A cloud chamber: modern physics tracks charged particles from radioactive decays
At tiny scales, energy comes in discrete packets called quanta 量子. Light is carried by photons 光子, each with energy set by its frequency:
$$E=hf,$$
where $h$ is Planck's constant. Wave–particle duality 波粒二象性: light and matter each show both wave behavior (interference, diffraction) and particle behavior (photons, electrons as localized hits). A particle also has a matter wavelength$\lambda=\dfrac{h}{p}$, and particles genuinely produce interference - a variation of Young's double-slit experiment 双缝实验 done with electrons builds up the same fringe pattern as light.
Electrons form a diffraction pattern, showing that particles have a wave nature
Worked example. Find the energy of a photon of orange light with frequency $5.0\times10^{14}\ \text{Hz}$ ($h=6.63\times10^{-34}\ \text{J s}$): $E=hf=6.63\times10^{-34}\times5.0\times10^{14}=3.3\times10^{-19}\ \text{J}$, which is about $2.1\ \text{eV}$ (dividing by $1.6\times10^{-19}$). Visible-light photons carry a few electronvolts – just right to trigger the chemistry of vision and photosynthesis.
The Bohr model 玻尔模型 pictures electrons orbiting the nucleus only in certain allowed energy levels 能级. An electron can jump between levels only by absorbing or emitting a photon whose energy exactly matches the gap:
A blackbody 黑体 emits a continuous spectrum that depends only on its temperature. Hotter objects glow brighter and peak at shorter wavelengths (red-hot to white-hot to blue-hot). Explaining this spectrum required quantized energy – a founding problem of quantum theory.
A hotter black body radiates more, and its peak shifts to shorter wavelengths
Shining light on a metal can eject electrons – the photoelectric effect 光电效应. Key facts (which only the photon picture explains): electrons come out only if the photon's frequency exceeds a threshold, no matter how bright a dimmer, lower-frequency light is. Energy conservation gives
$$K_{\max}=hf-\phi,$$
where $\phi$ is the metal's work function 逸出功 (the energy to free an electron).
The maximum kinetic energy of photoelectrons rises linearly with frequency
Worked example. Light of frequency $8.0\times10^{14}\ \text{Hz}$ falls on a metal with work function $\phi=3.0\times10^{-19}\ \text{J}$. The most energetic electrons come off with
In Compton scattering 康普顿散射, a photon collides with an electron like two particles, transferring some energy and momentum. The scattered photon comes out with less energy (longer wavelength). This is direct evidence that photons carry momentum and behave as particles.
The nucleus 原子核 stores enormous energy. Fission 裂变 splits a heavy nucleus into lighter ones, releasing energy (nuclear reactors, bombs). Fusion 聚变 joins light nuclei into a heavier one (the Sun's power). In energy-releasing fission or fusion, the final products have slightly less total rest mass; the difference supplies energy according to $E=mc^2$. Not every possible fission or fusion reaction releases energy.
Binding energy per nucleon peaks near iron, so both fusion and fission can release energy
Worked example. Even a tiny mass converts to a huge energy. If a nuclear reaction loses $1.0\times10^{-3}\ \text{kg}$ of mass, it releases $E=mc^2=1.0\times10^{-3}\times(3.0\times10^{8})^2=9.0\times10^{13}\ \text{J}$ – roughly the energy of $20\,000$ tonnes of TNT.
A nuclear power station: fission of heavy nuclei releases the energy that drives the plant
The penetrating power of alpha, beta, and gamma radiation
Alpha decay 阿尔法衰变 ($\alpha$): a helium nucleus – mass number drops by 4.
Beta decay 贝塔衰变 ($\beta$) comes in two kinds. In beta-minus ($\beta^-$) a neutron becomes a proton, emitting an electron and an antineutrino 反中微子: $n\rightarrow p+e^-+\bar{\nu}$. In beta-plus ($\beta^+$), inside a nucleus with enough available decay energy, a proton becomes a neutron, emitting a positron 正电子 and a neutrino 中微子: $p\rightarrow n+e^++\nu$.
Gamma decay 伽马衰变 ($\gamma$): a high-energy photon – the nucleus sheds excess energy.
The neutrino and antineutrino (symbols $\nu$, $\bar{\nu}$) are tiny, chargeless, almost-massless particles that carry off energy and keep beta decay balanced. Decay is random, but a sample halves every half-life 半衰期: after each half-life, half of the remaining nuclei have decayed. Every decay equation conserves three quantities: nucleon number (mass number), charge, and lepton number 轻子数 – an electron or neutrino counts as $+1$ lepton, a positron or antineutrino as $-1$, and the totals must match on both sides.
Worked example. A radioactive sample has a half-life of $8.0$ days. What fraction is left after $24$ days? That is $24/8.0=3$ half-lives, so the fraction remaining is $\left(\tfrac12\right)^3=\tfrac18$ – about $12.5\%$. In an alpha decay of uranium-238 ($^{238}_{\ 92}\text{U}$), the daughter has mass number $238-4=234$ and atomic number $92-2=90$: thorium-234.
Answer each before reading the answers below. Take $h=6.63\times10^{-34}\ \text{J s}$, $c=3.0\times10^{8}\ \text{m/s}$ and $1\ \text{eV}=1.6\times10^{-19}\ \text{J}$.
Find the energy of a photon of blue light of wavelength $450\ \text{nm}$, in joules and in electronvolts.
That light falls on a metal with work function $2.3\ \text{eV}$. Find the maximum kinetic energy of the ejected electrons.
In hydrogen the $n=3$ level is at $-1.51\ \text{eV}$ and the $n=2$ level at $-3.40\ \text{eV}$. Find the energy and the wavelength of the photon emitted when an electron drops from $n=3$ to $n=2$.
The activity of a sample falls from $1600\ \text{Bq}$ to $100\ \text{Bq}$ in $20$ hours. Find its half-life.
A student says: "Brighter light ejects photoelectrons with more kinetic energy." Correct the student.
15.8
答案
English
$E=\dfrac{hc}{\lambda}=\dfrac{(6.63\times10^{-34})(3.0\times10^{8})}{450\times10^{-9}}=4.4\times10^{-19}\ \text{J}$, which is $\dfrac{4.42\times10^{-19}}{1.6\times10^{-19}}=2.8\ \text{eV}$.
$E=-1.51-(-3.40)=1.89\ \text{eV}=3.0\times10^{-19}\ \text{J}$. Then $\lambda=\dfrac{hc}{E}=\dfrac{1.99\times10^{-25}}{3.02\times10^{-19}}=6.6\times10^{-7}\ \text{m}$, about $660\ \text{nm}$: the red line of hydrogen.
$1600\to800\to400\to200\to100$ is 4 half-lives in $20\ \text{h}$, so the half-life is $5.0\ \text{h}$.
Brighter light at the same frequency means more photons per second, so more electrons come off each second, but each photon still carries $hf$. The maximum kinetic energy $hf-\phi$ depends only on the frequency. Raise the frequency, not the brightness, to get faster electrons.
15.8
考试技巧
English
Photon energy is $E=hf$; below the threshold frequency no electrons are emitted however bright the light (the photoelectric effect).
Use $K_{\max}=hf-\phi$ for the fastest photoelectrons ($\phi$ = work function).
Electrons jump between discrete energy levels by absorbing/emitting a photon whose energy equals the level gap — the source of line spectra.
In fission and fusion the small mass lost becomes energy via $E=mc^2$.
Balance decay equations by conserving mass number, charge, and lepton number (write the antineutrino in $\beta^-$, the neutrino in $\beta^+$), and halve the sample every half-life ($(\tfrac12)^n$ after $n$ half-lives).
Common mistakes
Thinking brighter light gives faster photoelectrons. Intensity changes the number of photons; each photon's energy, and so the maximum kinetic energy, depends only on frequency: $K_{\max} = hf - \phi$.
Expecting electrons below the threshold frequency if the light is bright enough. Below $f_0 = \phi/h$ no electrons are emitted, however intense the light.
Mixing joules and electronvolts in $E = hf$. Use $h = 6.63\times10^{-34}\ \text{J s}$ for joules or $h = 4.14\times10^{-15}\ \text{eV s}$ for electronvolts; $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$.
Saying an atom can absorb any photon. It absorbs only photons whose energy matches the gap between two of its energy levels.
Saying that after two half-lives nothing is left. One quarter of the original nuclei remains; after $n$ half-lives, $(\tfrac12)^n$ remains.
Leaving out the mass in $\Delta E = \Delta m\,c^2$ units: $\Delta m$ in kilograms and $c = 3.0\times10^8\ \text{m/s}$ give joules.
