29.6 The Wave Nature of Matter

The least massive particle known to exist is the electron neutrino. Though scientists once believed that it had no mass, like the photon, they have now determined that this particle has an extremely low mass, equivalent to a few electron volts. Assuming a mass of 2.2 eV/c² (or 3.9 × 10^–36 kg) and a speed of 4.4 × 10⁶ m/s, which of the following values equals the neutrino’s de Broglie wavelength?

1. …

In a Davisson-Germer type of experiment, a crystal with a parallel-plane separation (d) of 9.1 × 10^–2 nm produces constructive interference with an electron beam at an angle of θ = 50°. Which of the following is the maximum de Broglie wavelength for these electrons?

1. 0.07nm
2. 0.09 nm
3. 0.14 nm
4. 0.21 nm

At what velocity will an electron have a wavelength of 1.00 m?

At what velocity does a proton have a 6.00-fm wavelength (about the size of a nucleus)? Assume the proton is nonrelativistic. (1 femtometer = $\text{10}^{- \text{15}} \text{m}.$)

Find the wavelength of a proton moving at 1.00% of the speed of light.

(a) Find the velocity of a neutron that has a 6.00-fm wavelength (about the size of a nucleus). Assume the neutron is nonrelativistic. (b) What is the neutron’s kinetic energy in MeV?

What is the kinetic energy of an electron in a TEM having a 0.0100-nm wavelength?

The velocity of a proton emerging from a Van de Graaff accelerator is 25.0% of the speed of light. (a) What is the proton’s wavelength? (b) What is its kinetic energy, assuming it is nonrelativistic? (c) What was the equivalent voltage through which it was accelerated?

(a) Assuming it is nonrelativistic, calculate the velocity of an electron with a 0.100-fm wavelength (small enough to detect details of a nucleus). (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?