere in space that tell us which direction a compass will point due to magnetic forces.

磁場是空間中無處不在的另一組矢量，它告訴我們由於磁力的作用，指南針會指向哪個方向。

According to the laws of electricity and magnetism, it's possible for electric and magnetic field vectors to vibrate back and forth together in a wave, called an electromagnetic wave.

根據電學和磁學定律，電場矢量和磁場矢量有可能在波中來回振動，這就是電磁波。

The light we see is an example of electromagnetic waves.

我們看到的光就是電磁波的一個例子。

Usually, electromagnetic waves are traveling waves, meaning they travel through space over time.

通常，電磁波是行波，即隨著時間的推移在空間中傳播。

The image you're seeing here is a snapshot of an electromagnetic wave at a single instant in time.

您現在看到的影像是電磁波在某一瞬間的快照。

But as time passes by, the wave will travel forward.

但隨著時間的流逝，波浪會向前移動。

Electromagnetic waves are transverse waves.

電磁波是橫波。

This means the direction of the wave oscillations is always perpendicular to the direction of travel.

這意味著波的振盪方向始終與行進方向垂直。

So if an electromagnetic wave is traveling in the z direction, the wave can oscillate in the x-y plane perpendicular to the direction of travel, but the wave cannot oscillate in the z direction, parallel to the direction of travel.

是以，如果電磁波沿 z 方向傳播，電磁波可以在垂直於傳播方向的 x-y 平面上振盪，但電磁波不能在與傳播方向平行的 z 方向上振盪。

The polarization of a given electromagnetic wave is the geometric orientation of the wave.

特定電磁波的偏振是指電磁波的幾何方向。

We only use the electric field E to define the polarization of an electromagnetic wave, so we ignore the magnetic field when talking about polarization.

我們只用電場 E 來定義電磁波的極化，是以在討論極化時，我們忽略了磁場。

Let's look at an electromagnetic wave propagating in the z direction.

讓我們來看看沿 Z 方向傳播的電磁波。

Again, here we're looking at a snapshot of a traveling wave at one instant in time.

同樣，在這裡我們看到的是行波在某一瞬間的快照。

When the electric field oscillates up and down along the y-axis, we call this vertical polarization.

當電場沿 y 軸上下襬動時，我們稱之為垂直極化。

If we rotate this wave by a quarter turn, it is now oscillating left and right along the x-axis, and we call this horizontal polarization.

如果我們將這個波旋轉四分之一圈，它現在就會沿著 x 軸左右擺動，我們稱之為水準偏振。

If we rotate the wave by another quarter turn, we end up with vertical polarization again, although with the wave shifted by a half cycle compared to the original wave.

如果我們將波浪再旋轉四分之一圈，就會再次出現垂直極化，不過與原始波浪相比，波浪偏移了半個週期。

Let's look more closely at a vertically polarized wave.

讓我們更仔細地觀察垂直極化波。

Remember, the electric field is a vector field, meaning it is represented as an arrow at every point in space and time.

請記住，電場是一個矢量場，這意味著它在空間和時間的每一點上都表示為一個箭頭。

The arrow has components in the x, y, and z directions, which we can write like this, using a linear combination of the x, y, z basis vectors.

箭頭在 x、y、z 三個方向上都有分量，我們可以用 x、y、z 基向量的線性組合來這樣寫。

We can also write this as a three-component column like this.

我們也可以把它寫成這樣一個由三部分組成的列。

For a vertically polarized wave, the electric field only exists along the y-axis, so the x and z components of the electric field go to zero.

對於垂直極化波，電場只沿 y 軸存在，是以電場的 x 和 z 分量為零。

If we were to write out the y-component of the electric field for this traveling wave, it would be an amplitude A times a cosine wave.

如果我們要寫出這個行波的電場 y 分量，它將是振幅 A 乘以餘弦波。

That depends on both time and space.

這取決於時間和空間。

Here, t is time, and z is the location along the z-axis.

這裡，t 是時間，z 是沿 Z 軸的位置。

Omega is the angular frequency, and k is the angular wave number.

Omega 是角頻率，k 是角波數。

I find traveling waves are easiest to visualize on a spacetime diagram, with time moving into the future as we move upward.

我發現行波最容易在時空圖上形象化，當我們向上移動時，時間就會向未來移動。

Let's say that we have a wave that travels in the positive z direction over time.

比方說，我們有一個波，它隨著時間的推移沿著正 Z 方向傳播。

We could draw out this traveling wave as a surface in our spacetime diagram.

我們可以在時空圖中將這種行進的波繪製成一個曲面。

The density of the waves in the spatial direction is given by the wave number, and the density of the waves in time is given by the frequency.

波在空間方向上的密度由波數表示，波在時間方向上的密度由頻率表示。

One thing we can do to modify our traveling wave is to add an extra phi value inside the sinusoid function.

要修改我們的行波，我們可以做的一件事就是在正弦函數中添加一個額外的 phi 值。

You can think of this as indicating the wave's starting value at the origin z equals zero when time t equals zero.

您可以認為這表示當時間 t 等於零時，波在原點 z 處的起始值等於零。

This starting value is called a phase, and changing the phase value will shift the wave back and forth along the axis of travel.

這個起始值被稱為相位，改變相位值將使波沿移動軸來回移動。

Applying a phase shift of negative pi over two to a cosine will shift the wave ahead a quarter cycle, which is equivalent to a sine wave.

對餘弦波進行負 pi 超過 2 的相移，會使波浪向前移動四分之一個週期，相當於正弦波。

Now something that can help us represent waves is remembering Euler's formula, which tells us that we can write e to the power of the imaginary i times theta as cosine theta plus i times sine theta.

歐拉公式告訴我們，我們可以把 e 的虛數 i 乘以 Theta 的冪寫成餘弦 Theta 加上 i 乘以正弦 Theta。

This is a useful property, because if we have e to the i theta and we want to add some angle phi to it, all we do is just multiply by e to the i times phi, and use the standard exponent rules to rewrite this as a single exponential with the exponents added.

這是一個非常有用的性質，因為如果我們有 e 到 i 的 theta，並想在其上添加某個角度 phi，我們只需用 e 到 i 乘以 phi，然後使用標準指數規則將其改寫為添加了指數的單指數。

If we then use Euler's formula to convert back to sine and cosine, we see that a phase phi has been added inside the sinusoid functions.

如果我們再用歐拉公式轉換回正弦和餘弦，就會發現正弦函數內部增加了一個相位 phi。

Now electromagnetic waves are described by real numbers.

現在，電磁波是用實數來描述的。

But for convenience, we can pretend that our Ey component is actually a complex number, where the real part of this complex number is the actual electric field we observe in real life, with the imaginary part being ignored.

但為了方便起見，我們可以假定我們的 Ey 分量實際上是一個複數，這個複數的實部就是我們在現實生活中觀察到的實際電場，虛部被忽略。

This means the full Ey component would have a real part with a cosine wave, and we can invent the fake imaginary part to be a sine wave, since we ignore it anyway.

這意味著全 Ey 分量的實部將是餘弦波，我們可以把假的虛部編成正弦波，因為無論如何我們都會忽略它。

Using Euler's formula, we can write this more compactly as a complex exponential.

利用歐拉公式，我們可以更簡潔地將其寫成復指數。

This also means we can pull the phase phi out using exponent rules, and write it as a multiplicative factor, e to the i phi.

這也意味著我們可以利用指數規則得出相位 phi，並將其寫成乘法因子 e 到 i phi。

This allows us to write travelling waves as three separate multiplicative factors, the amplitude, times the phase, times the actual travelling wave.

這樣，我們就可以將行波寫成三個獨立的乘法因子：振幅乘以相位，再乘以實際的行波。

So while the true electric field here is just the real part of this complex wave, we can write the electric field using complex exponentials for mathematical convenience.

是以，雖然這裡真正的電場只是這個複數波的實部，但為了數學上的方便，我們可以用複數指數來寫電場。

As I said before, if we take this vertically polarized wave and rotate it a quarter turn, we get a horizontally polarized wave.

正如我之前所說，如果我們把這個垂直極化波旋轉四分之一圈，就會得到一個水平極化波。

In this case, the wave only oscillates in the x direction, and so the y and z components of the field are zero.

在這種情況下，波只在 x 方向上振盪，是以場的 y 和 z 分量為零。

Once again, the true electric field is given by a cosine, but for convenience, we can write it as a complex exponential, with the amplitude and phase written separately in front.

同樣，真實的電場是由余弦給出的，但為了方便起見，我們可以把它寫成一個復指數，前面分別寫出振幅和相位。

So to review, with the vertically polarized waves, the y component of the electric field is a travelling wave, and the x and z components are zero.

是以，回顧一下，在垂直極化波中，電場的 y 分量是行波，而 x 和 z 分量為零。

And with a horizontally polarized wave, the x component of the electric field is a travelling wave, and the y and z components are zero.

而對於水平極化波，電場的 x 分量為行波，y 和 z 分量為零。

Now, we know from physical experiments that electromagnetic waves can be superimposed on top of each other.

現在，我們從物理實驗中得知，電磁波可以相互疊加。

So it's possible to add horizontally and vertically polarized waves together like this.

是以，可以像這樣將水準和垂直極化波疊加在一起。

Now notice that the two polarizations each have their own amplitude and their own phase.

現在請注意，兩個極化各有自己的振幅和相位。

But the actual travelling wave portion is identical in both, so we can factor it out and write it outside the column like this.

但實際的行波部分在兩者中是相同的，是以我們可以將其係數化，然後像這樣寫在列外。

Now here I want to state the main idea of this video.

現在，我想闡述一下本視頻的主旨。

When it comes to studying the polarizations of waves, all the information we need to look at is contained inside this column, the amplitudes and the phases.

在研究波的偏振時，我們需要查看的所有資訊都包含在這一列中，即振幅和相位。

The actual travelling wave portion of the formula can be ignored, since it's the same for all components.

公式中的實際行波部分可以忽略，因為它對所有元件都是一樣的。

We can even go one step further and ignore the z component altogether, since we can always choose coordinates where the z direction is the direction of wave propagation.

我們甚至可以更進一步，完全忽略 Z 分量，因為我們總是可以選擇 Z 方向為波傳播方向的座標。

And therefore the z component of the electric field will always be zero.

是以，電場的 Z 分量永遠為零。

The resulting two-by-one column of complex numbers is called a Jones vector, and it tells us everything we need to know about the polarization of a given wave.

由此得到的複數的二乘一列稱為瓊斯向量，它告訴了我們關於特定波的偏振的一切資訊。

Any light wave polarization can be written as a linear combination of a horizontally polarized wave and a vertically polarized wave, each with complex numbers in front denoting their respective amplitudes and phases.

任何光波的偏振都可以寫成水準偏振波和垂直偏振波的線性組合，每種偏振前面都有複數，表示各自的振幅和相位。

Moving forward, I'm going to write this horizontally polarized wave of amplitude 1 using the vector capital H, and I'll write this vertically polarized wave with amplitude 1 using the vector capital V.

接下來，我將用矢量大寫字母 H 來書寫振幅為 1 的水平極化波，用矢量大寫字母 V 來書寫振幅為 1 的垂直極化波。

So using the idea of Jones vectors, we can create new polarizations by selecting different values for the amplitude and phase parameters.

是以，利用瓊斯矢量的思想，我們可以通過選擇不同的振幅和相位參數值來創建新的極化。

For example, we can set both phases to zero, so that the horizontal and vertical polarizations are in phase with each other, and then set both amplitudes to one.

例如，我們可以將兩個相位都設為零，使水準和垂直極化互為同相，然後將兩個振幅都設為一。

If we think of H as a vector along the x-axis and V as a vector along the y-axis, this new Jones vector points diagonally to the upper right.

如果我們把 H 視為沿 x 軸的矢量，把 V 視為沿 y 軸的矢量，那麼這個新的瓊斯矢量就會斜指向右上方。

This is called diagonal polarization, denoted with a capital D.

這就是所謂的對角極化，用大寫字母 D 表示。

This is what we would get if we took a horizontally polarized wave and rotated it counterclockwise by 45 degrees.

如果我們將水準偏振波逆時針旋轉 45 度，就會得到這樣的結果。

Although, according to Pythagoras, the Jones vector H plus V has an amplitude of the square root of 2, so we usually divide the components by the square root of 2 to force the amplitude of D to be 1.

不過，根據畢達哥拉斯定律，瓊斯矢量 H 加 V 的振幅是 2 的平方根，是以我們通常會將各分量除以 2 的平方根，以迫使 D 的振幅為 1。

It's also possible to set the amplitude to plus 1 for H and negative 1 for V, and this We call this anti-diagonal polarization A.

也可以將振幅設置為正 1 表示 H，負 1 表示 V，我們稱之為反對角極化 A。

This is what we would get if we took a horizontally polarized wave and rotated it clockwise by 45 degrees.

如果我們將水準偏振波順時針旋轉 45 度，就會得到這樣的結果。

Once again, we normalize this by dividing by the square root of 2.

我們再一次通過除以 2 的平方根將其歸一化。

So we can get the D and A polarizations by adding H and V together in the right amounts.

是以，只要將 H 和 V 相加，就能得到 D 和 A 極化。

In fact, we can make the wave's polarization have any angle on this circle, if we set the coefficients in front of H and V properly.

事實上，只要我們適當設置 H 和 V 前面的係數，就可以使波的偏振在這個圓上具有任意角度。

But there are actually more polarizations that we still haven't seen yet.

但實際上，我們還沒有看到更多的兩極分化。

Let's again set both amplitudes to 1, and set the horizontal phase to be 0, and set the vertical phase to be pi over 2, which is a phase of a quarter wave cycle.

讓我們再次將兩個振幅都設為 1，水準相位設為 0，垂直相位設為 pi 大於 2，也就是四分之一波週期的相位。

This polarization is more difficult to visualize, so let's bring back our column notation with the traveling wave to understand it better.

這種偏振更難形象化，是以讓我們回到行波的列式符號，以便更好地理解它。

Let's pretend we're stuck at the position z equals 0, and the angular frequency of the wave is 1, and we're watching the wave travel around the xy plane over time.

假設我們停留在 z 等於 0 的位置，波的角頻率為 1，我們觀察波在 xy 平面上隨時間的變化。

What would this look like?

這看起來像什麼？

We can combine the exponentials, then use Euler's formula, and then take the real part to see that the electric field is given by a cosine, and a cosine with a phase factor of a quarter cycle, which is really the same thing as a negative sign.

我們可以將指數結合起來，然後使用歐拉公式，再求取實部，就可以看到電場是由余弦和相位係數為四分之一週期的餘弦給出的，這實際上與負號是一回事。

At time equals 0, this gives the coordinates 1, 0.

時間等於 0 時，座標為 1，0。

Then at t equals pi over 2, we get the coordinates 0, negative 1.

然後，在 t 等於圓周率大於 2 時，我們得到座標 0，負 1。

Then at t equals pi, we get negative 1, 0.

那麼在 t 等於 pi 時，我們得到負 1，0。

At t equals 3 pi over 2, we get 0, 1.

在 t 等於 3 pi 大於 2 時，我們得到 0、1。

And at t equals 2 pi, we get 1, 0 again.

而在 t 等於 2π 時，我們又得到 1，0。

So in this wave, the electric field vector travels around in a circle in a clockwise direction over time.

是以，在這個波中，電場矢量隨著時間的推移以順時針方向繞圈運動。

Bringing back the z-axis, if we take our left hand and point our thumb in the direction of wave travel along the z-axis, the electric field vector at the origin z equals 0 will follow the direction of our curled fingers.

回到 Z 軸，如果我們用左手的拇指沿著 Z 軸指向波的傳播方向，那麼原點 z 等於 0 處的電場矢量就會沿著我們蜷曲的手指的方向傳播。

Since the wave follows a circle according to our left hand, we call it left circular polarized.

由於該波按照我們左手的方向繞了一個圈，是以我們稱之為左圓極化波。

This wave would have a helix shape in three dimensions.

這種波在三維空間中呈螺旋狀。

Wikipedia user Dave3457 has made a great animation of this that he's put into the public domain.

維基百科用戶 Dave3457 製作了一個很棒的動畫，並將其發佈到公共領域。

We denote left circular polarized waves as capital L.

我們用大寫字母 L 表示左圓極化波。

It's also more common to write e to the i pi over 2 as just i, which is a quarter turn in the complex plane.

此外，更常見的做法是將 e 到 2 的 i pi 只寫成 i，即在複數平面上轉四分之一圈。

And once again, we normalize by dividing by the square root of 2.

我們再一次通過除以 2 的平方根來進行歸一化處理。

If we repeat this process from the start, but instead giving the vertical polarization a phase of negative pi over 2, we get a helix wave that corkscrews in the opposite direction, matching our right hand when the right thumb points in the direction of wave travel.

如果我們從頭開始重複這個過程，但讓垂直極化的相位為負 pi 大於 2，我們就會得到一個螺旋波，它向相反的方向旋轉，當右手拇指指向波的傳播方向時，就會與我們的右手相吻合。

So we call this right circular polarized, denoted by capital R.

是以，我們稱之為右圓極化，用大寫字母 R 表示。

Now you'll notice in this video, I've written my traveling waves with a positive time term and a negative z term.

現在你會注意到，在這段視頻中，我在寫行波時使用了正時間項和負 Z 項。

Some textbooks use the opposite sign convention, with a positive z term and a negative time term.

有些教科書使用相反的符號約定，即正的 Z 項和負的時間項。

And as a result, the Jones vectors for circularly polarized waves have opposite signs on the So make sure you know which sign convention you're using when talking about circularly polarized waves.

是以，圓極化波的瓊斯矢量在圓極化面上的符號是相反的。

So to summarize this video, we initially discovered the Jones vectors h and v, which represent horizontally and vertically polarized traveling waves.

是以，總結這段視頻，我們最初發現了瓊斯矢量 h 和 v，它們分別代表水準和垂直極化的行波。

We then found that writing them in different linear combinations could give us new polarizations of light.

然後我們發現，將它們寫成不同的線性組合，可以得到新的偏振光。

Like diagonal and anti-diagonal polarizations, and the left and right circular polarizations.

比如對角極化和反對角極化，以及左旋極化和右旋極化。

We're going to see in the next video that these Jones vectors that represent wave polarizations are actually spinors.

我們將在下一個視頻中看到，這些代表波極化的瓊斯矢量實際上是旋量。

At this point, there's no particular reason why we would decide these pairs of complex numbers are special objects that deserve the special name of spinner.

在這一點上，我們並沒有什麼特別的理由來認定這些複數對是特殊的對象，值得我們給它們起一個特殊的名字--"旋轉器"。

But in the next video, we're going to see how we can rotate these various polarizations into each other using special matrices called SU2 matrices.

不過，在下一個視頻中，我們將看到如何使用稱為 SU2 矩陣的特殊矩陣將這些不同的偏振相互旋轉。

And this will reveal a special angle doubling relationship between physical space and the space of wave polarizations.

這將揭示物理空間與波極化空間之間的特殊角度倍增關係。

We'll see that one rotation in physical space corresponds to two full rotations in the polarization space.

我們將看到，物理空間中的一次旋轉相當於偏振空間中的兩次完整旋轉。

The Jones vectors that live in polarization space require two full rotations to get back to the starting point.

極化空間中的瓊斯矢量需要旋轉兩圈才能回到起點。

And this is exactly what we would expect from spinors.

這正是我們對旋子的期望。

In the meantime, before the next video, I'd suggest trying to draw out the shapes of various polarizations for the Jones vectors shown on the screen here.

同時，在觀看下一個視頻之前，我建議大家試著畫出螢幕上顯示的瓊斯矢量的各種偏振形狀。

If you feel confused, just use the strategy I showed earlier of taking z equals zero and omega equals one, and plugging in different time values to see how the wave moves around in the xy plane.

如果你感到困惑，只需使用我之前展示的策略，即取 z 等於零，ω 等於 1，然後輸入不同的時間值，看看波在 xy 平面上是如何移動的。

The answers are linked in the description.

答案鏈接在說明中。