# 29.6 The wave nature of matter  (Page 5/6)

 Page 5 / 6

Using the definition of the de Broglie wavelength, explain how wavelike properties of matter increase with a decrease in mass or decrease in speed. Use as examples an electron (mass = 9.11 × 10 –31 kg) with a speed of 5.0 × 10 6 m/s and a proton (mass = 1.67 × 10 –27 kg) with a speed of 8.0 × 10 6 m/s.

## Test prep for ap courses

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

(c)

In a Davisson-Germer experiment, electrons with a speed of 6.5 × 10 6 m/s exhibit third-order ( n = 3) constructive interference for a crystal with unknown plane separation, d . Given an angle of incidence of θ = 45°, compute the value for d . Compare the de Broglie wavelength to electromagnetic radiation with the same wavelength. (Recall that the mass of the electron is 9.11 × 10 –31 kg.)

## Section summary

• Particles of matter also have a wavelength, called the de Broglie wavelength, given by $\lambda =\frac{h}{p}$ , where $p$ is momentum.
• Matter is found to have the same interference characteristics as any other wave.

## Conceptual questions

How does the interference of water waves differ from the interference of electrons? How are they analogous?

Describe one type of evidence for the wave nature of matter.

Describe one type of evidence for the particle nature of EM radiation.

## Problems&Exercises

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

$7.28×{\text{10}}^{–4}\phantom{\rule{0.25em}{0ex}}\text{m}$

What is the wavelength of an electron moving at 3.00% of the speed of light?

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 = ${\text{10}}^{-\text{15}}\phantom{\rule{0.25em}{0ex}}\text{m.}$ )

$6.62×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 cm (large enough for it to interfere with other billiard balls)?

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

$1.32×{\text{10}}^{–13}\phantom{\rule{0.25em}{0ex}}\text{m}$

Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. (a) What is the wavelength of such a neutron? (b) What is its kinetic energy in eV?

(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?

(a) $6.62×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

(b) $22.9 MeV$

What is the wavelength of an electron accelerated through a 30.0-kV potential, as in a TV tube?

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

15.1 keV

(a) Calculate the velocity of an electron that has a wavelength of $1\text{.}\text{00 μm.}$ (b) Through what voltage must the electron be accelerated to have this velocity?

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) 5.29 fm

(b) $4\text{.}\text{70}×{\text{10}}^{-\text{12}}\phantom{\rule{0.25em}{0ex}}\text{J}$

(c) 29.4 MV

The kinetic energy of an electron accelerated in an x-ray tube is 100 keV. Assuming it is nonrelativistic, what is its wavelength?

Unreasonable Results

(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?

(a) $7.28×{\text{10}}^{\text{12}}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

(b) This is thousands of times the speed of light (an impossibility).

(c) The assumption that the electron is non-relativistic is unreasonable at this wavelength.

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