Name: 光子動量和有效品質 (Photon Momentum and Effective Mass)
Uploaded: 2024-10-02T10:38:17.000Z
Duration: 6 min 57 s

In this video, we're going to talk about how to calculate the momentum of a photon.

在本視頻中，我們將討論如何計算光子的動量。

The formula for it is momentum is equal to Planck's constant divided by the wavelength of a photon.

它的計算公式是：動量等於普朗克常數除以光子的波長。

Now for those of you who want to see the derivation for the formula, it starts from this equation.

現在，如果你們想了解公式的推導過程，可以從這個等式開始。

E squared is equal to the rest mass of a photon times c squared squared plus the momentum times c squared.

E 的平方等於光子的靜止品質乘以 c 的平方再加上動量乘以 c 的平方。

Now a photon is defined as not having any rest mass.

現在，光子被定義為沒有任何靜止品質。

It does have effective mass, but not any rest mass.

它確實有有效品質，但沒有靜止品質。

Now if we take the square root of both sides, we get that the energy of a photon is equal to the momentum times the speed of light.

現在，如果我們把兩邊都取平方根，就可以得到光子的能量等於動量乘以光速。

Dividing both sides by c, we get that E over c is equal to the momentum.

將兩邊都除以 c，我們可以得出，E 大於 c 等於動量。

Now E is equal to hf, it's Planck's constant times the frequency.

現在，E 等於 hf，是普朗克常數乘以頻率。

And keep in mind the speed of light is equal to the wavelength of a photon times its frequency.

請記住，光速等於光子的波長乘以頻率。

So if we were to isolate lambda, it would be c over f.

是以，如果我們要將 lambda 分離出來，那就是 c 大於 f。

If we were to raise both sides to the negative one, we get that f over c is one over lambda.

如果我們把兩邊都提升到負一，就會得到 f over c 是 one over lambda。

So we can write this as h times f over c and then replace f over c with what we have here.

是以，我們可以將其寫成 h 乘以 f over c，然後用這裡的內容替換 f over c。

So we get that the momentum of a photon is Planck's constant divided by the wavelength.

是以，我們可以得出光子的動量是普朗克常數除以波長。

So whenever the wavelength of a photon changes, the momentum of that photon will change as well.

是以，只要光子的波長髮生變化，光子的動量也會隨之變化。

So to finish this problem, let's replace h with this value, 6.626 times 10 to the negative 34 joules times seconds.

是以，為了完成這個問題，讓我們用這個值來代替 h，即 6.626 乘以 10 的負 34 焦耳乘以秒。

And we're going to divide it by the wavelength.

So for part A, we have a 700 nanometer wavelength, or 700 times 10 to the negative 9 meters.

是以，對於 A 部分，我們的波長是 700 納米，即 700 乘以 10 到負 9 米。

And so the momentum is going to be 9.466 times 10 to the negative 28 kilograms times meters per second.

是以，動量將是每秒 9.466 乘以 10 的負 28 千克乘以米。

So that's the momentum of a photon of red light.

What is the momentum of a 350 nanometer photon of UV light?

350 納米紫外線光子的動量是多少？

So notice that the momentum has been reduced by a factor of 2.

所以請注意，動量已經減少了 2 倍。

What happens to the, I mean the wavelength was reduced by a factor of 2.

So what happens to the momentum of a photon if we reduce the wavelength?

那麼，如果我們減小波長，光子的動量會發生什麼變化呢？

Notice that it's on the bottom of the fraction.

Anytime you decrease the denominator of a fraction, the value of the whole fraction goes up.

只要減小分數的分母，整個分數的值就會增加。

So if you decrease the wavelength, the momentum is going to increase.

是以，如果減小波長，動量就會增加。

So let's use the same formula to calculate the new momentum.

是以，讓我們用同樣的公式來計算新動量。

So all we've got to do is change the wavelength from 700 to 350 nanometers.

是以，我們要做的就是把波長從 700 納米改為 350 納米。

Of course, we could simply double our first answer, and that will give us the same answer as well.

當然，我們可以簡單地將第一個答案加倍，這樣也能得到相同的答案。

So this is going to be 1.893 times 10 to the negative 27 kilograms times meters per second.

是以，這將是每秒 1.893 乘以 10 的負 27 千克乘以米。

So that's the momentum of the photon of UV light.

What is the effective mass of a 450 nanometer photon of blue light?

450 納米藍光光子的有效品質是多少？

Now we could use this equation to get that answer.

現在，我們可以用這個等式得到答案。

Momentum is equal to Planck's constant divided by the wavelength.

動量等於普朗克常數除以波長。

Now we know that momentum is mass times velocity.

現在我們知道，動量是品質乘以速度。

So when using this equation, this will be treated as effective mass times velocity.

是以，在使用該方程時，這將被視為有效品質乘以速度。

The velocity of a photon is going to be the speed of light.

And that's going to equal Planck's constant times the wavelength.

So the effective mass is going to be, if we divide both sides by c, is Planck's constant divided by the wavelength times the speed of light.

是以，如果我們把兩邊都除以 c，有效品質就是普朗克常數除以波長乘以光速。

So this is the formula that we could use to calculate the effective mass of a photon.

是以，我們可以用這個公式來計算光子的有效品質。

And notice that the only thing that can change here is the wavelength.

注意，這裡唯一可以改變的是波長。

Planck's constant and the speed of light, they're both constant values.

So the only thing that changes is the wavelength.

As the wavelength of the photon increases, the effective mass of that photon decreases.

隨著光子波長的增加，光子的有效品質也隨之減小。

Anytime the wavelength goes up, the frequency of the photon goes down.

只要波長上升，光子的頻率就會下降。

The energy that that photon carries goes down as well.

So we have Planck's constant, which is this number.

And then that's going to be divided by the wavelength, which is 450 nanometers, or 450 times 10 to the negative 9 meters.

然後再除以波長，波長是 450 納米，即 450 乘以 10 得到負 9 米。

And then times the speed of light, 3 times 10 to the 8 meters per second.

再乘以光速，3乘以10，就是每秒8米。

所以，讓我們繼續把它輸入進去。

So we get this value for the effective mass, 4.908.

是以，我們得出有效品質為 4.908。

我想我們可以把它四捨五入為 4.91。

 But that is the effective mass of a 450 nanometer photon of blue light.

但這只是 450 納米藍光光子的有效品質。