29.8 The Particle-Wave Duality Reviewed

Which of the following describes one of the main features of wave-particle duality?

1. As speed increases, the wave nature of matter becomes more evident.
2. As momentum decreases, the particle nature of matter becomes more evident.
3. As energy increases, the wave nature of matter becomes easier to observe.
4. As mass increases, the wave nature of matter is less easy to observe.

The 54.0-eV electron in the reference has a 0.167-nm wavelength. If such electrons are passed through a double slit and have their first maximum at an angle of $\text{25} . 0º$, what is the slit separation $d$?

A certain heat lamp emits 200 W of mostly IR radiation averaging 1500 nm in wavelength. (a) What is the average photon energy in joules? (b) How many of these photons are required to increase the temperature of a person’s shoulder by $2 . 0º \text{C}$, assuming the affected mass is 4.0 kg with a specific heat of $0 .\text{83 kcal} /\text{kg} \cdot º\text{C}$. Also assume no other significant heat transfer. (c) How long does this take?

(a) Calculate the amount of microwave energy in joules needed to raise the temperature of 1.00 kg of soup from $\text{20} . 0º \text{C}$ to $\text{100} º\text{C}$. (b) What is the total momentum of all the microwave photons it takes to do this? (c) Calculate the velocity of a 1.00-kg mass with the same momentum. (d) What is the kinetic energy of this mass?

(a) What is $\gamma$ for a proton having an energy of 1.00 TeV, produced by the Fermilab accelerator? (b) Find its momentum. (c) What is the proton’s wavelength?

(a) Calculate the velocity of electrons that form the same pattern as 450-nm light when passed through a double slit. (b) Calculate the kinetic energy of each and compare them. (c) Would either be easier to generate than the other? Explain.

(a) What is the separation between double slits that produces a second-order minimum at $\text{45} . 0º$ for 650-nm light? (b) What slit separation is needed to produce the same pattern for 1.00-keV protons.

One problem with x rays is that they are not sensed. Calculate the temperature increase of a researcher exposed in a few seconds to a nearly fatal accidental dose of x rays under the following conditions. The energy of the x-ray photons is 200 keV, and $4 . \text{00} \times \text{10}^{\text{13}}$ of them are absorbed per kilogram of tissue, the specific heat of which is $0 . \text{830 kcal}/\text{kg} \cdot º\text{C}$. (Note that medical diagnostic x-ray machines cannot produce an int…

The momentum of light is exactly reversed when reflected straight back from a mirror, assuming negligible recoil of the mirror. Thus the change in momentum is twice the photon momentum. Suppose light of intensity $1 . \text{00 kW}/\text{m}^{2}$ reflects from a mirror of area $2 . \text{00 m}^{2}$. (a) Calculate the energy reflected in 1.00 s. (b) What is the momentum imparted to the mirror? (c) Using the most general form of Newton’s second law,…

A photon of light with a wavelength of 550.0 nm is involved in a collision with an electron. Use $6.63 \times 10^{−34} \text{m}^{2} \text{kg}/\text{s}$ for Planck’s constant. (a) How much momentum does the photon have before the collision? (b) If the wavelength of the photon leaving the collision is 650.0 nm, how much momentum does the photon have? (c) Does the momentum have to be conserved in every direction? (d) Does a longer wavelength imply mor…