De Broglie wavelength

Also called Matter wavelength

Every moving particle has a de Broglie wavelength equal to Planck's constant divided by its momentum. It grows as momentum falls, and wave behavior only matters when the wavelength is comparable to the size of the system.

The sheet prints λ=h/p\lambda = h/p, and 15.1.A.4.i adds the direction of the dependence: the de Broglie wavelength increases as the momentum of the particle decreases.

Run it on something you can hold and the answer explains itself. A 0.15 kg ball at 40 m/s has p=6.0 kg⋅m/sp = 6.0 \, \text{kg} \cdot \text{m/s}, so

λ=6.63×10−346.0≈1×10−34 m\lambda = \frac{6.63 \times 10^{-34}}{6.0} \approx 1 \times 10^{-34} \, \text{m}

That is roughly nineteen orders of magnitude smaller than an atomic nucleus. Nothing exists for such a wave to diffract around, so the ball behaves as a particle and always will. This is why duality never appears in a mechanics question.

An electron is the other case. At 1×1061 \times 10^6 m/s its momentum is 9.1×10−25 kg⋅m/s9.1 \times 10^{-25} \, \text{kg} \cdot \text{m/s} and λ≈7×10−10\lambda \approx 7 \times 10^{-10} m, which is atomic scale. That is the regime named in 15.1.A.4.ii, where quantum theory becomes necessary.

Momentum is the input, not speed, so a slow neutron and a fast electron can share a wavelength. Read backwards, the same equation gives a photon its momentum from its wavelength, which is how the Compton collision is set up.

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