Mistake Master
Colour sets the energy, brightness sets the count
Two knobs, and they are independent. Frequency sets the energy of each photon, $E = hf$. Intensity sets how many arrive per second. A dim blue beam delivers few but energetic photons; a bright red beam delivers many weak ones. Every threshold in this unit is a threshold in the first knob, and turning the second one up never gets you across it.
§1
Energy per photon and photons per second.
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$$E = hf = \frac{hc}{\lambda}.$$
Nothing about brightness appears in that. At $500$ nm every photon carries $2.5$ eV whether the source is a candle or a searchlight, and doubling the lamp's power doubles the number of photons per second.
- Want more energetic photons? Change the colour.
- Want more photons? Change the power.
Those are separate quantities, and separate experiments measure them: a stopping-voltage measurement reads the energy per photon, and a photocurrent reads the arrival rate. Collapsing the two makes every threshold in this unit look like a brightness threshold, which is exactly the prediction classical wave theory made and experiment refuted.
§2
Pick a matched unit pair and stay in it.
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$E = hf$ and $E = hc/\lambda$ give the same answer only when the constant and the wavelength carry matching units. There are two consistent sets:
- $h = 6.63\times10^{-34}$ J$\cdot$s, with $\lambda$ in metres, giving an answer in joules.
- $hc = 1240$ eV$\cdot$nm, with $\lambda$ in nanometres, giving an answer in electronvolts.
Blending them, or feeding nanometres into the metre formula, throws the exponent about nineteen orders of magnitude off with nothing in the algebra to flag it.
So convert first, or pick the matched pair. For $400$ nm light: $4.00\times10^{-7}$ m gives $4.97\times10^{-19}$ J, and $1240/400$ gives $3.1$ eV. The same energy, since $1$ eV $= 1.60\times10^{-19}$ J.
The size check that catches every version of this: visible photons sit near a few eV. An answer far from that range is a unit slip rather than a physics result.
§3
One description, two questions you can ask of it.
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Light does not switch between being a wave and being a particle. There is one description, and which behaviour shows up depends on what the experiment measures.
- Ask about a pattern on a screen, and the wave description answers: interference through two slits.
- Ask about energy delivered to one electron, and the photon description answers: the photoelectric effect, Compton scattering.
The same beam does both, and nothing about the light changes between the two questions. The experiment that closes off the switching story: send photons through a double slit one at a time, and each arrives as a single dot while the accumulated dots build the interference pattern.
Electrons behave the same way, diffracting through a double slit and still landing as single dots. So this is not a peculiarity of light; it is how everything works at this scale.
§4
de Broglie applies to everything, and it is inverse.
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$$\lambda = \frac{h}{p}.$$
Every particle with momentum has one. Restricting it to electrons because that is where you first met it leaves out protons, neutrons and whole molecules, all of which have been diffracted in the laboratory.
And it is inverse: tripling the momentum cuts the wavelength to a third, so a faster electron diffracts less. Speeding a particle up to spread its fringes runs the relation backwards.
The reason wave behaviour is invisible in everyday life falls straight out of the same formula. A $0.1$ kg ball at $10$ m/s has
$$\lambda = \frac{6.63\times10^{-34}}{1} \approx 7\times10^{-34}\ \text{m},$$
which is far below the size of anything it could pass through. So the working test is a comparison: compare $\lambda$ with the size of the system, and if it is vastly smaller, the classical description is enough.
§5
Skill Check.
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Ten scenarios. Pick the chips that match your answer, then check. A scenario marks complete the first time every part is right. Progress saves on this device.