9. Quantum Mechanics

The speed of sound in dry air at 20 °C is 343 m/s and the lowest frequency sound wave that the human ear can detect is approximately 20 Hz. (b) What would be the frequency of electromagnetic radiation with the same wavelength?

Use the de Broglie relationship to determine the wavelengths of the following objects: (a) an 85-kg person skiing at 50 km/hr

Use the de Broglie relationship to determine the wavelengths of the following objects: (d) an ozone 1O32 molecule in the upper atmosphere moving at 550 m/s.

Among the elementary subatomic particles of physics is the muon, which decays within a few microseconds after formation. The muon has a rest mass 206.8 times that of an electron. Calculate the de Broglie wavelength associated with a muon traveling at 8.85 * 105 cm/s.

The resolution limit of a microscope is roughly equal to the wavelength of light used in producing the image. Electron microscopes use an electron beam (in place of photons) to produce much higher resolution images, about 0.20 nm in modern instruments. Assuming that the resolution of an electron microscope is equal to the de Broglie wavelength of the electrons used, to what speed must the electrons be accelerated to obtain a resolution of 0.20 nm?

Neutron diffraction is an important technique for determining the structures of molecules. Calculate the velocity of a neutron needed to achieve a wavelength of 125 pm. (Refer to the inside cover for the mass of the neutron.)

The smallest atoms can themselves exhibit quantum-mechanical behavior. Calculate the de Broglie wavelength (in pm) of a hydrogen atom traveling at 475 m/s.

A proton in a linear accelerator has a de Broglie wavelength of 122 pm. What is the speed of the proton?

Calculate the de Broglie wavelength of a 143-g baseball traveling at 95 mph. Why is the wave nature of matter not important for a baseball?

A 0.22-caliber handgun fires a 1.9-g bullet at a velocity of 765 m>s. Calculate the de Broglie wavelength of the bullet. Is the wave nature of matter significant for bullets?

Protons and electrons can be given very high energies in particle accelerators. What is the wavelength in meters of an electron (mass = 9.11 * 10-31 kg) that has been accelerated to 5% of the speed of light? In what region of the electromagnetic spectrum is this wavelength?

What is the de Broglie wavelength in meters of a baseball weighing 145 g and traveling at 156km/h? Why do we not observe this wavelength?

At what speed in meters per second must a 145 g baseball be traveling to have a de Broglie wavelength of 0.500 nm?

What velocity would an electron (mass = 9.11 * 10-31 kg) need for its de Broglie wavelength to be that of red light (750 nm)?

The first 25 years of the twentieth century were momentous for the rapid pace of change in scientists' understanding of the nature of matter. (b) In what ways is de Broglie's hypothesis, as it applies to electrons, consistent with J. J. Thomson's conclusion that the electron has mass? In what sense is it consistent with proposals preceding Thomson's work that the cathode rays are a wave phenomenon?

X rays with a wavelength of 1.54 * 10-10 m are produced when a copper metal target is bombarded with high-energy electrons that have been accelerated by a voltage difference of 30,000 V. The kinetic energy of the electrons equals the product of the voltage difference and the electronic charge in coulombs, where 1 volt-coulomb = 1 J. (a) What is the kinetic energy in joules and the de Broglie wavelength in meters of an electron that has been accel-erated by a voltage difference of 30,000 V?

(b) Light with a wavelength of 2.50 * 10-7 m falls on a piece of chromium in an evacuated glass tube. What is the minimum de Broglie wavelength of the emitted elec-trons? (Note that the energy of the incident light must be conserved; that is, the photon's energy must equal the sum of the energy needed to eject the electron plus the kinetic energy of the electron.)

(c) What is the velocity of an electron with a de Broglie wavelength equal to (b)?