Chapter 1 Quiz: The Quantum Revolution

Instructions: This quiz covers the core concepts from Chapter 1. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. The ultraviolet catastrophe refers to the classical prediction that:

(a) Ultraviolet light cannot be produced by blackbodies (b) A blackbody should radiate infinite total energy at any finite temperature (c) Ultraviolet photons have too much energy to exist (d) The Rayleigh-Jeans law fails at low frequencies

Q2. In the photoelectric effect, increasing the intensity of incident light (above the threshold frequency) will:

(a) Increase the maximum kinetic energy of ejected electrons (b) Increase the number of ejected electrons per second (c) Decrease the threshold frequency (d) Increase the work function of the metal

Q3. The Compton effect demonstrates that photons have:

(a) Mass (b) Charge (c) Momentum (d) Spin

Q4. In the Bohr model, the quantization condition requires that the electron's angular momentum equals:

(a) $nh$ where $n = 1, 2, 3, \ldots$ (b) $n\hbar$ where $n = 1, 2, 3, \ldots$ (c) $n\hbar/2$ where $n = 1, 2, 3, \ldots$ (d) $(n + 1/2)\hbar$ where $n = 0, 1, 2, \ldots$

Q5. In the double-slit experiment with single electrons, adding a which-path detector at the slits causes the pattern on the screen to:

(a) Become brighter overall (b) Shift to the left (c) Change from an interference pattern to a sum of single-slit patterns (d) Remain unchanged — detection at the slits doesn't affect the screen

Q6. The de Broglie wavelength of a particle is given by $\lambda = h/p$. If you double the kinetic energy of a non-relativistic particle, its de Broglie wavelength:

(a) Doubles (b) Halves (c) Decreases by a factor of $\sqrt{2}$ (d) Increases by a factor of $\sqrt{2}$

Q7. The Stern-Gerlach experiment with silver atoms produced exactly two spots on the detector screen. This is evidence for:

(a) The electron having spin-1/2 (b) The electron having orbital angular momentum $l = 1$ (c) Silver atoms being bosons (d) The magnetic field being too weak

Q8. Planck's radiation law reduces to the Rayleigh-Jeans law in the limit:

(a) $h\nu \gg k_BT$ (high frequency) (b) $h\nu \ll k_BT$ (low frequency) (c) $T \to 0$ (low temperature) (d) $\nu \to \infty$ (infinite frequency)

Q9. Which of the following has the longest de Broglie wavelength?

(a) An electron moving at $10^6$ m/s (b) A proton moving at $10^6$ m/s (c) An electron moving at $10^4$ m/s (d) A proton moving at $10^4$ m/s

Q10. The energy levels of hydrogen in the Bohr model are $E_n = -13.6/n^2$ eV. The ionization energy of hydrogen from the ground state is:

(a) 3.4 eV (b) 10.2 eV (c) 13.6 eV (d) 27.2 eV

True/False with Justification (4 questions)

Q11. TRUE or FALSE: In the photoelectric effect, if the frequency of incident light is below the threshold, ejecting electrons is possible by using a sufficiently intense light source.

Justification:

Q12. TRUE or FALSE: The Bohr model correctly predicts the energy levels of all atoms, not just hydrogen.

Justification:

Q13. TRUE or FALSE: The double-slit interference pattern builds up even when particles are sent through the slits one at a time, proving that each individual particle interferes with itself.

Justification:

Q14. TRUE or FALSE: A macroscopic object like a tennis ball has a de Broglie wavelength, but it is too small to observe experimentally.

Justification:

Short Answer (4 questions)

Q15. Explain why Planck's quantization hypothesis $E = nh\nu$ resolves the ultraviolet catastrophe. Specifically, why does quantization suppress the contribution of high-frequency modes to the total energy?

Q16. Describe the key difference between Einstein's interpretation of the photon and Planck's original interpretation of energy quantization. Why was Einstein's view more radical?

Q17. The sequential Stern-Gerlach experiment ($z \to x \to z$) shows that a measurement of spin along $x$ "erases" the previously known spin along $z$. Explain why this result is incompatible with the classical idea that measurement merely reveals pre-existing properties.

Q18. De Broglie's hypothesis assigns a wavelength to every particle. Why do we never observe wave-like behavior for macroscopic objects? Provide a quantitative argument involving a specific example.

Applied Scenarios (2 questions)

Q19. Designing a Photoelectric Experiment

You are designing a photoelectric effect experiment to measure Planck's constant. You have access to a sodium cathode ($\phi = 2.28$ eV) and a tunable laser that can produce light at wavelengths between 300 nm and 700 nm.

(a) Over what range of wavelengths will the photoelectric effect occur with this cathode? (3 points)

(b) Calculate the stopping potential $V_0$ for wavelengths of 350 nm, 400 nm, and 450 nm. (6 points)

(c) Describe how you would plot your data to extract $h$. What quantity goes on each axis? What is the expected slope? (3 points)

(d) If your measured slope turns out to be $4.0 \times 10^{-15}$ V$\cdot$s, what value of $h$ does this give? What is the percent error relative to the accepted value? (3 points)

Q20. Neutron Diffraction

Thermal neutrons from a nuclear reactor are used for crystal structure determination, similar to X-ray diffraction.

(a) Calculate the de Broglie wavelength of a neutron at $T = 300$ K (use $K = \frac{3}{2}k_BT$, $m_n = 1.675 \times 10^{-27}$ kg). (3 points)

(b) Typical crystal lattice spacings are $d \sim 0.2$–$0.5$ nm. Is the neutron wavelength from part (a) suitable for crystal diffraction? Explain. (2 points)

(c) "Cold neutrons" from a moderator at $T = 25$ K are also used. What is their wavelength, and what advantage do they offer? (3 points)

(d) Compare the energy of a thermal neutron ($T = 300$ K) with the energy of an X-ray photon having the same wavelength. Which carries more energy? By what factor? (4 points)

Answer Key

Multiple Choice

  1. (b) — The Rayleigh-Jeans law predicts $u \propto \nu^2$, which diverges when integrated, giving infinite total energy.
  2. (b) — More intensity means more photons, hence more photoelectrons (more current), but each photon still has the same energy.
  3. (c) — Compton treated the photon as carrying momentum $p = h/\lambda$ and derived the wavelength shift from energy-momentum conservation.
  4. (b) — Bohr's quantization condition: $L = n\hbar$, $n = 1, 2, 3, \ldots$
  5. (c) — Which-path information destroys quantum interference; the pattern becomes the classical sum $P_1 + P_2$.
  6. (c) — $p = \sqrt{2mK}$, so doubling $K$ multiplies $p$ by $\sqrt{2}$, and $\lambda = h/p$ decreases by $\sqrt{2}$.
  7. (a) — Two spots correspond to $m_s = \pm 1/2$, characteristic of spin-1/2.
  8. (b) — When $h\nu \ll k_BT$, the quantum discreteness is unimportant and classical equipartition applies.
  9. (c) — Longest $\lambda = h/p$ means smallest $p = mv$. The electron at $10^4$ m/s has the smallest momentum: $m_e \times 10^4 \ll m_p \times 10^4 \ll m_e \times 10^6 < m_p \times 10^6$.
  10. (c) — Ionization from $n=1$: $E_\infty - E_1 = 0 - (-13.6) = 13.6$ eV.

True/False

  1. FALSE. Each photon below threshold has insufficient energy to eject an electron ($h\nu < \phi$). Intensity increases the number of photons, but each individual photon-electron interaction still lacks sufficient energy. Multi-photon absorption is negligible at ordinary intensities.
  2. FALSE. The Bohr model works only for hydrogen and hydrogen-like ions (one electron). It fails for helium and all multi-electron atoms because it cannot account for electron-electron interactions.
  3. TRUE. Since particles arrive one at a time with no possibility of inter-particle interaction, the interference pattern must arise from each particle's probability amplitude passing through both slits and interfering with itself.
  4. TRUE. The de Broglie relation $\lambda = h/mv$ applies to all objects. For a tennis ball ($m \approx 0.06$ kg, $v \approx 50$ m/s), $\lambda \approx 2 \times 10^{-34}$ m — vastly smaller than any measurable length scale.

Short Answer (Key Points)

  1. In the classical picture, each electromagnetic mode gets energy $k_BT$ regardless of frequency. With quantization, a mode of frequency $\nu$ requires at least energy $h\nu$ to be excited. When $h\nu \gg k_BT$, the probability of excitation is exponentially small ($\sim e^{-h\nu/k_BT}$), so high-frequency modes contribute negligible energy. This prevents the divergence.
  2. Planck quantized the oscillators in the cavity walls — the emitters and absorbers of radiation. He hoped quantization was a property of matter, not of radiation itself. Einstein went further: he proposed that the electromagnetic field itself is quantized into photons. This was more radical because it challenged Maxwell's wave theory of light directly.
  3. Classically, a measurement reveals a property that already existed before the measurement. If the atom "had" a definite $z$-spin before the $x$-measurement, it should still have that same $z$-spin afterward. The fact that the $z$-measurement after the $x$-measurement gives random results means the $x$-measurement changed (or created) the $z$-spin — measurement is not passive revelation but active transformation.
  4. For a $70$ kg person walking at $1.5$ m/s, $\lambda = h/(mv) \approx 6.3 \times 10^{-36}$ m. This is $10^{-21}$ times smaller than a proton. No slit, crystal, or detector exists that could resolve structures at this scale. Wave effects require $\lambda$ comparable to obstacle/slit sizes; since all relevant objects are at least $10^{25}$ times larger than this wavelength, diffraction and interference are utterly unobservable.

Applied Scenarios

  1. (a) $\lambda_0 = hc/\phi = 1240/2.28 = 544$ nm. PE effect occurs for $\lambda < 544$ nm, so from 300 nm to 544 nm. (b) $V_0 = hc/(e\lambda) - \phi/e$. At 350 nm: $V_0 = (1240/350) - 2.28 = 3.54 - 2.28 = 1.26$ V. At 400 nm: $V_0 = 3.10 - 2.28 = 0.82$ V. At 450 nm: $V_0 = 2.76 - 2.28 = 0.48$ V. (c) Plot $V_0$ vs. $\nu$. Slope $= h/e$, intercept $= -\phi/e$. (d) $h = (4.0 \times 10^{-15})(1.602 \times 10^{-19}) = 6.41 \times 10^{-34}$ J$\cdot$s. Error: $(6.626 - 6.41)/6.626 = 3.3\%$.
  2. (a) $K = \frac{3}{2}k_BT = 6.21 \times 10^{-21}$ J. $p = \sqrt{2m_nK} = 4.56 \times 10^{-24}$ kg$\cdot$m/s. $\lambda = 1.45 \times 10^{-10}$ m $= 0.145$ nm. (b) Yes — $0.145$ nm is within the $0.2$–$0.5$ nm lattice spacing range, making diffraction possible. (c) At $25$ K: $\lambda = 0.145 \sqrt{300/25} = 0.145 \times 3.46 = 0.50$ nm. Larger wavelength probes larger structures and has better angular resolution. (d) Neutron $K = 0.039$ eV. Photon $E = hc/\lambda = 1240/0.145 = 8550$ eV. The photon has about $2.2 \times 10^5$ times more energy. This is why neutrons probe structure without damaging samples.