Exercises 7.7 Problems
1.
The so-named Paschen series of radiation emitted by hydrogen corrersponds to transitions from states of principle quantum numbers \(n = 4\text{,}\) 5, 6, …, to the state \(n = 3\text{.}\)
Calculate the frequencies and wavelengths of the three lowest energy photons emitted from hydrogen in the Paschen series.
In what region of the electromagnetic spectrum are these photons?
2.
An electron in a hydrogen atom is in a state of principle quantum number \(n = 4\text{.}\) Using the notation (\(n, l, m_l, m_s\)) to specify a state, write all of the possible states for this electron. [ HINT: There should be a total of \(32\) states.]
3.
Find the maximum possible magnitude for the orbital angular momentum \(|\vec{L}|\) of an electron in the state of principle quantum number \(n = 7\) of a hydrogen atom.
4.
The orbital angular momentum of the electron in a hydrogen atom has a magnitude \(|\vec{L}| = 2.585 \times 10^{-34} \Xunits{J \cdot s}\text{.}\) What is the minimum possible energy for this electron in the hydrogen atom?
5.
Determine the principle quantum number \(n\) and orbital quantum number \(l\) for a hydrogen atom whose electron has energy \(-0.850 \Xunits{eV}\) and orbital angular momentum \(|\vec{L}| = \sqrt{12} \hbar\text{.}\)
6.
Show that the probability that a state with energy \(\Delta E\) above the Fermi energy will be occupied is equivalent to the probability that a state with energy \(\Delta E\) below the Fermi energy will be unoccupied. Explain in 1 or 2 sentences why this result makes sense intuitively, when considering how conduction works in a semiconductor.
7.
A one-dimensional periodic potential composed of 10 side-by-side finite square well potentials has a valence band with 10 energy levels from a low of \(0.81\Xunits{ eV}\) to a high of \(1.10\Xunits{eV}\) and a conduction band with 10 levels ranging from \(3.19\Xunits{eV}\) to \(4.39\Xunits{eV}\) . The Fermi energy (in the middle of the band gap) is \(2.145\Xunits{eV}\) for this system. Assume that there are 20 electrons in this system (this is analogous to 10 atoms with 2 valence electrons each).
Calculate the probability that an electron will be found in the energy state \(E_{11} = 3.19\Xunits{eV}\) (i.e., in the lowest conduction band state) for \(T = 300\Xunits{K}\)(room temperature), \(3,000\Xunits{K}\text{,}\) and \(30,000\Xunits{K}\text{.}\)
Calculate the values of \(k_BT\) for each of the temperatures from part (a). Do the probabilities that you calculated in (a) make sense, given these values for \(k_BT\text{?}\)
Now, download the Excel sheet “FDForTenWells.xlsx” from the calendar page. This worksheet calculates Fermi-Dirac probabilities for electrons to occupy any of the 10 conduction band energies. The sheet also calculates an expected average number of electrons in each of these levels (by multiplying the probability by 2, since there are two energy levels per state) and adds up the total number of electrons expected in the conduction band. Use this worksheet to calculate the estimated number of conduction band electrons for \(T = 300\text{,}\) 3000, and \(30000\Xunits{K}\text{.}\) For each of these temperatures, would you expect this system to be an electrical insulator or electrical conductor?
if there were only 10 electrons (e.g., 10 atoms with only 1 valence electron instead of 2), how would that affect the conduction properties of this system?
8.
Lead sulfide (PbS) has a band gap energy of \(0.37\Xunits{eV}\text{.}\)
Calculate the value of \(k_BT\) for temperatures \(30\Xunits{K}\text{,}\) \(300\Xunits{K}\) (room temperature), \(3,000\Xunits{K}\) and \(30,000\Xunits{K}\text{.}\) At these temperatures would you expect PbS to be a poor, moderate or good electrical conductor?
Use the Fermi-Dirac relation to calculate the probability of the lowest conduction band energy state being occupied for temperatures 30, 300, 3,000 and \(30,000\Xunits{K}\text{.}\) Are these results consistent with your answers from (a)? (Note: it's not necessary for every conduction band level to be filled for a material to be a reasonable conductor, but the probabilities should at least be on the order of \(10^{-5}\) or better.)
9.
Some recent studies have investigated the possibility of using DNA strands as fundamental building blocks of nanotechnology. One possibility is the use of DNA for nanoelectronic devices. Experimental studies of a particular DNA sequence (Poly(dA)*Poly(dT) with a B-type structure 1 ) measured a band gap of \(2.7\Xunits{eV}\) between valence and conduction bands. Would you expect this DNA strand to be a good electrical conductor at room temperature? Explain. (A “\(k_BT\)” analysis is sufficient — you don't need to use Fermi-Dirac distributions here.) If it is a good conductor, approximate how low a temperature would be needed to make it a poor conductor. If it is a poor conductor, approximate how large a temperature would be needed to make it a good conductor.
10.
A silicon photodiode has a band gap energy of \(1.17\Xunits{eV}\text{.}\)
Calculate the maximum wavelength of electromagnetic radiation that this photodiode can detect.
Why can't a silicon photodiode detect EM radiation with wavelengths larger than the value calculated in part (a)?
Give an argument as to how a silicon photodiode can detect EM radiation with wavelengths smaller than the value calculated in part (a).
11.
Calculate the wavelength (and state the color — or “IR” or “UV” if infrared or ultraviolet) of electromagnetic radiation emitted from a light-emitting diode (LED) made from the following materials:
Gallium arsenide (band gap \(1.43\Xunits{eV}\));
Gallium (III) Phosphide (band gap \(2.26 \Xunits{eV}\));
Gallium nitride (band gap \(3.4\Xunits{eV}\)).
12.
White light LED bulbs can be made by using blue or violet LEDs that are coated with phosphor (fluorescent) coatings. (a) Explain in a few sentences how this kind of approach can produce light that appears white. (b) Why would it not be possible to make a white light LED with a red LED as the source? (Note: white-light LED bulbs can also be made with coatings of quantum dots or can be made with combinations of red, green and blue LEDs.)
13.
Draw a sketch of a p-n junction in the absence of any imposed electric fields. In the sketch, show the depletion zone (DZ) and sketch mobile carriers within the p- and n-type semiconductors away from the DZ. Make sure that you have the correct type of charge carriers (+ or -) in the correct material.
Now, make a similar sketch, but for the case with an applied electric field pointing from the n- to the p-type semiconductor. Is the DZ larger or smaller in this case than for the case with \(E = 0\) from part (a) Explain why. Do you expect this p-n junction to conduct electricity readily in the direction of this electric field? Explain.
Repeat part (b), but for an electric field pointing from the p- to the n-type semiconductor, answering the same questions.
Does this device act like a diode; i.e., a device that passes current readily in one direction but not in the other? In what ways does a p-n junction deviate from an “ideal” diode?
14.
Transistors. Figure 7.11(a) is a sketch of a sandwich made of 2 n-type semiconductors with a thin p-type layer in between.
A wire is connected to each of the ends of this npn sandwich. Give a short explanation as to why this npn device would not be expected to conduct electricity in either direction. (Hint: consider what happens to the DZ at each junction if you apply an electric field in either direction.)
A third wire is connected to the right pn junction. If a very small current is fed into the device through this third wire, there can be a significant conduction of electricity between the two end wires if the current is going to the left. Explain why. (Hint: consider the depletion zone and what happens if you inject some electrons or holes into it.)
Instead of a third wire, you can make a phototransistor that turns on and off the current between the ends by shining light on the p-type layer. Explain why light can turn on the current. (Hint: this will only work if the light has a wavelength smaller than a particular critical value.)
15.
The wavefunction for the ground state of the hydrogen atom \hbox{(\(n=1, l=0, m_l=0\))} is
where \(a_0\) is a constant which can be written in terms of fundamental constants as \(a_0 = \frac{\hbar^2}{m_e k e^2}\text{.}\)
-
Show that this wavefunction is a solution to the 3D Schrödinger equation
\begin{equation*} -\frac{\hbar^2}{2mr^2} \left[ \frac{\partial}{\partial r} \left( r^2 \frac{\partial \psi}{\partial r} \right) + \frac{1}{\sin{\theta}} \frac{\partial}{\partial \theta} \left( \sin{\theta} \frac{\partial \psi}{\partial \theta} \right) + \frac{1}{\sin^2{\theta}} \frac{\partial^2 \psi}{\partial \phi^2} \right] - \frac{k e^2}{r} \psi = E \psi \end{equation*}(Hint: since there are no \(\theta\) or \(\phi\) terms in the wavefunction, the partial derivatives \(\partial \psi/\partial \theta\) and \(\partial \psi/\partial \phi\) are both zero. That will simplify things a lot.)
Determine the value of \(E\) required for \(\psi_{100}\) to be a solution. Compare your result with (7.4).
16.
The ground state wavefunction of the electron in the hydrogen atom is spherically symmetric which means that the wavefunction \(\psi(r)\) can be written solely in terms of the radial coordinate \(r\) representing the distance between the proton and electron.
What does the quantity \(|\psi(r)|^2\) mean physically?
Show that the volume of a thin spherical shell of radius \(r\) and thickness \(dr\) is \(4\pi r^2\, dr\text{.}\) (You can use the approximation for small \(dr\) that the volume is the surface area of the sphere times \(dr\text{.}\))
-
In spherical coordinates, the ground state solution of the Schrödinger equation for the hydrogen atom is
\begin{equation*} \psi_{100} = \frac{1}{\sqrt{\pi a_0^3}}\, e^{-r/a_0}\text{,} \end{equation*}where \(a_0\) is the same constant as from the previous problem. Use the result of part (b) to write an expression for the probability that the electron is in a spherical shell of radius \(r\) and thickness \(dr\text{.}\)
Calculate the radius of the shell (of constant thickness \(dr\)) where the electron is most likely to be found.