Chapter 1 Exercises: The Quantum Revolution

Part A: Conceptual Questions (⭐)

These questions test your understanding of the core ideas. No calculations required.

A.1 Explain in your own words why the ultraviolet catastrophe is called a "catastrophe." What does classical physics predict that is physically absurd?

A.2 A friend claims: "The photoelectric effect proves light is a particle, and the double-slit experiment proves light is a wave. So physics contradicts itself." How would you respond? Is your friend right that physics contradicts itself, or is something more subtle going on?

A.3 In the photoelectric effect, increasing the intensity of light above the threshold frequency increases the photocurrent but does not increase the maximum kinetic energy of the ejected electrons. Explain why this is natural in the photon picture but inexplicable in the classical wave picture.

A.4 Why can't we observe the de Broglie wavelength of a baseball? Give a quantitative argument, not just "it's too small."

A.5 The Bohr model gets the right energy levels for hydrogen but is fundamentally wrong. In what specific ways does it fail, and what aspects of it survive in the full quantum mechanical treatment?

A.6 Explain the difference between the statement "classical physics is imprecise at small scales" and the statement "classical physics is conceptually wrong at small scales." Which statement is correct, and what evidence from this chapter supports your answer?

A.7 In the sequential Stern-Gerlach experiment ($z$-filter → $x$-filter → $z$-filter), the final $z$-measurement gives a 50-50 split even though the atoms were initially prepared as spin-up in $z$. Why is this result impossible to explain classically? What does it tell us about the nature of quantum measurements?

A.8 Why did Planck call his quantization hypothesis "an act of desperation"? What was he desperate about, and why was quantization the only way out?

Part B: Applied Problems (⭐⭐)

These problems require direct application of the chapter's key equations.

B.1: Photoelectric Effect Calculations

The work function of platinum is $\phi = 5.65$ eV.

(a) What is the threshold wavelength for the photoelectric effect in platinum?

(b) Light of wavelength $\lambda = 150$ nm strikes a platinum surface. What is the maximum kinetic energy of the ejected electrons, in eV?

(c) What is the maximum speed of these ejected electrons?

(d) What is the stopping potential — the voltage needed to bring the fastest electrons to rest?

B.2: Compton Scattering

An X-ray photon with wavelength $\lambda = 0.0500$ nm undergoes Compton scattering from a free electron.

(a) Calculate the wavelength of the scattered photon for scattering angles $\theta = 30°$, $90°$, and $180°$.

(b) For $\theta = 180°$ (backscattering), what fraction of the photon's energy is transferred to the electron?

(c) At what angle is the wavelength shift equal to one Compton wavelength ($\Delta\lambda = \lambda_C$)?

B.3: De Broglie Wavelengths

Calculate the de Broglie wavelength for each of the following:

(a) An electron accelerated through a potential difference of $V = 54$ V (the Davisson-Germer experiment).

(b) A proton ($m_p = 1.673 \times 10^{-27}$ kg) with kinetic energy $K = 1.0$ MeV.

(c) A helium atom ($m = 6.646 \times 10^{-27}$ kg) at room temperature ($T = 300$ K), using $K = \frac{3}{2}k_BT$.

(d) A C$_{60}$ molecule (buckyball, $m = 1.2 \times 10^{-24}$ kg) traveling at $v = 200$ m/s. (Interference of C$_{60}$ molecules was observed experimentally in 1999 by Arndt et al.)

B.4: Bohr Model

(a) Calculate the first three energy levels ($n = 1, 2, 3$) of the hydrogen atom in eV using the Bohr formula.

(b) What wavelength photon is emitted in the transition $n = 4 \to n = 2$ (the H-$\beta$ line)? What color is this?

(c) What is the ionization energy of hydrogen — the energy needed to remove the electron from the ground state entirely ($n = 1 \to n = \infty$)?

(d) Calculate the radius of the $n = 1$, $n = 2$, and $n = 3$ Bohr orbits. Express your answers in units of the Bohr radius $a_0$.

B.5: Blackbody Radiation

(a) The surface temperature of the Sun is approximately $T = 5778$ K. At what frequency does the solar spectrum peak? What wavelength does this correspond to? What part of the electromagnetic spectrum is this?

(b) The cosmic microwave background has a temperature of $T = 2.725$ K. At what wavelength does its spectrum peak? (Use Wien's displacement law: $\lambda_{\max} T = 2.898 \times 10^{-3}$ m$\cdot$K.)

(c) A blast furnace operates at $T = 1800$ K. At what wavelength does its radiation peak? Is this in the visible range?

B.6: Photon Momentum

(a) Calculate the momentum of a photon of red light ($\lambda = 650$ nm).

(b) How does this compare to the momentum of an electron with the same de Broglie wavelength of 650 nm? What is the electron's kinetic energy?

(c) A laser pointer emits $5$ mW of light at $\lambda = 532$ nm. How many photons per second does it emit? What is the total momentum carried by these photons per second (i.e., the radiation pressure force)?

Part C: Computational Problems (⭐⭐–⭐⭐⭐)

These problems involve quantitative analysis that benefits from numerical computation.

C.1: Planck vs. Rayleigh-Jeans (⭐⭐)

Using the code in code/example-01-setup.py as a starting point:

(a) Plot the Planck spectral energy density $u(\nu, T)$ and the Rayleigh-Jeans approximation on the same axes for $T = 3000$ K, $5000$ K, and $8000$ K. Use a frequency range of $0$ to $3 \times 10^{15}$ Hz.

(b) For $T = 5000$ K, at what frequency does the Rayleigh-Jeans prediction exceed the Planck prediction by a factor of 2? By a factor of 10?

(c) Numerically integrate the Planck spectrum over all frequencies to verify the Stefan-Boltzmann law: $\int_0^\infty u(\nu, T) \, d\nu = aT^4$, where $a = 4\sigma/c$ and $\sigma = 5.67 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$.

C.2: Photoelectric Stopping Potential (⭐⭐)

In Millikan's photoelectric experiment, the stopping potential $V_0$ is measured as a function of photon frequency $\nu$. Simulated data (with realistic scatter):

(a) Plot $V_0$ vs. $\nu$ and perform a linear fit: $V_0 = (h/e)\nu - \phi/e$.

(b) From your fit, extract the slope $h/e$ and determine Planck's constant $h$. Compare with the accepted value.

(c) Determine the work function $\phi$ from the intercept. What metal might this be?

C.3: De Broglie Wavelength Explorer (⭐⭐⭐)

Write a Python program that:

(a) Takes as input a particle mass (in kg or in atomic mass units) and either velocity or kinetic energy.

(b) Calculates and reports the de Broglie wavelength.

(c) Compares the wavelength to relevant length scales: atomic diameter ($\sim 10^{-10}$ m), nuclear diameter ($\sim 10^{-15}$ m), Planck length ($\sim 10^{-35}$ m), and prints whether quantum effects are expected to be significant.

(d) Use your program to explore: at what kinetic energy does a neutron's de Broglie wavelength equal the typical nuclear diameter ($\sim 1$ fm)? At what speed does a C$_{60}$ molecule have a wavelength equal to its own diameter ($\sim 1$ nm)?

Part D: Synthesis Problems (⭐⭐⭐)

These problems require integrating multiple concepts and thinking beyond direct formulas.

D.1: The Quantum-Classical Boundary

Consider a dust grain of mass $m = 10^{-15}$ kg floating in air at room temperature ($T = 300$ K).

(a) Calculate its thermal de Broglie wavelength.

(b) The typical spacing between air molecules is about $3$ nm. Is the dust grain's de Broglie wavelength comparable to this? To the grain's own size (about $1$ $\mu$m)?

(c) Now consider cooling the grain to $T = 10^{-6}$ K (achievable in modern experiments). Recalculate the de Broglie wavelength. Is the situation qualitatively different?

(d) Based on your calculations, discuss the conditions under which macroscopic objects might exhibit quantum behavior. What experimental challenges does this present?

D.2: Photon Pressure and the Sun

The Sun's luminosity is $L = 3.846 \times 10^{26}$ W, and its surface temperature is $T = 5778$ K.

(a) Using the Stefan-Boltzmann law, verify that this luminosity is consistent with a blackbody of the Sun's radius ($R = 6.96 \times 10^8$ m).

(b) How many photons per second does the Sun emit? (Assume the average photon energy corresponds to the peak of the spectrum.)

(c) What is the total momentum carried away by photons per second? What force does this exert on the Sun?

(d) Compare this radiation pressure force to the gravitational force holding the Sun together. Is radiation pressure significant for the Sun? For what kind of star would it become important?

D.3: The Double-Slit Thought Experiment

In a double-slit experiment with electrons ($V = 100$ V accelerating potential), the slit separation is $d = 1$ $\mu$m and the screen is $L = 2$ m away.

(a) Calculate the fringe spacing $\Delta y = \lambda L / d$.

(b) Now imagine gradually increasing the electron mass while keeping the kinetic energy fixed. At what mass does the fringe spacing become smaller than the electron's classical point-like detection spot (say, $\Delta y < 10$ nm)? Would you still see interference?

(c) Discuss what happens in the limit $m \to \infty$ with fixed kinetic energy. How does the quantum result approach the classical expectation?

(d) Some physicists have proposed sending viruses through a double slit to test the quantum-classical boundary. A typical virus has mass $\sim 10^{-20}$ kg and speed $\sim 100$ m/s. Calculate the fringe spacing you would need to detect and discuss the experimental challenges.

D.4: Connecting the Experiments

Construct a timeline of the key experiments from this chapter (1887–1927) and for each one, identify: (a) the specific prediction of classical physics that failed, (b) the quantum concept that resolved the failure, and (c) how that concept connects to the formal postulates of quantum mechanics (which you can look up or speculate about). Present your analysis as a table or concept map.

Part E: Research and Exploration (⭐⭐⭐⭐)

These problems go beyond the chapter's direct content and require independent investigation.

E.1: Modern Double-Slit Experiments

Research the 2012 experiment by Bach et al. ("Controlled double-slit electron diffraction," New Journal of Physics) that directly imaged the buildup of the electron double-slit pattern one electron at a time.

(a) Describe the experimental setup and how they achieved single-electron detection.

(b) How did their results compare with the theoretical predictions?

(c) What was the electron energy, and what was the measured fringe spacing?

(d) The authors created a movie showing the pattern building up. Describe what you would see in such a movie and explain why it is so conceptually disturbing.

E.2: Matter-Wave Interference with Large Molecules

Research the experiments by Markus Arndt's group in Vienna on matter-wave interference with increasingly large molecules (C$_{60}$, C$_{70}$, and eventually molecules with over 800 atoms).

(a) What is the largest molecule for which interference has been observed? What was its de Broglie wavelength?

(b) What experimental techniques were required to observe interference at such small wavelengths?

(c) These experiments probe the quantum-classical boundary. What is the current theoretical understanding of why large objects don't show quantum interference? (Look up "decoherence.")

E.3: Planck's Original Derivation

Read Planck's original 1900 paper (available in English translation) or a detailed historical account (e.g., in Kuhn's Black-Body Theory and the Quantum Discontinuity).

(a) What was Planck's original reasoning for the quantization hypothesis? Did he believe the quantization was "real" or merely a mathematical convenience?

(b) How did Planck's interpretation differ from Einstein's later interpretation of quantization?

(c) When did the physics community generally accept that quantization was a fundamental feature of nature, not a computational trick?

E.4: The Stern-Gerlach Experiment and Quantum Computing

Research how the Stern-Gerlach concept — a two-state quantum system — connects to modern quantum computing.

(a) What is a qubit, and how does it relate to the spin-1/2 system observed in the Stern-Gerlach experiment?

(b) What physical systems are used to implement qubits in modern quantum computers? (List at least three.)

(c) The sequential Stern-Gerlach experiment demonstrates that measurement in one basis disturbs the state in another basis. How does this relate to the concept of "quantum gates" in quantum computing?

Part M: Mixed Practice

These problems mix concepts from Chapter 1 with prerequisite physics (classical mechanics, electromagnetism, thermodynamics) to build fluency.

M.1: A hydrogen atom in the $n = 2$ state absorbs a photon and transitions to $n = 5$. (a) What is the wavelength of the absorbed photon? (b) If the atom then decays from $n = 5$ to $n = 1$, how many possible decay paths are there? (Hint: the atom can go $5 \to 1$ directly, or $5 \to 4 \to 1$, or $5 \to 3 \to 1$, etc.) (c) For the direct $5 \to 1$ transition, what is the photon wavelength and what series does it belong to?

M.2: The gravitational attraction between an electron and a proton is $F_g = Gm_em_p/r^2$. The electrostatic attraction is $F_e = e^2/(4\pi\epsilon_0 r^2)$. (a) What is the ratio $F_e/F_g$? (b) Could gravity, rather than electrostatics, hold a "gravitational atom" together? If so, what would the "gravitational Bohr radius" be? (Replace $e^2/4\pi\epsilon_0$ with $Gm_em_p$ in the Bohr radius formula.)

M.3: In classical electrodynamics, an accelerating charge radiates power $P = e^2 a^2 / (6\pi\epsilon_0 c^3)$ (Larmor formula). For an electron in a classical circular orbit at the Bohr radius $a_0$: (a) Calculate the centripetal acceleration. (b) Calculate the radiated power. (c) Estimate how long it would take for the electron to radiate away its kinetic energy and spiral into the nucleus. (This is why classical atoms are unstable.)

M.4: A photon and an electron have the same de Broglie wavelength $\lambda = 0.1$ nm. (a) What is the photon's energy? (b) What is the electron's kinetic energy? (Use the non-relativistic formula.) (c) Which has more energy? (d) At what wavelength would the photon and electron have the same energy?