7.2.1 The Wave Nature of Matter

Question 1

The Wave Nature of Matter

Answer 1

• In 1924, de Broglie postulated that matter, just like light, can have both wave and particle properties.
• Electrons produce an interference pattern when passed through a crystal.
• Electrons are standing waves with quantized energy levels.

Question 2

note

Answer 2

• In 1924, de Broglie postulated that matter, just like light, can have both wave and particle properties.
• Specifically, de Broglie calculated that the wavelength of a particle is equal to Planck’s constant (h) divided by the momentum ( ) of the particle. Momentum is equal to mass multiplied by velocity. When two particles with the same mass travel at different velocities, the slower particle has a longer wavelength. When two particles with different masses travel at the same velocity, the larger particle has a smaller wavelength.
• Using this formula, the de Broglie wavelength of an electron traveling at 1.0 x 10 6 m/s can be calculated. Planck’s constant is 6.63 x 10 –34 kg · m2, and the mass of an electron is 9.109 x 10 –31 kg. Plugging in these values yields a wavelength of 7.3 x 10 –10 m. This is on the order of the size of an atom.
• In 1927, Davisson and Germer proved that particles
  (specifically electrons) have wave properties.
• When waves are passed through small slits, an interference pattern is formed. Davisson and Germer passed electrons through a crystal, and found an interference pattern. The wave nature of the electron was exactly as de Broglie had predicted.
• A standing wave vibrates in a fixed region. Standing waves (such as guitar strings) can only have certain frequencies (the fundamental, second harmonic, third harmonic, etc). The frequencies are therefore quantized.
• Electrons are standing waves. If the wave reinforces itself (such as in the example on the left), that energy level is permitted. However, if the wave does not reinforce itself (such as the example on the right), that energy level is not permitted.
• The energy of the electron is quantized because of the wave properties associated with it.

Question 3

What will happen when two identical waves that are out of phase with each other combine?

Answer 3

The resulting wave will have zero amplitude.

Question 4

Which of the following is the best explanation of why the energy of electrons is quantized?

Answer 4

Wavelengths of electrons that result in destructive interference are disallowed near the nucleus.

Question 5

A string has a length of 84 cm. The string is stretched taut, and both ends are restricted to be nonmoving. Touching the string at which of the following points will not produce a standing wave when the string is plucked?

Answer 5

36 cm from one end

Question 6

Which of these particles will have the shortest de Broglie wavelength?

Answer 6

A proton with a velocity of 9.0 × 10^5 m / s

Question 7

In which of the following ways does an electron in a hydrogen atom differ from a radio wave?
(The mass of an electron is 9.11 × 10^−31 kg.)

Answer 7

The electron will have a slower velocity.

Question 8

Suppose a beam of neutrons is fired at a crystal that has a spacing of 0.215 nm between atoms. Photographic film is used to record the places where the neutrons come through the crystal.

Which of the following is the best estimate of the slowest speed at which the neutrons can be fired at the crystal and still result in an interference pattern on the film? (The mass of a neutron is 1.67 × 10−27 kg.)

Answer 8

1.85 × 10^3 m / s

Question 9

What would the results of Davisson and Germer’s experiment have been if they had used X-rays with a wavelength of 1 angstrom instead of a beam of electrons? (In their experiment, electrons were fired at a crystal of magnesium oxide, and photographic film was used to record the places where the electrons came through the crystal.)

Answer 9

An interference pattern would have been seen on the film.

Question 10

An atom of helium
has a de Broglie wavelength of
4.3 × 10^−12 meters.

What is its velocity?

Answer 10

2.3 × 10^4 m / s

Question 11

A molecule of oxygen gas (O2 ) has a velocity of 2800 m / s.

What is its de Broglie wavelength?

Answer 11

4.5 × 10^−12 m

Question 12

Suppose the two waves shown below were to combine in phase with one another, what will the resulting wave look like?

Answer 12

When two identical waves combine in phase, they constructively interfere with one another. In other words, one wave reinforces the other. The frequency and wavelength will not change, but the amplitudes of the two waves (the height of their peaks) will add together. Thus, the amplitude of the resulting wave will be twice that of the original waves.