6.6 Wave-Particle Duality

For the situation described in the example, find the angular position of the fifth-order bright fringe on the viewing screen.

Suppose that the diameter of the aperture in the example is halved. How does it affect the resolving power?

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

Give at least one argument in support of the particle-nature of radiation.

Does the Heisenberg uncertainty principle allow a particle to be at rest in a designated region in space?

Do the photons of red light produce better resolution in a microscope than blue light photons? Explain.

An AM radio transmitter radiates 500 kW at a frequency of 760 kHz. How many photons per second does the emitter emit?

Find the Lorentz factor $\gamma$ and de Broglie’s wavelength for a 1.0-TeV proton in a particle accelerator.

If electron is to be diffracted significantly by a crystal, its wavelength must be about equal to the spacing, d, of crystalline planes. Assuming $d = 0.250 \text{nm} ,$ estimate the potential difference through which an electron must be accelerated from rest if it is to be diffracted by these planes.

Solar wind (radiation) that is incident on the top of Earth’s atmosphere has an average intensity of $1.3 \text{kW} / \text{m}^{2} .$ Suppose that you are building a solar sail that is to propel a small toy spaceship with a mass of 0.1 kg in the space between the International Space Station and the moon. The sail is made from a very light material, which perfectly reflects the incident radiation. To assess whether such a projec…

Show that Wien’s displacement law results from Planck’s radiation law. (Hint: substitute $x = h c / \lambda k T$ and write Planck’s law in the form $I \left(x , T\right) = A x^{5} / \left(e^{x} - 1\right) ,$ where $A = 2 \pi \left(k T\right)^{5} / \left(h^{4} c^{3}\right) .$ Now, for fixed T, find the position of the maximum in I(x,T) by solving for x in the equation $d I \left(x , T\right) / d x = 0 .$)

Determine the power intensity of radiation per unit wavelength emitted at a wavelength of 500.0 nm by a blackbody at a temperature of 10,000 K.

A quantum mechanical oscillator vibrates at a frequency of 250.0 THz. What is the minimum energy of radiation it can emit?

A sodium lamp emits 2.0 W of radiant energy, most of which has a wavelength of about 589 nm. Estimate the number of photons emitted per second by the lamp.

If the work function of a metal is 3.2 eV, what is the maximum wavelength that a photon can have to eject a photoelectron from this metal surface?

A 400-nm laser beam is projected onto a calcium electrode. The power of the laser beam is 2.00 mW and the work function of calcium is 2.31 eV. (a) How many photoelectrons per second are ejected? (b) What net power is carried away by photoelectrons?

A laser with a power output of 2.00 mW at a 400-nm wavelength is used to project a beam of light onto a calcium photoelectrode. (a) How many photoelectrons leave the calcium surface per second? (b) What power is carried away by ejected photoelectrons, given that the work function of calcium is 2.31 eV? (c) Calculate the photocurrent. (d) If the photoelectrode suddenly becomes electrically insulate…

(a) Calculate the wavelength of a photon that has the same momentum as a proton moving with 1% of the speed of light in a vacuum. (b) What is the energy of this photon in MeV? (c) What is the kinetic energy of the proton in MeV?

The momentum of light, as it is for particles, is exactly reversed when a photon is reflected straight back from a mirror, assuming negligible recoil of the mirror. The change in momentum is twice the photon’s incident momentum, as it is for the particles. Suppose that a beam of light has an intensity $1.0 \text{kW} / \text{m}^{2}$ and falls on a $−2.0 -\text{m}^{2}$ area of a mirror and reflects from it. (a) Calculate the energy reflected in 1…

A 0.75-nm photon is scattered by a stationary electron. The speed of the electron’s recoil is $1.5 \times 10^{6} \text{m}/\text{s} .$ (a) Find the wavelength shift of the photon. (b) Find the scattering angle of the photon.

A photon of wavelength 700 nm is incident on a hydrogen atom. When this photon is absorbed, the atom becomes ionized. What is the lowest possible orbit that the electron could have occupied before being ionized?

Singly ionized atomic helium $\text{He}^{+ 1}$ is a hydrogen-like ion. (a) What is its ground-state radius? (b) Calculate the energies of its four lowest energy states. (c) Repeat the calculations for the $\text{Li}^{2 +}$ ion.

In extreme-temperature environments, such as those existing in a solar corona, atoms may be ionized by undergoing collisions with other atoms. One example of such ionization in the solar corona is the presence of $\text{C}^{5 +}$ ions, detected in the Fraunhofer spectrum. (a) By what factor do the energies of the $\text{C}^{5 +}$ ion scale compare to the energy spectrum of a hydrogen atom? (b) What is the wavelength of the fir…

Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. Find the wavelength of such an ultracold neutron and its kinetic energy.

The spacing between crystalline planes in the NaCl crystal is 0.281 nm, as determined by X-ray diffraction with X-rays of wavelength 0.170 nm. What is the energy of neutrons in the neutron beam that produces diffraction peaks at the same locations as the peaks obtained with the X-rays?

Calculate the velocity of a $1.0 -\mu \text{m}$ electron and a potential difference used to accelerate it from rest to this velocity.

Find the de Broglie wavelength of an electron accelerated from rest in an X-ray tube in a potential difference of 100 keV. (Rest mass energy of an electron is $E_{0} = 511 \text{keV} .)$

Compare the wavelength shift of a photon scattered by a free electron to that of a photon scattered at the same angle by a free proton.

Consider a hydrogen-like ion where an electron is orbiting a nucleus that has charge $q = + Z e .$ Derive the formulas for the energy $E_{n}$ of the electron in nth orbit and the orbital radius $r_{n} .$

An atom can be formed when a negative muon is captured by a proton. The muon has the same charge as the electron and a mass 207 times that of the electron. Calculate the frequency of the photon emitted when this atom makes the transition from $n = 2$ to the $n = 1$ state. Assume that the muon is orbiting a stationary proton.