Useful Equations:

01. The photo below shows the interference pattern produced by an electron double-slit experiment. In this experiment, the electrons were sent through a double-slit apparatus with an effective slit separation of 200 nm. The detector screen was 79.0 cm from the double slits. The image has been magnified by a factor of 100.

(a) Use Young’s double-slit equation to determine the wavelength of the electrons.
(b) Use the de Broglie wavelength equation to determine the momentum and velocity for the electrons passing through the apparatus.
(c) The electrons were accelerated by an electric field. Calculate the potential difference needed to produce these results.

02. The resolving power of imaging devices is limited by the wavelength of radiation used. Optical microscopes use visible light, so they can only resolve objects down to a size of about 200 nm. Electron microscopes can resolve much smaller objects because the wavelength of the electrons can be made much shorter than the wavelength of visible light.

(a) A typical transmission electron microscope (TEM) accelerates the electrons through a potential difference of 30 kV. Calculate the velocity of the electrons incident on the sample.
(b) Determine the de Broglie wavelength for these electrons.
(c) Compare the electron wavelength to the wavelength for green light (550 nm).
(d) If resolving power depended only on wavelength, what would the resolving power of this TEM be?
(e) Using the Internet, research the resolving power for a typical electron microscope.

03. A standard He-Ne laser produces about 1.0 mW of light at a wavelength of 633 nm. To create a single-photon interference experiment the laser is shone through a series of filters that reduce the beam to a small fraction of the original number of photons.

(a) Calculate the number of photons produced by the laser every second.
(b) Determine the time taken for the photons to travel 0.30 m from the filters to the detector.
(c) Each filter absorbs 96% of the photons. How many photons per second pass through after seven filters?
(d) Compare the time taken by each photon to travel 0.30 m with the time between successive photons emerging from the final filter (assume the photons are equally spaced). Express your answer as a fraction. This fraction describes the chance that there is more than one photon in flight between the filters and the detector at any one time.

04. The experiment demonstrating interference of buckminsterfullerene, C₆₀, had the molecules moving at 210 m/s. Each molecule has an atomic mass of 720 atomic units and a diameter of 1 nm. The molecules passed through slits with widths of 50 nm and separations of 100 nm. After the slits, the molecules travelled 1.25 m before being detected.

(a) What is the mass of one molecule?
(b) What is the momentum?
(c) What is its wavelength?
(d) How does this wavelength compare with the size of the molecule?
(e) How does this wavelength compare with the size of the slits?
(f) What would the distance between fringe maxima be if the screen was 5.0 m from the slits?