State the de Broglie wavelength relation.

What happens to a particle's de Broglie wavelength when its momentum increases?

Why can electrons diffract from crystals?

An electron has momentum \(2.00\times10^{-24}\,\mathrm{kg\,m\,s^{-1}}\). Find its de Broglie wavelength.

A particle has de Broglie wavelength \(1.50\times10^{-10}\,\mathrm m\). Find its momentum.

An electron is accelerated from rest through \(100\,\mathrm V\). Find its kinetic energy in electronvolts and joules.

An electron is accelerated through \(150\,\mathrm V\). Estimate its nonrelativistic de Broglie wavelength.

A nonrelativistic electron's kinetic energy is quadrupled. By what factor does its de Broglie wavelength change?

Use \(K=p^2/(2m)\) to derive \(\lambda=h/\sqrt{2mK}\).

A crystal has plane spacing \(0.200\,\mathrm{nm}\). Electrons of wavelength \(0.100\,\mathrm{nm}\) scatter from it. Find the first-order Bragg angle.

Derive the wavelength of an electron accelerated from rest through potential difference \(V\).

What accelerating voltage gives an electron a nonrelativistic wavelength of \(0.100\,\mathrm{nm}\)?

A proton and an electron have the same speed. Which has the shorter de Broglie wavelength?

A proton and an electron have the same nonrelativistic kinetic energy. Which has the shorter de Broglie wavelength?

Why are macroscopic matter waves usually unobservable?

Show that the de Broglie relation gives a standing-wave condition for a circular orbit.

Explain why observing electron diffraction supports the wave-particle model rather than a purely particle model.

An electron wavelength is comparable to atomic spacing. Explain why this matters for microscopy.

A nonrelativistic electron wavelength calculation gives a speed close to \(c\). What should you change?

Derive how electron wavelength scales with accelerating voltage and state the physical meaning.