---
title: "The Quantum World: Photons and Energy Levels"
book: "High School Physics"
subject: physics
language: en
chapter: 34
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/34-the-quantum-world-photons-and-energy-levels
---

# Chapter 34 — The Quantum World: Photons and Energy Levels

An [infrared](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) heat lamp can bathe your skin all evening and never mark it; ten minutes of [thin](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) mountain sunlight leaves a burn. The lamp delivers far more [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) — but [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) is not what marks skin. Light, this chapter shows, arrives in grains, and only a grain that hits hard enough does chemistry: one [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) grain snaps a bond that a billion [infrared](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) grains only warm. Follow the grains and a century opens: metals leaking electrons, [atoms](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) singing in [spectral lines](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-emission), electrons interfering like [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave).

## 34.1 Light arrives in grains: the photon

**Definition 34.1 (The photon).**

Light of [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) $\nu$ exchanges [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) with matter only in whole grains called *photons*, each of [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy)

$$
E = h\nu = \frac{hc}{\lambda}, \qquad h = 6.63 \times 10^{-34}\,\mathrm{J}\,\mathrm{s},
$$

where $h$ is *Planck’s constant* (Planck, 1900; Einstein, 1905) and $\lambda$ the [wavelength in vacuum](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#def-g12-light-as-wave-lightwave). A beam’s intensity measures the *number* of photons per second; each grain’s [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) is fixed by the [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) alone.

**Notation 34.2 (The electronvolt).**

Atomic physics uses the *electronvolt*, $1\,\mathrm{eV} = 1.60 \times 10^{-19}\,\mathrm{J}$: the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) one [elementary charge](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-elementary) gains crossing $1\,\mathrm{V}$ ([Chapter 14](https://one-course.com/books/physics/2/en/chapter/14-electric-and-gravitational-fields#ch-g11-electric-gravitational-fields)). Handy: $E \,(\mathrm{eV}) \approx 1240/\lambda\,(\mathrm{nm})$.

![One scale for the electromagnetic family (): wavelength (top), energy per photon hc/ (blue), from radio grains of a microelectronvolt to the MeV gamma grains of — yet a 1.0\, mW pointer pours 3 × 1015 grains per second: everyday light flows as smoothly as poured sand.](https://one-course.com/images/onecourse/chapters/physics-2/g12-quantum-world/fig-b099902e5a47.svg)

*One scale for the electromagnetic family ([Chapter 22](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#ch-g12-light-as-wave)): [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) (top), [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) per [photon](#def-g12-quantum-world-photon) $hc/\lambda$ (blue), from radio grains of a microelectronvolt to the $\mathrm{MeV}$ gamma grains of [Chapter 19](https://one-course.com/books/physics/2/en/chapter/19-the-nucleus-and-radioactivity#ch-g11-nucleus-radioactivity) — yet a $1.0\,\mathrm{mW}$ pointer pours $3 \times 10^{15}$ grains per second: everyday light flows as smoothly as poured sand.*

## 34.2 The photoelectric effect

Shine light on a clean metal plate in vacuum and electrons pop out — the *photoelectric effect*. Its rules broke classical physics.

**Proposition 34.3 (What the experiment shows).**

For each metal there is a *threshold frequency* $\nu_0$:

1. below $\nu_0$ , no electron leaves, however intense the light;
2. above $\nu_0$ , emission starts without [delay](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-delay) , however faint;
3. intensity raises the *number* of electrons per second, not their maximum [kinetic energy](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#def-g11-mechanical-energy-kinetic) ; [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) raises $E_{k,\max}$ .

None of it fits a continuous [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), which any color should let charge up — over months in faint light ([Exercise 34.14](#exo-g12-quantum-world-14)) — and whose intensity should set the electrons’ [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy): all three predictions fail.

**Proof.** *Admitted at this level.* ∎

**Theorem 34.4 (Einstein’s photoelectric relation).**

An electron needs at least the *work function* $W_0$ of the metal to escape its surface. One [photon](#def-g12-quantum-world-photon) is absorbed whole by one electron, so

$$
h\nu = W_0 + E_{k,\max}, \qquad \text{hence } \nu_0 = \frac{W_0}{h}.
$$

**Proof.** [Energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) bookkeeping: $h\nu$ in, $W_0$ spent on escape, the rest kinetic — at most $h\nu - W_0$, less for electrons starting deeper. Below $\nu_0$ one grain cannot pay the exit toll, whatever their number. ∎

**Definition 34.5 (Stopping potential).**

To measure $E_{k,\max}$, a reverse [voltage](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-voltage) brakes the electrons ([Chapter 14](https://one-course.com/books/physics/2/en/chapter/14-electric-and-gravitational-fields#ch-g11-electric-gravitational-fields)); the *stopping potential* $U_s$ is the smallest [voltage](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-voltage) canceling the current: the fastest electrons just fail to cross, so $E_{k,\max} = e\,U_s$.

![Measured E_k, = eU_s against frequency (sodium): a straight line of slope h, cutting the axis at _0; extrapolated (dashes), it reveals -W_0 at = 0.](https://one-course.com/images/onecourse/chapters/physics-2/g12-quantum-world/fig-5c7a29c03b62.svg)

*Measured $E_{k,\max} = eU_s$ against [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) (sodium): a straight line of slope $h$, cutting the axis at $\nu_0$; extrapolated (dashes), it reveals $-W_0$ at $\nu = 0$.*

**Method 34.6 (Measuring Planck’s constant).**

1. Illuminate a photocell through [filters](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-subtractive) of known [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) ; compute each $\nu = c/\lambda$ ; record each [stopping potential](#def-g12-quantum-world-stopping) .
2. Plot $E_{k,\max} = eU_s$ against $\nu$ : Einstein promises the straight line $h\nu - W_0$ .
3. Read the slope: it *is* $h$ ; the intercepts give $\nu_0$ and $-W_0$ .

## 34.3 Energy levels: why atoms emit lines

**Definition 34.7 (Energy levels).**

The [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) of an electron bound in an [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) is *quantized*: restricted to a discrete ladder of *energy levels* $E_1 < E_2 < \dots < 0$, counted negative from the free electron. The lowest rung is the *ground state*, the others *excited states*; reaching $0$ is *ionization*.

**Proposition 34.8 (The hydrogen ladder).**

The [energy levels](#def-g12-quantum-world-levels) of the hydrogen [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) are

$$
E_n = -\frac{13.6\,\mathrm{eV}}{n^2}, \qquad n = 1, 2, 3, \dots
$$

so $E_1 = -13.6\,\mathrm{eV}$, $E_2 = -3.40\,\mathrm{eV}$, $E_3 =
-1.51\,\mathrm{eV}$, $E_4 = -0.85\,\mathrm{eV}$, crowding toward $0$.

**Proof.** *Admitted at this level.* ∎

**Remark 34.9 (Where the ladder comes from).**

Why these numbers is quantum mechanics, kept for the university volumes — as is de Broglie’s formula: the electron’s [matter wave](#def-g12-quantum-world-debroglie) ([Definition 34.15](#def-g12-quantum-world-debroglie) below) must close on itself around the [nucleus](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder), and only certain [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) fit — like a guitar string, an [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) has a discrete set of notes.

**Theorem 34.10 (Photon of a transition).**

An [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) changes level only by emitting or absorbing one [photon](#def-g12-quantum-world-photon) carrying exactly the difference: dropping from level $p$ to a lower level $n$ emits $h\nu = E_p - E_n$; climbing from $n$ to $p$ absorbs that same [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy).

**Proof.** [Energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) conservation, one grain at a time: the [photon](#def-g12-quantum-world-photon) is the only currency light accepts, and the [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder)’s books must balance. ∎

**Remark 34.11 (Line spectra explained).**

Here is the answer promised by [Chapter 2](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#ch-g10-light-spectra): a hot gas emits only the frequencies $(E_p - E_n)/h$ of its own ladder — a barcode of bright lines — and a cool gas steals exactly those frequencies from [white light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white), printing matching dark absorption lines. Each element has its own ladder, hence its own barcode — how we read starlight.

**Example 34.12 (The visible lines of hydrogen).**

Transitions ending on $n = 2$ fall in the visible. With $E \,(\mathrm{eV}) =
1240/\lambda\,(\mathrm{nm})$: $3 \to 2$: $1.89\,\mathrm{eV}$, $656\,\mathrm{nm}$ (red); $4 \to 2$: $2.55\,\mathrm{eV}$, $486\,\mathrm{nm}$ (blue-green); $5 \to 2$: $2.86\,\mathrm{eV}$, $434\,\mathrm{nm}$ (violet). Transitions to $n = 1$ exceed $10\,\mathrm{eV}$: all [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength).

![The hydrogen ladder, to scale: the visible lines all end on n = 2; the drop to the ground state is deep ultraviolet.](https://one-course.com/images/onecourse/chapters/physics-2/g12-quantum-world/fig-92bede013813.svg)

*The hydrogen ladder, to scale: the visible lines all end on $n = 2$; the drop to the [ground state](#def-g12-quantum-world-levels) is deep [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength).*

**Remark 34.13 (Lasers, in one breath).**

A [photon](#def-g12-quantum-world-photon) of exactly $E_p - E_n$ passing an *excited* [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) can trigger it to emit an identical [photon](#def-g12-quantum-world-photon) — same [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy), same direction, in step. Keep more [atoms](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) up than down and one [photon](#def-g12-quantum-world-photon) becomes an avalanche of clones: *stimulated emission* — and a [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) is little more.

## 34.4 Matter waves

**Proposition 34.14 (Electrons interfere).**

Send electrons one at a time through a double slit ([Chapter 22](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#ch-g12-light-as-wave)): each lands as a single sharp dot — a particle. But as thousands accumulate, the dots draw [interference fringes](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#def-g12-light-as-wave-fringes): each electron, alone, passes as a [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) through *both* slits. Crystals diffract electron beams too (Davisson and Germer, 1927).

**Proof.** *Admitted at this level.* ∎

![One electron, one dot; thousands, fringes. Where the next dot lands is random — like one nucleus’s decay () — but the wave fixes the probability: bright fringe, likely; dark, nearly never. Individual chance ruled by an exact wave: the heart of quantum physics.](https://one-course.com/images/onecourse/chapters/physics-2/g12-quantum-world/fig-4637721e37dc.svg)

*One electron, one dot; thousands, fringes. Where the next dot lands is random — like one [nucleus](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder)’s decay ([Chapter 19](https://one-course.com/books/physics/2/en/chapter/19-the-nucleus-and-radioactivity#ch-g11-nucleus-radioactivity)) — but the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) fixes the *probability*: bright fringe, likely; dark, nearly never. Individual chance ruled by an exact [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave): the heart of quantum physics.*

**Definition 34.15 (De Broglie wavelength).**

Every particle of [momentum](https://one-course.com/books/physics/2/en/chapter/25-newtons-laws#def-g12-newtons-laws-momentum) $p = mv$ travels as a *matter wave* of *de Broglie wavelength*

$$
\lambda = \frac{h}{p} = \frac{h}{mv}.
$$

**Proof.** *Admitted at this level.* ∎

**Example 34.16 (Why you never diffract).**

An electron accelerated through $100\,\mathrm{V}$ has $E_k = 1.60 \times 10^{-17}\,\mathrm{J}$, $p = \sqrt{2m_e E_k} = 5.4 \times 10^{-24}\,\mathrm{kg}\,\mathrm{m}/\mathrm{s}$, so $\lambda \approx
0.12\,\mathrm{nm}$ — the spacing of [atoms](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) in a crystal, which therefore acts as its double slit. A $58\,\mathrm{g}$ tennis ball at $25\,\mathrm{m}/\mathrm{s}$: $\lambda =
6.63\times10^{-34}/1.45 \approx 4.6 \times 10^{-34}\,\mathrm{m}$, some $10^{19}$ times smaller than a [nucleus](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) — no slit will ever show it fringing. $h$ is so small that quantum graininess surfaces only at the [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder)’s scale.

**Example 34.17 (The electron microscope).**

[Diffraction](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#def-g12-light-as-wave-diffraction) forbids resolving detail much smaller than the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) used ([Chapter 22](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#ch-g12-light-as-wave)): visible light stops near $0.5\,\text{µ}\mathrm{m}$. Electrons at $10\,\mathrm{kV}$ have $\lambda \approx 12\,\mathrm{pm}$ — tens of thousands of times shorter: electron microscopes image viruses, cell machinery, even single [atoms](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder). The [matter wave](#def-g12-quantum-world-debroglie) handed physics a new eye.

## 34.5 Exercises

**Exercise 34.1 ★.**

A green [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) pointer emits $1.0\,\mathrm{mW}$ at $530\,\mathrm{nm}$. Compute one [photon](#def-g12-quantum-world-photon)’s [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) in [joules](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-work) and electronvolts, then the [photons](#def-g12-quantum-world-photon) per second.

**Solution of Exercise 34.1.**

$E = hc/\lambda = 1.99 \times 10^{-25}/5.30 \times 10^{-7} = 3.75 \times 10^{-19}\,\mathrm{J} =
2.35\,\mathrm{eV}$. $N = 1.0 \times 10^{-3}/3.75 \times 10^{-19} \approx 2.7 \times 10^{15}$ [photons](#def-g12-quantum-world-photon) per second.

**Exercise 34.2 ★.**

An FM station broadcasts at $100\,\mathrm{MHz}$. Compute its [photon](#def-g12-quantum-world-photon) [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) in $\mathrm{eV}$ and compare it to a visible [photon](#def-g12-quantum-world-photon): why is radio graininess undetectable?

**Solution of Exercise 34.2.**

$E = h\nu = 6.63\times10^{-34} \times 10^{8} = 6.63 \times 10^{-26}\,\mathrm{J}
\approx 4.1 \times 10^{-7}\,\mathrm{eV}$ — about five million times less than a visible [photon](#def-g12-quantum-world-photon). Any detectable radio [signal](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-signal) involves so many [photons](#def-g12-quantum-world-photon) per second that the flow is perfectly smooth.

**Exercise 34.3 ★.**

Cesium has $W_0 = 1.9\,\mathrm{eV}$. Compute its [threshold frequency](#prop-g12-quantum-world-facts) and [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength). Does a red [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) at $633\,\mathrm{nm}$ eject electrons from cesium — and if so, with what maximum [kinetic energy](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#def-g11-mechanical-energy-kinetic)?

**Solution of Exercise 34.3.**

$\nu_0 = W_0/h = 3.04 \times 10^{-19}/6.63 \times 10^{-34} \approx
4.6 \times 10^{14}\,\mathrm{Hz}$; $\lambda_0 = c/\nu_0 \approx 654\,\mathrm{nm}$. At $633\,\mathrm{nm}$, $E = 1240/633 = 1.96\,\mathrm{eV} > 1.9\,\mathrm{eV}$: yes, barely, with $E_{k,\max} \approx 0.06\,\mathrm{eV} \approx 1 \times 10^{-20}\,\mathrm{J}$.

**Exercise 34.4 ★.**

Using $E_n = -13.6/n^2$ $\mathrm{eV}$, compute the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) and [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) of the [photon](#def-g12-quantum-world-photon) emitted in the hydrogen transition $3 \to 2$. Its color?

**Solution of Exercise 34.4.**

$E_3 - E_2 = -1.51 + 3.40 = 1.89\,\mathrm{eV}$; $\lambda = 1240/1.89 \approx 656\,\mathrm{nm}$: the red line.

**Exercise 34.5 ★.**

From the $E_{k,\max}$-versus-$\nu$ graph of the course, read off sodium’s [threshold frequency](#prop-g12-quantum-world-facts) and [work function](#thm-g12-quantum-world-einstein) (in $\mathrm{eV}$); what constant is the slope?

**Solution of Exercise 34.5.**

$\nu_0 \approx 5.6 \times 10^{14}\,\mathrm{Hz}$; $W_0 = h\nu_0 \approx
3.7 \times 10^{-19}\,\mathrm{J} \approx 2.3\,\mathrm{eV}$ (the $-W_0$ intercept). The slope is [Planck’s constant](#def-g12-quantum-world-photon) $h$.

**Exercise 34.6 ★★.**

Light of [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $400\,\mathrm{nm}$ falls on potassium ($W_0 = 2.3\,\mathrm{eV}$). Compute the [photon](#def-g12-quantum-world-photon) [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy), the maximum [kinetic energy](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#def-g11-mechanical-energy-kinetic) of the electrons (in $\mathrm{eV}$ and [joules](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-work)), and the [stopping potential](#def-g12-quantum-world-stopping).

**Solution of Exercise 34.6.**

$E = 1240/400 = 3.10\,\mathrm{eV}$; $E_{k,\max} = 3.10 - 2.3 =
0.80\,\mathrm{eV} = 1.3 \times 10^{-19}\,\mathrm{J}$; $U_s = E_{k,\max}/e =
0.80\,\mathrm{V}$.

**Exercise 34.7 ★★.**

A photocell is lit above threshold. What happens to (a) the current, (b) the [stopping potential](#def-g12-quantum-world-stopping), when the intensity is doubled at fixed [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator)? When the [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) is raised? Which fact contradicts the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) picture, and why?

**Solution of Exercise 34.7.**

Intensity doubled: current doubles (twice the [photons](#def-g12-quantum-world-photon)), $U_s$ unchanged (same [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) per [photon](#def-g12-quantum-world-photon)). [Frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) raised: $U_s$ rises, $E_{k,\max} = h\nu - W_0$. The [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) picture ties the electrons’ [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) to the intensity, not the [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) — exactly what is *not* observed.

**Exercise 34.8 ★★.**

What minimum [photon](#def-g12-quantum-world-photon) [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) ionizes a hydrogen [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) from its [ground state](#def-g12-quantum-world-levels)? From $n = 2$? Compute both threshold [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) and name their spectral domains.

**Solution of Exercise 34.8.**

From $n = 1$: $13.6\,\mathrm{eV}$, $\lambda = 1240/13.6 \approx
91\,\mathrm{nm}$, far [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength). From $n = 2$: $3.40\,\mathrm{eV}$, $\lambda \approx 365\,\mathrm{nm}$, near [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength).

**Exercise 34.9 ★★.**

[White light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white) crosses a jar of cool hydrogen (all [atoms](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) in the [ground state](#def-g12-quantum-world-levels)). Explain why the transmitted [spectrum](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-timbre) shows *no* dark line in the visible, yet Balmer absorption lines appear in the spectra of hot stars.

**Solution of Exercise 34.9.**

From the [ground state](#def-g12-quantum-world-levels) the smallest jump costs $E_2 - E_1 =
10.2\,\mathrm{eV}$ ($122\,\mathrm{nm}$): every absorption line is [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength), none visible. In a hot stellar atmosphere collisions keep some [atoms](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) in $n = 2$, from which visible Balmer [photons](#def-g12-quantum-world-photon) ($1.89\text{ to }3.40\,\mathrm{eV}$) can be absorbed.

**Exercise 34.10 ★★.**

Compute the [de Broglie wavelength](#def-g12-quantum-world-debroglie) of (a) an electron accelerated through $100\,\mathrm{V}$, (b) a $0.10\,\mathrm{g}$ fly at $1.0\,\mathrm{m}/\mathrm{s}$. Compare each to the size of an [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) ($\sim 10^{-10}\,\mathrm{m}$); conclude.

**Solution of Exercise 34.10.**

(a) $E_k = 1.6 \times 10^{-17}\,\mathrm{J}$, $p = \sqrt{2m_e E_k} =
5.4 \times 10^{-24}\,\mathrm{kg}\,\mathrm{m}/\mathrm{s}$, $\lambda = h/p \approx 1.2 \times 10^{-10}\,\mathrm{m}$ — atom-sized: crystals diffract it. (b) $p = 1.0 \times 10^{-4} \times 1.0 =
1.0 \times 10^{-4}\,\mathrm{kg}\,\mathrm{m}/\mathrm{s}$, $\lambda \approx 6.6 \times 10^{-30}\,\mathrm{m}$ — $10^{20}$ times smaller than an [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder): the fly’s [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) is forever invisible.

**Exercise 34.11 ★★.**

A dark-adapted eye responds to about $90$ [photons](#def-g12-quantum-world-photon) of $505\,\mathrm{nm}$ light. Compute that [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy), compare it to the [kinetic energy](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#def-g11-mechanical-energy-kinetic) of a $2.5\,\mathrm{mg}$ mosquito flying at $0.50\,\mathrm{m}/\mathrm{s}$, and interpret the ratio.

**Solution of Exercise 34.11.**

One [photon](#def-g12-quantum-world-photon): $1.99 \times 10^{-25}/5.05 \times 10^{-7} = 3.94 \times 10^{-19}\,\mathrm{J}$; ninety: $\approx 3.5 \times 10^{-17}\,\mathrm{J}$. Mosquito: $\tfrac12 \times 2.5 \times 10^{-6}
\times 0.50^2 = 3.1 \times 10^{-7}\,\mathrm{J}$ — $10^{10}$ times more. The eye works within a hundred grains of the absolute limit: almost a [photon](#def-g12-quantum-world-photon) counter.

**Exercise 34.12 ★★★.**

A photocell gives $U_s = 1.19\,\mathrm{V}$ at $405\,\mathrm{nm}$ and $U_s =
0.40\,\mathrm{V}$ at $546\,\mathrm{nm}$. From these two points measure $h$, then the [work function](#thm-g12-quantum-world-einstein) in $\mathrm{eV}$. Which metal of this chapter is the cathode?

**Solution of Exercise 34.12.**

$\nu_1 = 7.41 \times 10^{14}\,\mathrm{Hz}$, $\nu_2 = 5.49 \times 10^{14}\,\mathrm{Hz}$; $h = e\,\Delta U_s/\Delta\nu = 1.60 \times 10^{-19} \times 0.79 /
1.91 \times 10^{14} \approx 6.6 \times 10^{-34}\,\mathrm{J}\,\mathrm{s}$. $W_0 = h\nu_2 - eU_{s,2} =
3.63 \times 10^{-19} - 0.64 \times 10^{-19} = 3.0 \times 10^{-19}\,\mathrm{J} \approx
1.9\,\mathrm{eV}$: cesium.

**Exercise 34.13 ★★★.**

A hydrogen lamp shows lines at $486\,\mathrm{nm}$ and $434\,\mathrm{nm}$. Convert each to a [photon](#def-g12-quantum-world-photon) [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) and identify the two transitions by matching differences of the levels $E_n = -13.6/n^2$ $\mathrm{eV}$.

**Solution of Exercise 34.13.**

$1240/486 = 2.55\,\mathrm{eV} = E_4 - E_2 = 3.40 - 0.85$: transition $4 \to 2$. $1240/434 = 2.86\,\mathrm{eV} = E_5 - E_2 = 3.40 - 0.54$: transition $5 \to 2$.

**Exercise 34.14 ★★★.**

Classical [delay](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-delay) estimate: faint light of intensity $1.0 \times 10^{-6}\,\mathrm{W}/\mathrm{m}^{2}$ falls on potassium ($W_0 = 2.3\,\mathrm{eV}$). If an [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) collected only the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) crossing its cross-section (radius $1.0 \times 10^{-10}\,\mathrm{m}$), how long would it need to gather $W_0$? Compare with the observed [delay](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-delay) (under $10^{-9}\,\mathrm{s}$).

**Solution of Exercise 34.14.**

Cross-section $\pi r^2 \approx 3.1 \times 10^{-20}\,\mathrm{m}^{2}$, collected [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) $3.1 \times 10^{-26}\,\mathrm{W}$; $t = 3.7 \times 10^{-19}/3.1 \times 10^{-26} \approx
1.2 \times 10^{7}\,\mathrm{s}$ — four months, against under a nanosecond observed. The [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) cannot be spread over the [wavefront](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavefront): it travels concentrated, in grains.

**Exercise 34.15 ★★★.**

In an electron microscope the electrons are accelerated through $5.0\,\mathrm{kV}$. Compute their [de Broglie wavelength](#def-g12-quantum-world-debroglie), compare it to green light ($550\,\mathrm{nm}$), and explain via [diffraction](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#def-g12-light-as-wave-diffraction) ([Chapter 22](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#ch-g12-light-as-wave)) why this buys resolution.

**Solution of Exercise 34.15.**

$E_k = 8.0 \times 10^{-16}\,\mathrm{J}$, $p = \sqrt{2m_e E_k} =
3.8 \times 10^{-23}\,\mathrm{kg}\,\mathrm{m}/\mathrm{s}$, $\lambda = h/p \approx 1.7 \times 10^{-11}\,\mathrm{m} =
17\,\mathrm{pm}$ — about $3 \times 10^{4}$ times shorter than $550\,\mathrm{nm}$. [Diffraction](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#def-g12-light-as-wave-diffraction) blurs detail smaller than $\sim\lambda$, so shrinking $\lambda$ by $10^4$ buys $10^4$ in resolution.

## 34.6 Problem: The Solar-Panel Physics Lab

**Problem 34.1.**

Weekend problem — the solar-panel physics lab: the $10^{21}$ grains a rooftop drinks each second, a photocell that measures [Planck’s constant](#def-g12-quantum-world-photon), a hydrogen lamp read line by line — one weekend, one $h$

Your class instruments the school’s rooftop solar panel for a weekend. Data: solar irradiance at noon $1.0 \times 10^{3}\,\mathrm{W}/\mathrm{m}^{2}$; mean solar [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $550\,\mathrm{nm}$; panel area $1.0\,\mathrm{m}^{2}$, [efficiency](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-efficiency) $20\%$; the day delivers the equivalent of $5.0\,\mathrm{h}$ of full sun.

**Part I — [Photon](#def-g12-quantum-world-photon) bookkeeping.**

1. Compute the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) of one mean solar [photon](#def-g12-quantum-world-photon) , in [joules](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-work) and electronvolts.
2. Deduce the number of [photons](#def-g12-quantum-world-photon) striking the panel per second at noon.
3. Compute the panel’s [electrical power](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#prop-g11-circuits-and-power-power) at noon, then the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) produced over the day, in $\mathrm{kW}\,\mathrm{h}$ .
4. The panel’s current looks perfectly smooth. Using question 2, explain why the grain-by-grain arrival is invisible.
5. Sunlight carries more [infrared](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) than [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) , yet only [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) burns skin (bonds $\sim 3\text{ to }4\,\mathrm{eV}$ ). Explain in grains.

**Part II — Measuring $h$ with the kit’s photocell.** The kit’s photocell has an unknown cathode. [Filters](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-subtractive) give:

| $\lambda$ ($\mathrm{nm}$) | 365 | 405 | 436 | 546 |
| --- | --- | --- | --- | --- |
| $U_s$ ($\mathrm{V}$) | 1.50 | 1.16 | 0.94 | 0.37 |

6. What does the [stopping potential](#def-g12-quantum-world-stopping) measure? Relate $U_s$ , $\nu$ , $W_0$ , $h$ .
7. Convert the four [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) to frequencies (in $10^{14}\,\mathrm{Hz}$ ).
8. Why should plotting $eU_s$ against $\nu$ give a straight line? Check the alignment of the points.
9. From the extreme points, compute the slope: your measured $h$ .
10. Deduce the cathode’s [work function](#thm-g12-quantum-world-einstein) ( $\mathrm{eV}$ ), [threshold frequency](#prop-g12-quantum-world-facts) and threshold [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) .
11. Which part of the sunlight could eject electrons from this cathode — and which half of the [spectrum](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-timbre) is useless for it?
12. Compare your $h$ to $6.63 \times 10^{-34}\,\mathrm{J}\,\mathrm{s}$ : relative error in percent?

**Part III — The hydrogen calibration lamp.** The kit’s [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) scale is checked against a hydrogen lamp, $E_n = -13.6/n^2$ $\mathrm{eV}$.

13. Compute $E_2$ , $E_3$ , $E_4$ and $E_5$ .
14. Compute the [photon](#def-g12-quantum-world-photon) energies of the transitions $3 \to 2$ , $4 \to 2$ , $5 \to 2$ .
15. Deduce the three [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) and give each line’s color.
16. Why does the lamp emit *lines* rather than a continuous rainbow? One sentence, naming the key idea.
17. Could the lamp’s hydrogen absorb light at $500\,\mathrm{nm}$ ? Justify with the ladder.

**Part IV — Inspecting the cells: the electron [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave).** Micro-cracks in a cell lie far below the $0.5\,\text{µ}\mathrm{m}$ reach of light microscopes; the lab’s scanning electron microscope runs at $10\,\mathrm{kV}$.

18. Compute the electrons’ speed; check it stays below $0.25\,c$ (classical formula).
19. Compute their [momentum](https://one-course.com/books/physics/2/en/chapter/25-newtons-laws#def-g12-newtons-laws-momentum) and [de Broglie wavelength](#def-g12-quantum-world-debroglie) .
20. Resolution scales with [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) : compute the gain over green light ( $550\,\mathrm{nm}$ ); conclude with the weekend’s two numbers — your $h$ and this gain.

**Solution of Problem 34.1.**

**1.** $E = hc/\lambda = 1.99 \times 10^{-25}/5.50 \times 10^{-7} =
3.62 \times 10^{-19}\,\mathrm{J} = 2.26\,\mathrm{eV}$.

**2.** $N = 1.0 \times 10^{3}/3.62 \times 10^{-19} \approx 2.8 \times 10^{21}$ [photons](#def-g12-quantum-world-photon) per second.

**3.** $P = 0.20 \times 1.0 \times 10^{3}\,\mathrm{W} = 200\,\mathrm{W}$; $E = 200 \times 5.0 = 1.0\,\mathrm{kW}\,\mathrm{h}$ ($3.6 \times 10^{6}\,\mathrm{J}$).

**4.** With $10^{21}$ grains per second, the statistical flicker is fantastically small: graininess averages into a perfectly smooth current, as sand poured fast pours like water.

**5.** Damage needs one grain of $3\text{ to }4\,\mathrm{eV}$: [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) grains qualify, [infrared](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) grains ($\sim 0.1\,\mathrm{eV}$) never do — numbers cannot substitute for punch per grain.

**6.** $U_s$ is the reverse [voltage](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-voltage) that just stops the fastest electrons: $eU_s = E_{k,\max} = h\nu - W_0$.

**7.** $\nu = c/\lambda$: $8.22$, $7.41$, $6.88$ and $5.49$ $\times\,10^{14}\,\mathrm{Hz}$.

**8.** $eU_s = h\nu - W_0$ is affine in $\nu$ with slope $h$. Successive slopes: $(0.94 - 0.37)/1.39 = 0.41$ and $(1.50 - 0.94)/1.34 = 0.42$ $\mathrm{eV}$ per $10^{14}\,\mathrm{Hz}$ — the three points align.

**9.** $h = e(U_{s,1}-U_{s,4})/(\nu_1-\nu_4) = 1.60 \times 10^{-19}
\times 1.13/2.72 \times 10^{14} = 6.64 \times 10^{-34}\,\mathrm{J}\,\mathrm{s}$.

**10.** At $546\,\mathrm{nm}$: $W_0 = h\nu - eU_s = 3.64 \times 10^{-19} -
0.59 \times 10^{-19} = 3.05 \times 10^{-19}\,\mathrm{J} = 1.90\,\mathrm{eV}$; $\nu_0 = W_0/h = 4.60 \times 10^{14}\,\mathrm{Hz}$; $\lambda_0 \approx 653\,\mathrm{nm}$.

**11.** Only $\lambda < 653\,\mathrm{nm}$ ejects: the violet-to-red visible and the [ultraviolet](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength). The whole [infrared](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength) side — roughly half the Sun’s [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) — is useless to this cathode.

**12.** $(6.64 - 6.63)/6.63 \approx 0.2\%$ — a two-permille physics-lab measurement of a universal constant.

**13.** $E_2 = -3.40\,\mathrm{eV}$, $E_3 = -1.51\,\mathrm{eV}$, $E_4 = -0.85\,\mathrm{eV}$, $E_5 = -0.54\,\mathrm{eV}$.

**14.** $3 \to 2$: $1.89\,\mathrm{eV}$; $4 \to 2$: $2.55\,\mathrm{eV}$; $5 \to 2$: $2.86\,\mathrm{eV}$.

**15.** $\lambda = 1240/E$: $656\,\mathrm{nm}$ (red), $486\,\mathrm{nm}$ (blue-green), $434\,\mathrm{nm}$ (violet).

**16.** The [atom](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder)’s energies are [quantized](#def-g12-quantum-world-levels), so it can only emit the discrete differences $E_p - E_n$: a ladder makes lines, not a rainbow.

**17.** No: $500\,\mathrm{nm}$ means $2.48\,\mathrm{eV}$, which matches no level difference ($E_2$ upward gives $1.89$, $2.55$, $2.86$, $3.40\,\mathrm{eV}$; from $E_1$ everything costs at least $10.2\,\mathrm{eV}$) — the gas is transparent there.

**18.** $eU = \tfrac12 m_e v^2$: $v = \sqrt{2 \times
1.60 \times 10^{-19} \times 1.0 \times 10^{4}/9.11 \times 10^{-31}} \approx
5.9 \times 10^{7}\,\mathrm{m}/\mathrm{s} \approx 0.20\,c$ — classical treatment acceptable.

**19.** $p = m_e v = 5.4 \times 10^{-23}\,\mathrm{kg}\,\mathrm{m}/\mathrm{s}$; $\lambda = h/p = 1.2 \times 10^{-11}\,\mathrm{m} = 12\,\mathrm{pm}$.

**20.** Gain $= 5.50 \times 10^{-7}/1.23 \times 10^{-11} \approx 4.5 \times 10^{4}$. The weekend’s two numbers: $h = 6.64 \times 10^{-34}\,\mathrm{J}\,\mathrm{s}$, measured to $0.2\%$ with [filters](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-subtractive) and a voltmeter, and a microscope forty-five-thousand times sharper because electrons [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave).