Modern Physics: Rapid-Fire Drill Pack — Wave-Particle Duality

Objective: Master calculations involving photon momentum, Compton scattering, de-Broglie wavelength, and Heisenberg uncertainty.
Target: 1.5 minutes per problem. If you stall >3 minutes, skip and mark it.
Total problems: 36
Estimated time: 50–60 minutes

Cheat Sheet (Memorize First)

Constants (Memorize or keep handy)

Symbol Value Meaning
$h$ $6.63 \times 10^{-34}$ J·s Planck's constant
$c$ $3.00 \times 10^{8}$ m/s Speed of light
$m_e$ $9.11 \times 10^{-31}$ kg Electron rest mass
$1 \text{ eV}$ $1.602 \times 10^{-19}$ J Electron-volt conversion

Key Formulas

Photon Momentum $$p = \frac{h}{\lambda} = \frac{hf}{c} = \frac{E}{c}$$

Compton Effect (Wavelength Shift) $$\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos \theta)$$

  • $\lambda'$ = scattered photon wavelength
  • $\lambda$ = incident photon wavelength
  • $\theta$ = scattering angle

de-Broglie Wavelength $$\lambda = \frac{h}{p} = \frac{h}{mv}$$

de-Broglie from Kinetic Energy $$\lambda = \frac{h}{\sqrt{2m(KE)}}$$

Heisenberg Uncertainty Principle $$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$$

Quick-Identification Rules

  • Compton shift increases with angle: Max at $\theta = 180°$ ($\Delta \lambda_{max} = \frac{2h}{m_e c}$)
  • Higher KE → shorter wavelength: Fast particles behave less like waves
  • Smaller mass → more wave-like: Electrons show wave nature; baseballs don't
  • Uncertainty trade-off: Pinpoint position → momentum unknown; know momentum well → position smeared

Part A: Photon Momentum

Target: 60 seconds per problem.

Set A1 — Basic Photon Momentum (6 problems)

Calculate the momentum of photons with the following wavelengths.

  1. $\lambda = 5.0 \times 10^{-7}$ m (visible light)
  2. $\lambda = 1.0 \times 10^{-10}$ m (X-ray)
  3. $\lambda = 6.0 \times 10^{-7}$ m (orange light)
  4. $\lambda = 2.5 \times 10^{-7}$ m (UV)
  5. $\lambda = 1.0 \times 10^{-12}$ m (gamma ray)
  6. $\lambda = 4.0 \times 10^{-7}$ m (violet light)

Score: ___/6

Set A2 — Photon Momentum from Energy/Frequency (5 problems)

  1. A photon has energy $E = 3.0 \times 10^{-19}$ J. Find its momentum.
  2. A photon has frequency $f = 5.0 \times 10^{14}$ Hz. Find its momentum.
  3. A photon has energy $E = 2.0 \times 10^{-15}$ J. Find its momentum.
  4. A photon has frequency $f = 1.0 \times 10^{18}$ Hz. Find its momentum.
  5. Find the wavelength of a photon with momentum $p = 1.0 \times 10^{-27}$ kg·m/s.

Score: ___/5

Part B: Compton Effect

Target: 90 seconds per problem.

Set B1 — Compton Wavelength Shift (6 problems)

Use $\Delta \lambda = \frac{h}{m_e c}(1 - \cos \theta)$ where $\frac{h}{m_e c} = 2.43 \times 10^{-12}$ m (Compton wavelength).

  1. An X-ray with $\lambda = 1.0 \times 10^{-11}$ m scatters at $\theta = 90°$. Find the scattered wavelength $\lambda'$.
  2. An X-ray with $\lambda = 2.0 \times 10^{-11}$ m scatters at $\theta = 60°$. Find $\Delta \lambda$.
  3. An X-ray with $\lambda = 1.5 \times 10^{-11}$ m scatters at $\theta = 180°$. Find $\lambda'$.
  4. Find the scattering angle $\theta$ if $\Delta \lambda = 1.215 \times 10^{-12}$ m.
  5. At what angle does $\Delta \lambda = \frac{h}{m_e c}$ (i.e., maximum shift)?
  6. An X-ray photon scatters at $\theta = 120°$. If the incident wavelength is $3.0 \times 10^{-12}$ m, find $\lambda'$.

Score: ___/6

Set B2 — Compton Effect — Percentage/Reverse Problems (4 problems)

  1. The scattered wavelength is 2% longer than the incident wavelength. Find the scattering angle $\theta$.
  2. The Compton shift is $\Delta \lambda = 1.0 \times 10^{-12}$ m. At what angle did the photon scatter?
  3. An X-ray with $\lambda = 0.100$ nm scatters at $\theta = 45°$. By what percentage does the wavelength increase?
  4. A photon loses 50% of its momentum in a Compton scattering event. Find $\theta$.

Score: ___/4

Part C: de-Broglie Waves

Target: 90 seconds per problem.

Set C1 — de-Broglie Wavelength from Velocity (6 problems)

Use $\lambda = \frac{h}{mv}$ for electrons ($m_e = 9.11 \times 10^{-31}$ kg).

  1. Find the de-Broglie wavelength of an electron moving at $v = 1.0 \times 10^{6}$ m/s.
  2. Find the de-Broglie wavelength of an electron moving at $v = 5.0 \times 10^{6}$ m/s.
  3. An electron has $\lambda = 1.0 \times 10^{-9}$ m. Find its velocity.
  4. An electron has $\lambda = 5.0 \times 10^{-10}$ m. Find its velocity.
  5. Find the de-Broglie wavelength of a proton ($m_p = 1.67 \times 10^{-27}$ kg) moving at $v = 2.0 \times 10^{5}$ m/s.
  6. A neutron ($m_n = 1.67 \times 10^{-27}$ kg) has $\lambda = 1.0 \times 10^{-10}$ m. Find its velocity.

Score: ___/6

Set C2 — de-Broglie from Kinetic Energy (5 problems)

Use $\lambda = \frac{h}{\sqrt{2m(KE)}}$. Convert eV to joules first if needed.

  1. An electron has $KE = 100$ eV. Find its de-Broglie wavelength.
  2. An electron has $KE = 50$ eV. Find its de-Broglie wavelength.
  3. Find the kinetic energy (in eV) of an electron with $\lambda = 0.10$ nm.
  4. A proton has $KE = 1.0 \times 10^{-17}$ J. Find its de-Broglie wavelength.
  5. Find the de-Broglie wavelength of an electron accelerated through 200 V.

Score: ___/5

Part D: Heisenberg Uncertainty Principle

Target: 60 seconds per problem.

Set D1 — Position-Momentum Uncertainty (4 problems)

Use $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$. Use the minimum equality $\Delta x \cdot \Delta p = \frac{h}{4\pi}$ for calculations.

  1. An electron's position is measured with uncertainty $\Delta x = 1.0 \times 10^{-10}$ m. Find the minimum uncertainty in its momentum $\Delta p$.
  2. An electron has momentum uncertainty $\Delta p = 1.0 \times 10^{-25}$ kg·m/s. Find the minimum position uncertainty $\Delta x$.
  3. A proton is confined to a nucleus of size $\Delta x = 1.0 \times 10^{-14}$ m. Find the minimum momentum uncertainty.
  4. An electron is confined in an atom with $\Delta x = 5.0 \times 10^{-11}$ m. Find the minimum velocity uncertainty $\Delta v$.

Score: ___/4

Final Scorecard

Part Sets Problems Raw Score
A — Photon Momentum A1, A2 11 ___/11
B — Compton Effect B1, B2 10 ___/10
C — de-Broglie Waves C1, C2 11 ___/11
D — Heisenberg Uncertainty D1 4 ___/4
TOTAL 36 ___/36

Proficiency Benchmarks

  • 25/36 (70%) — Proficient. You can handle standard exam problems.
  • 31/36 (85%) — Solid. Fast and accurate.
  • 34/36 (95%) — Exam-ready. Any mistake is a careless slip.

Speed Benchmarks

  • 40 minutes: Excellent mechanical fluency.
  • 55 minutes: Good. Review missed patterns.
  • 75 minutes: Drill the specific sets you scored lowest on again tomorrow.

Error Log Template

After grading, list every wrong problem number with a one-word reason:

Problem Reason
e.g. 4 forgot unit conversion

Re-solve all wrong problems immediately with notes, then again in 24 hours without notes.

Answer Key

Set A1 — Basic Photon Momentum

  1. $p = 1.33 \times 10^{-27}$ kg·m/s
  2. $p = 6.63 \times 10^{-24}$ kg·m/s
  3. $p = 1.11 \times 10^{-27}$ kg·m/s
  4. $p = 2.65 \times 10^{-27}$ kg·m/s
  5. $p = 6.63 \times 10^{-22}$ kg·m/s
  6. $p = 1.66 \times 10^{-27}$ kg·m/s

Set A2 — Photon Momentum from Energy/Frequency

  1. $p = 1.0 \times 10^{-27}$ kg·m/s
  2. $p = 1.11 \times 10^{-27}$ kg·m/s
  3. $p = 6.67 \times 10^{-24}$ kg·m/s
  4. $p = 2.21 \times 10^{-27}$ kg·m/s
  5. $\lambda = 6.63 \times 10^{-7}$ m

Set B1 — Compton Wavelength Shift

  1. $\lambda' = 1.024 \times 10^{-11}$ m
  2. $\Delta \lambda = 1.215 \times 10^{-12}$ m
  3. $\lambda' = 1.986 \times 10^{-11}$ m
  4. $\theta = 60°$
  5. $\theta = 180°$
  6. $\lambda' = 4.865 \times 10^{-12}$ m

Set B2 — Compton Effect — Percentage/Reverse

  1. $\theta = 47.2°$
  2. $\theta = 49.9° \approx 50°$
  3. $\approx 0.63%$
  4. $\theta = 120°$

Set C1 — de-Broglie Wavelength from Velocity

  1. $\lambda = 7.27 \times 10^{-10}$ m
  2. $\lambda = 1.45 \times 10^{-10}$ m
  3. $v = 7.27 \times 10^{5}$ m/s
  4. $v = 1.45 \times 10^{6}$ m/s
  5. $\lambda = 1.98 \times 10^{-12}$ m
  6. $v = 3.97 \times 10^{3}$ m/s

Set C2 — de-Broglie from Kinetic Energy

  1. $\lambda = 1.23 \times 10^{-10}$ m
  2. $\lambda = 1.74 \times 10^{-10}$ m
  3. $KE = 150$ eV
  4. $\lambda = 3.63 \times 10^{-12}$ m
  5. $\lambda = 8.69 \times 10^{-11}$ m

Set D1 — Heisenberg Uncertainty

  1. $\Delta p = 5.28 \times 10^{-25}$ kg·m/s
  2. $\Delta x = 5.28 \times 10^{-10}$ m
  3. $\Delta p = 5.28 \times 10^{-21}$ kg·m/s
  4. $\Delta v = 1.16 \times 10^{5}$ m/s

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