Problem Statement
Solve the quantum/modern physics problem: A photon of $\lambda = 0.0600$ nm is scattered at $\theta = 120°$. Find the scattered wavelength and electron kinetic energy. (a) $\Delta\lambda = \lambda_C(1-\cos120°) = 0.00243(1.5) = 0.00364$ nm $$\lambda’ = 0.0600 + 0.00364 = 0.0636 \text{ nm}$$ (b) $T = hc(1/\lambda – 1/\lambda’) = 1240\text{ e
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
Solve the quantum/modern physics problem: A photon of $\lambda = 0.0600$ nm is scattered at $\theta = 120°$. Find the scattered wavelength and electron kinetic energy. (a) $\Delta\lambda = \lambda_C(1-\cos120°) = 0.00243(1.5) = 0.00364$ nm $$\lambda’ = 0.0600 + 0.00364 = 0.0636 \text{ nm}$$ (b) $T = hc(1/\lambda – 1/\lambda’) = 1240\text{ e
Given Information
- Frequency $\nu$ or wavelength $\lambda$ of radiation
- Work function $\phi$ of metal (if photoelectric)
- Planck’s constant $h = 6.626\times10^{-34}\,\text{J s}$
- Speed of light $c = 3\times10^8\,\text{m/s}$
Physical Concepts & Formulas
Einstein’s photoelectric effect established that light comes in discrete quanta (photons) each carrying energy $E = h\nu$. When a photon strikes a metal surface, it gives all its energy to one electron. If $h\nu > \phi$ (work function), the electron is ejected with maximum kinetic energy $KE_{\max} = h\nu – \phi$. de Broglie’s relation $\lambda = h/p$ extends wave-particle duality to matter. Heisenberg’s uncertainty principle $\Delta x \Delta p \geq h/(4\pi)$ sets a fundamental limit on simultaneous knowledge of position and momentum — not an instrument limitation but a property of nature.
- $E = h\nu = hc/\lambda$ — photon energy
- $KE_{\max} = h\nu – \phi$ — Einstein photoelectric equation
- $\lambda_{\text{dB}} = h/p = h/(mv)$ — de Broglie wavelength
- $\Delta x\,\Delta p \geq \dfrac{h}{4\pi}$ — Heisenberg uncertainty
- $E_n = -13.6/n^2\,\text{eV}$ — hydrogen energy levels
Step-by-Step Solution
Step 1 — Compute photon energy:
$$E = \frac{hc}{\lambda}$$
Step 2 — Compare to work function: If $E > \phi$, emission occurs.
Step 3 — Maximum KE of emitted electron:
$$KE_{\max} = E – \phi = \frac{hc}{\lambda} – \phi$$
Step 4 — Stopping potential: $eV_s = KE_{\max}$
Worked Calculation
Substituting all values with units:
UV light $\lambda = 200\,\text{nm}$ on zinc ($\phi = 4.3\,\text{eV}$):
$$E = \frac{6.626\times10^{-34}\times3\times10^8}{200\times10^{-9}} = \frac{1.988\times10^{-25}}{2\times10^{-7}} = 9.94\times10^{-19}\,\text{J} = 6.21\,\text{eV}$$
$$KE_{\max} = 6.21 – 4.3 = 1.91\,\text{eV}$$
Answer
$$\boxed{KE_{\max} = h\nu – \phi = 1.91\,\text{eV}}$$
Physical Interpretation
1.91 eV is a modest but significant energy — enough to accelerate an electron to about 820 km/s. The fact that increasing light intensity does not increase $KE_{\max}$ (more photons, same energy each) was inexplicable classically but perfectly natural in quantum theory. This was the observation that convinced physicists of light’s particulate nature and earned Einstein the 1921 Nobel Prize.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{KE_{\max} = h\nu – \phi = 1.91\,\text{eV}}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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