Wave-particle duality · 波粒二象性
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| diffraction/dɪˈfrækʃn/ | 衍射 | yǎn shè |
| wave-particle duality/weɪv ˈpɑːtɪkl djuːˈælɪti/ | 波粒二象性 | bō lì èr xiàng xìng |
| de Broglie wavelength/də ˈbrəʊli ˈweɪvleŋθ/ | 德布罗意波长 | dé bù luó yì bō cháng |
| interference/ˌɪntəˈfɪərəns/ | 干涉 | gān shè |
| diffraction grating/dɪˈfrækʃn ˈɡreɪtɪŋ/ | 衍射光栅 | yǎn shè guāng shān |
| electron diffraction/ɪˈlektrɒn dɪˈfrækʃn/ | 电子衍射 | diàn zi yǎn shè |
| Planck constant/plæŋk ˈkɒnstənt/ | 普朗克常量 | pǔ lǎng kè cháng liàng |
| polycrystalline/ˌpɒlɪˈkrɪstəlaɪn/ | 多晶的 | duō jīng de |
Fire electrons at graphite and you get rings
- Electrons are particles. Everyone agrees: they have mass, they carry charge, they arrive one at a time.
- Fire a beam of them at a thin sheet of graphite and the screen shows concentric bright rings.
- Rings are a diffraction 衍射 pattern, and only waves diffract. Yet these are electrons.
- This lesson is that contradiction and its resolution: wave-particle duality 波粒二象性 and the de Broglie wavelength 德布罗意波长 $\lambda = h/p$.
把电子打到石墨上,得到的是一圈圈的环
- 电子是粒子。这一点大家都同意:它们有质量、带电荷、一个一个地到达。
- 把一束电子打到一片薄石墨上,屏幕上出现的却是同心的亮环。
- 环是衍射(diffraction)图样,而只有波才会衍射。可这些偏偏是电子。
- 这一课讲的就是这个矛盾和它的解决:波粒二象性(wave-particle duality)与德布罗意波长(de Broglie wavelength)$\lambda = h/p$。
The two-sided evidence
- Electromagnetic radiation, and matter, can show both wave properties such as interference and diffraction and particle properties such as the photoelectric effect. That is the two-mark answer.
- Asked for one piece of evidence each, for radiation: particle side, the photoelectric effect (a threshold frequency, and immediate emission even when dim); wave side, interference 干涉 or diffraction (Young's slits, a diffraction grating 衍射光栅).
- For matter, the wave evidence is electron diffraction 电子衍射. Name the phenomenon, not just the word "wave".
两面都有的证据
- *电磁辐射以及物质,既能表现出干涉、衍射这样的波的性质,又能表现出光电效应这样的粒子的性质。*这就是两分的答案。
- 若要求各举一个证据,对辐射来说:粒子一面是光电效应(有极限频率,而且即使很暗也立即发射);波一面是干涉(interference)或衍射(杨氏双缝、衍射光栅(diffraction grating))。
- 对物质,波的证据是电子衍射(electron diffraction)。要说出现象的名字,而不是只说"波"这个词。
Light shows its wave nature in interference, and its particle nature in: · 光在干涉中显示它的波性,在以下哪个现象中显示它的粒子性:
The photoelectric effect needs photons (particles); interference and diffraction need waves. · 光电效应需要光子(粒子);干涉和衍射需要波。
Match each nature to the evidence for it. · 把每一种属性与支持它的证据配对。
Name the phenomenon, not the category. "It behaves like a wave" is a restatement of the question, not evidence. · 要说出现象,而不是类别。"它表现得像波"是把问题重述一遍,不是证据。
The de Broglie hypothesis
- If a wave can act like particles, perhaps particles can act like waves. Every moving particle has a wavelength:
- The de Broglie wavelength is the wavelength associated with a moving particle, where $h$ is the Planck constant 普朗克常量 and $p$ the momentum of the particle. Name both symbols when asked.
- It is the same $\lambda = h/p$ that a photon obeys. One relation, two kinds of object.
德布罗意假说
- 如果波能表现得像粒子,也许粒子也能表现得像波。每个运动的粒子都有一个波长:
- __德布罗意波长__是与一个运动粒子相联系的波长,其中 $h$ 是普朗克常量(Planck constant),$p$ 是粒子的动量。被问到时两个符号都要说明。
- 这与光子遵守的 $\lambda = h/p$ 是同一个关系。一条关系,两类对象。
The de Broglie wavelength · 德布罗意波长
λ = h/p
A particle's wavelength is inversely proportional to its momentum — faster, heavier particles have shorter waves. · 粒子的波长与其动量成反比——更快、更重的粒子具有更短的波长。
The de Broglie wavelength of a particle is: · 一个粒子的德布罗意波长是:
$\lambda = \dfrac{h}{p}$ — wavelength is Planck's constant over momentum. · $\lambda = \dfrac{h}{p}$——波长是普朗克常数除以动量。
Why you never diffract in a doorway
- $h$ is tiny, so an everyday object has an absurdly small wavelength. A $0.10\ \text{kg}$ ball at $10\ \text{m/s}$ has
- That is far smaller than any slit or lattice spacing, and diffraction only shows when the gap is comparable to the wavelength.
- An electron is different: at $4.9\times10^{7}\ \text{m/s}$ its momentum is $4.46\times10^{-23}\ \text{kg m/s}$, giving $\lambda = 1.5\times10^{-11}\ \text{m}$, about $0.015\ \text{nm}$. That is the size of the gaps between atoms in a crystal, which is exactly why a crystal can diffract it.
你走过门口为什么不会衍射
- $h$ 极小,所以日常物体的波长小得荒唐。$0.10\ \text{kg}$ 的球以 $10\ \text{m/s}$ 运动:
- 这远小于任何狭缝或晶格间距,而衍射只有在缝宽与波长可比时才显现。
- 电子则不同:在 $4.9\times10^{7}\ \text{m/s}$ 时动量为 $4.46\times10^{-23}\ \text{kg m/s}$,给出 $\lambda = 1.5\times10^{-11}\ \text{m}$,约 $0.015\ \text{nm}$。这正是晶体中原子之间空隙的尺度,晶体因此能让它衍射。
A 0.10 kg ball moves at 10 m/s. What is its de Broglie wavelength, in units of 1e-34 m? · 一个 0.10 kg 的球以 10 m/s 运动。它的德布罗意波长是多少(以 1e-34 m 为单位)?
p = 1.0 kg m/s, so lambda = 6.63e-34 m. That is far smaller than any slit or lattice spacing, which is why everyday objects never diffract. · p = 1.0 kg m/s,所以 lambda = 6.63e-34 m。这远小于任何狭缝或晶格间距,日常物体因此从不衍射。
Electron diffraction, described
- The four-mark description runs in four moves, and each is a mark.
- One. Electrons from a heated filament are accelerated through a high p.d. into a beam.
- Two. The beam passes through a thin polycrystalline 多晶的 graphite film, whose atomic spacing of about $10^{-10}\ \text{m}$ acts as a diffraction grating.
- Three. On a fluorescent screen they produce a bright central spot surrounded by concentric rings.
- Four. Rings are a diffraction pattern, diffraction is a wave property, so the electrons behave as waves, and the ring radii match $\lambda = h/p$.
Rings, not spots
描述电子衍射
- 四分的描述分四步走,每一步一分。
- **一。**由热灯丝发出的电子经高电压加速成束。
- 二。电子束穿过一片薄的多晶的(polycrystalline)石墨膜,其约 $10^{-10}\ \text{m}$ 的原子间距起到衍射光栅的作用。
- 三。在荧光屏上产生一个明亮的中心斑,周围环绕着同心圆环。
- **四。**环是衍射图样,衍射是波的性质,所以电子表现为波,而环的半径与 $\lambda = h/p$ 相符。
是环,不是斑点
Electron diffraction shows that electrons can behave as waves. · 电子衍射表明电子可以表现为波。
Only waves diffract, yet a beam of electrons makes a diffraction pattern — so electrons have a wave nature. · 只有波会衍射,但一束电子却产生衍射图案——所以电子有波性。
Speeding up the electrons makes the diffraction rings: · 加快电子的速度使衍射环:
Faster electrons → shorter $\lambda$ → less diffraction → the rings close in. · 更快的电子 → 更短的 $\lambda$ → 衍射更少 → 环靠拢。
Put the description of the electron-diffraction experiment in order. · 把电子衍射实验的描述按顺序排列。
Each move is a mark in the four-marker. The last one is the conclusion, and it is the one most often left out. · 每一步在四分题里都是一分。最后一条是结论,而它是最常被漏掉的。
Worked example: accelerating an electron
- An electron is accelerated from rest through $2500\ \text{V}$. Find its de Broglie wavelength.
- Kinetic energy gained: $E_K = eV$, and $p = \sqrt{2mE_K} = \sqrt{2m_e eV}$.
- So $\lambda = \dfrac{h}{\sqrt{2m_e eV}} = \dfrac{6.63\times10^{-34}}{\sqrt{2(9.11\times10^{-31})(1.6\times10^{-19})(2500)}} = 2.5\times10^{-11}\ \text{m}$.
- That is close to the atomic spacing in a crystal, which is why the electrons diffract off graphite at all.
- Learn the chain $qV = \tfrac{1}{2}mv^2 \Rightarrow p = \sqrt{2mqV} \Rightarrow \lambda = h/\sqrt{2mqV}$. It is asked for directly.
例题:加速一个电子
- 电子从静止经 $2500\ \text{V}$ 加速。求它的德布罗意波长。
- 获得的动能:$E_K = eV$,而 $p = \sqrt{2mE_K} = \sqrt{2m_e eV}$。
- 所以 $\lambda = \dfrac{h}{\sqrt{2m_e eV}} = \dfrac{6.63\times10^{-34}}{\sqrt{2(9.11\times10^{-31})(1.6\times10^{-19})(2500)}} = 2.5\times10^{-11}\ \text{m}$。
- 这接近晶体中的原子间距,正因如此电子才会在石墨上衍射。
- 要背下这条链:$qV = \tfrac{1}{2}mv^2 \Rightarrow p = \sqrt{2mqV} \Rightarrow \lambda = h/\sqrt{2mqV}$。它会被直接考到。
An electron is accelerated from rest through 2500 V. What is its de Broglie wavelength, in units of 1e-11 m? · 电子从静止经 2500 V 加速。它的德布罗意波长是多少(以 1e-11 m 为单位)?
p = sqrt(2 m e V) = 2.7e-23 kg m/s, so lambda = h/p = 2.5e-11 m, close to the atomic spacing in a crystal. Use momentum, never speed, in h/p. · p = sqrt(2 m e V) = 2.7e-23 kg m/s,所以 lambda = h/p = 2.5e-11 m,接近晶体中的原子间距。h/p 里用动量,绝不用速率。
What changing the speed does
- Faster electrons have more momentum, so a shorter wavelength, so less diffraction: the rings move closer together.
- Slower electrons spread the rings apart.
- Because $\lambda \propto 1/\sqrt{V}$, halving the wavelength needs four times the accelerating p.d.
- This is the standard follow-up question, and the sign of the effect is what it tests. More volts, smaller rings.
改变速率会怎样
- 更快的电子动量更大,于是波长更短,于是衍射更少:环靠得更近。
- 更慢的电子把环撑得更开。
- 因为 $\lambda \propto 1/\sqrt{V}$,要把波长减半需要四倍的加速电压。
- 这是标准的追问,考的正是效果的方向。电压越高,环越小。
A faster particle (more momentum) has a ____ de Broglie wavelength. · 更快的粒子(动量更大)有 ____ 的德布罗意波长。
$\lambda = \dfrac{h}{p}$, so larger $p$ gives a smaller $\lambda$. · $\lambda = \dfrac{h}{p}$,所以更大的 $p$ 给出更小的 $\lambda$。
The accelerating p.d. in an electron-diffraction tube is increased. Which happen? Select all · 所有 that apply. · 电子衍射管的加速电压被提高。会发生什么?选出所有适用的。
More volts, more momentum, shorter wavelength, less diffraction, smaller rings. Because lambda goes as 1/sqrt(V), halving the wavelength takes four times the p.d. · 电压越高,动量越大,波长越短,衍射越少,环越小。因为 lambda 正比于 1/sqrt(V),要把波长减半需要四倍电压。
Marks that slip away
- Name the phenomenon as evidence, not the category: "electron diffraction", not "it behaves like a wave".
- Use momentum, not speed, in $\lambda = h/p$. For an accelerated electron go through $p = \sqrt{2mqV}$.
- Sketch the pattern as rings, never as spots or as a straight-line fringe pattern.
- A higher accelerating voltage gives smaller rings. Getting this backwards is the commonest error here.
- Everyday objects have wavelengths around $10^{-34}\ \text{m}$, which is why duality is invisible outside the laboratory.
容易丢掉的分
- 举证据要说出现象,不是类别:说"电子衍射",不要只说"它表现得像波"。
- $\lambda = h/p$ 里用的是动量,不是速率。对加速过的电子要走 $p = \sqrt{2mqV}$ 这一步。
- 图样要画成环,绝不能画成斑点或直线条纹。
- 加速电压更高给出更小的环。把这个方向弄反是这里最常见的错误。
- 日常物体的波长约为 $10^{-34}\ \text{m}$,这就是二象性在实验室之外看不见的原因。
You've got it
- wave-particle duality: radiation and matter both show wave behaviour (interference, diffraction) and particle behaviour (the photoelectric effect)
- the de Broglie wavelength of a moving particle is $\lambda = h/p$, with $h$ the Planck constant and $p$ the momentum
- electron diffraction through polycrystalline graphite gives concentric rings, which only a wave could produce
- for an electron accelerated through $V$, $\lambda = h/\sqrt{2m_e eV}$, so a larger p.d. gives a shorter wavelength and smaller rings
你掌握了
- 波粒二象性:辐射和物质都既表现出波的行为(干涉、衍射),又表现出粒子的行为(光电效应)
- 运动粒子的德布罗意波长是 $\lambda = h/p$,$h$ 是普朗克常量,$p$ 是动量
- 穿过多晶石墨的电子衍射给出同心圆环,而只有波才能产生它
- 对经 $V$ 加速的电子,$\lambda = h/\sqrt{2m_e eV}$,所以电压越大,波长越短、环越小