PART 1: INVESTIGATING THE HYDROGEN ATOM
A simple model for the hydrogen atom is shown below. In this model, the electron orbits around the proton in circular shells. The first shell represents the lowest energy level and is called the ground state. When the hydrogen atom is in an excited state, the electron will occupy one of the higher shells. To drop down to a lower energy level the electron must emit a photon that has the same amount of energy as the electron needs to lose.
01. The ground state for hydrogen is –13.6 eV. The average radius of the ground state is 5.29 x 10–11 m. Calculate the wavelength of the electron when it is in the ground state. (Hint: Etotal = Ekinetic + Epotential)
02. The wavelength of light emitted by the electron when it drops from the second excited state to the ground state is 102.4 nm. Determine the energy of the electron when it is in the second excited state.
03. The average radius of the second excited state is 4.76 x 10-10 m. Calculate the wavelength of the electron when it is in this state.
04. The energy levels allowed are those that have integer values for the electron wavelengths. How many complete wavelengths of the electron fit around the orbits of the ground state and the second excited state?
PART 2: FRANCK-HERTZ EXPERIMENT
The Franck–Hertz experiment demonstrates the quantum nature of electrons and light. In this Nobel Prize winning experiment, electrons are accelerated through a low pressure gas. As the electrons accelerate through the electric field they gain kinetic energy. At very specific distances, we observe bands of monochromatic light. Changing the potential difference, gas pressure, tube length, or gas used will produce changes in the colour or location of the glowing bands.
The photons are produced by the excited electrons in the gas molecules when they drop back down after colliding with the accelerated electrons. The energy gained by each electron as it accelerates through the electric field can be determined by
where
EK = the kinetic energy of the electron (J)
q = the charge on the electron (C)
V = the potential difference applied across the tube (V)
Δx = the distance between successive bands (m)
L = the distance from anode to cathode (m)
01. Show how the above equation can be derived from the work done on the electron by the electric field.
02. A Franck–Hertz tube containing low-pressure neon has a potential difference of 22.0 V applied between anode and cathode. The gap is 12.5 mm. The distance from the anode to the first bright band of light (and any successive bands of light) is 1.19 mm.
(a) Determine the kinetic energy of an electron when it reaches the first bright band.
(b) Calculate the frequency of light being emitted. What wavelength and colour does this equate to?
(c) What happens to the accelerated electron after it loses all of its kinetic energy to the gas molecule? (Hint: Why is there a series of bright monochromatic bands?)
(d) What will happen to the spacing between the light bands if the voltage is doubled?
03. The neon gas is replaced with mercury. When 13.4 V is applied across the 12.5 mm gap between anode and cathode, the wavelength of light emitted is 253 nm. How many bands will be produced? (Note: these bands would not be visible because 253 nm is outside the visible spectrum.)
04. What does the Franck–Hertz experiment tell us about the structure of the atom?
05. What does the Franck–Hertz experiment tell us about the quantum nature of light?
PART 3: INVESTIGATING THE LIMIT OF QUANTUM OBSERVATION
The observation of quantum interference is limited by an experimenter’s ability to isolate the experiment from the environment. This activity investigates how thermal interactions would affect the interference pattern in a hypothetical double-slit experiment. The experimental set-up consists of three distinct components:
• A hot filament emits electrons in all directions.
• A series of collimating filters creates a uniform “pencil” beam of electrons travelling in one direction.
• The electrons in the “pencil” beam gain a small, perpendicular drift velocity, with the average velocity related to the drift temperature, Tdrift.
01. The electron source is heated to 7500 K and ejects electrons with a velocity of 5.84 x 105 m/s. What is the de Broglie wavelength of the electrons in the beam?
02. The electrons emerge from the collimating filters as a uniform “pencil” beam. They pass through slits that are 200 nm apart and then produce an interference pattern on the detector screen 1.0 m away. Use equation 2 to determine the spacing of the interference maxima, Δx = Δxinterference.
03. The electrons in the uniform “pencil” beam will gain a small amount of kinetic energy from unavoidable interactions with the environment expressed by equation 4. For this exercise only the component of the drift velocity that is perpendicular to the original direction of travel (Figure 4.4) will be considered. Use equations 3 and 4 to derive the drift velocity equation:
04. The magnitude of the average drift velocity, and resulting drift distance, Δxdrift, is dependent on the environmental interactions. Show that
and then calculate the drift distance, Δxdrift, if the drift temperature is 0.25 K.
05. Use a sketch to explain how the interference pattern will change if the drift velocity gets too large. Explain why the interference pattern is washed out when
06. Prove that the drift temperature that will just wash out the interference pattern is given by:
07. Calculate the drift temperature that will just wash out the interference pattern using the data from question 02.
08. Consider the relationship derived in question 06. Why does the mass of the object limit the visibility of an interference pattern in a realistic double-slit experiment?
