Exercises 4.7 Problems
1.
Draw a sketch of the wavefunction \(\psi(x)\) for the \(3^\text{ rd }\) lowest energy state for an electron trapped in a one-dimensional, infinite square well potential (i.e., the “particle in a box”).
Interpreting your drawing from part (a) as a standing wave pattern, determine the wavelength \(\lambda\) (in terms of the width \(L\) of the box) of this standing wave.
Substitute the wavefunction \(\psi(x) = \sqrt{2/L}\sin(2\pi x/\lambda)\) with your value of \(\lambda\) from (b) into Schrödinger's Equation for the infinite square well potential. Verify that this is a solution for the region \(0 \lt x \lt L\text{,}\) and determine the energy of the state. Compare your result with what you get when using the formula from (4.6).
2.
Find the energies (in eV) of an electron in (a) the ground state; and (b) the first excited state of a \(1.0 \Xunits{nm}\) one-dimensional infinite square well potential.
3.
A particle is in the ground state of an infinite square well between \(x = 0\) and \(x = L\text{.}\) What is the probability of finding the particle in the region between \(x=0\) and \(x= L/3\) ? (NOTE: You may want to make use of the Table of Integrals in Appendix A of your Wolfson text for this problem!)
4.
Sketch the probability density for the \(n=2\) state (first excited state) of an infinite square well extending from \(x=0\) to \(x=L\text{.}\) In the vicinity of what position(s) is the particle most likely to be found?
5.
Let's apply what we have learned about infinite square wells to a macromolecule confined to a biological cell. Consider a protein of mass \(250,000 \Xunits{u}\) (where \(1\Xunits{u} = 1.661 \times 10^{-27}\Xunits{kg}\)) confined to a \(10 \Xunits{ \mu m}\)-diameter cell. Treating this as a particle in a one-dimensional square well, find the energy difference between the ground state and the first excited state. Given that biochemical reactions typically involve energies on the order of \(1\Xunits{eV}\text{,}\) what so you conclude about the role of quantization in these reactions?
6.
The potential energy for the one-dimensional finite square well is shown in Figure 4.21, with dotted lines representing the energies of the ground state and the first excited state.
Using general principles developed in Examples 4.8 and Example 4.9, sketch the ground state wavefunction versus position, including regions in which \(E\lt U\text{.}\)
Compared to the ground state wavefunction of the infinite square well with the same width, is the wavelength in the classically-allowed region longer or shorter? Is the energy larger or smaller?
Repeat a) and b) for the first excited state.
If you did (a) - (c) correctly, you should notice that the wavefunction isn't zero for \(x \lt 0\) or for \(x > L\text{,}\) so there is a non-zero probability that the particle could be located in these regions. Why is this a violation of classical physics? (Hint: what can you say about the kinetic energy and speed of the particle when it is in one of these regions?)
7.
The wavefunction solution to the semi-infinite square well was given in (4.19) for the two regions \(0 \lt x \lt L\) and \(x > L\text{:}\)
In order for the wavefunction solution to be continuous and smooth across the boundary at \(x = L\text{,}\) the value of the wavefunction and its first derivative for the two solutions must match up at the boundary at \(x = L\text{.}\)
Using the solutions given in (4.19), write an equation that states that the two wavefunction solutions are equal at the boundary \(x = L\text{.}\) (Your equation should only contain the symbols \(k, \kappa, A, D\) and numerical constants.)
Using the solutions given in (4.19), write an equation that states that the {derivatives} of the two wavefunction solutions are equal at the boundary \(x = L\text{.}\) (Your equation should only contain the symbols \(k, \kappa, A, D\) and numerical constants.)
-
Using the Schrödinger equation, it can be shown that the quantities \(k\) and \(\kappa\) are related to the energy \(E\) of the particle according to
\begin{equation} k^2 = \frac{2 m E}{\hbar^2} \hspace{0.5in} \mbox{and} \hspace{0.5in} \kappa^2 = \frac{2m \left( U_0 - E \right)}{\hbar^2}\tag{4.25} \end{equation}where \(m\) is the mass of the particle and \(U_0\) is the height of the potential well. Given these relations and the results of parts (a) and (b), do you think that the particle can have any value of energy \(E\) in the well?
8.
Semi-infinite square-well potential. Download the Excel worksheet semi-finite.xls from either the Handouts page or from the Calendar page. This sheet shows the calculations for determining the wavefunctions for a potential well that is infinite at \(x=0\) but of finite magnitude on the right side of the well (which is at \(x=5\) in this problem). You'll see two graphs: the top one shows the semi-infinite potential well (in purple) along with a non-normalized plot of the calculated wavefunction so you can see it along with the potential. The bottom graph shows the normalized wavefunction, corresponding to the second-to-last column in the worksheet.
When you bring up the worksheet, the energy will be set for the value for the ground state. Some questions:
Sketch or print out (just the first page!) the wavefunctions that are displayed for the ground state along with at least two of the excited states. To display the 1\(^{ st}\) and \(2^{ nd}\) excited states, type in 0.64282 and 1.4144 respectively in the framed box for energy.
What happens if you type in an energy that isn't one of the well-defined energies for the problem? Try it out, and comment on what happens. Had we not told you what the allowed energies were, how might you figure them out? (You'll be doing this in lab later this semester.)
9.
For the potential energy function \(U(x)\) and total mechanical energy \(E\) in Figure 4.11:
Use classical physics to argue that if a particle starts out in the region \(0 \lt x \lt L_1\text{,}\) it will remain there forever.
Use quantum arguments (specifically, refer to wavefunction) to argue that a particle that starts in the region \(0 \lt x \lt L_1\) can escape (tunnel). Can you predict the precise moment when the particle will escape? (Hint: no.)
10.
Given the solution to Schrödinger's Equation for an electron in the semi-infinite square well potential (see Examples 4.8 and Example 4.9). Assume that \(U_0 = 12\Xunits{eV}\) and the electron is in a state with an energy \(E = 10\Xunits{eV}\text{.}\)
Calculate the value of the constant \(\kappa\) in the exponent \(e^{-\kappa x}\) for the wavefunction in the classically-forbidden region.
Calculate the distance into the classically-forbidden region beyond \(x = L\) where the probability density is a factor of 2 smaller than that at \(x = L\) (i.e., calculate the value of \(d\) such that \(|\psi(L+d)|^2/|\psi(L)|^2 = e^{-2\kappa d} = 0.5\text{.}\)
Now, let's assume that the particle is a ball with mass \(0.5\Xunits{kg}\text{,}\) and assume that \(U_0\) and \(E\) have everyday values of, say, \(100\Xunits{J}\) and \(50\Xunits{J}\text{,}\) respectively. Repeat the calculations from parts (a) and (b).
Now consider the potential energy function in Figure 4.11. What do you think is needed to get a reasonable (not ridiculously small) probability for the particle to tunnel through the barrier? Why do you think we never experience quantum tunneling for everyday objects?
11.
Consider a proton (\(m_p = 1.67 \times 10^{-27}\Xunits{kg}\)) confined in an infinite potential well of size \(L = 10^{-14}\Xunits{m}\text{.}\) Calculate the energy of a photon emitted when the proton makes a transition from the \(n=4\) state to the \(n=3\) state.
12.
Electrons in an ensemble of \(10\Xunits{nm}\) wide one-dimensional infinite square-well potentials are all initially in the \(n=4\) state. Find the wavelengths of all possible photons emitted as electrons make transitions to the ground state.
13.
Calculate the radius of a quantum dot made from CdSe (band gap energy \(1.74\Xunits{eV}\) and \(\eta = 16.5\)) so that the largest emitted wavelength will be \(629\Xunits{nm}\text{.}\)
14.
Ozone (O\(_3\)) in the atmosphere absorbs ultraviolet radiation that can damage the skin and cause cancer. It does this via a photochemistry process \(\mbox{O} _3 + \gamma \rightarrow \mbox{O} _2 + \mbox{O}\text{,}\) where \(\gamma\) is a photon. If the dissociation energy of ozone is \(3.94\Xunits{eV}\text{,}\) calculate the maximum wavelength of UV radiation that is absorbed by this photochemical process.
Diatomic oxygen (O\(_2\)) also absorbs UV radiation with smaller wavelengths via the process \(\mbox{O} _2 + \gamma \rightarrow \mbox{O} + \mbox{O}\text{.}\) Given a dissociation energy of \(5.17\Xunits{eV}\) for an O\(_2\) molecule, calculate the maximum wavelength of UV radiation that is absorbed by this process.
15.
Titanium dioxide is one of many different possible ingredients used in sunblock lotions. It has a band gap energy of \(3.2\Xunits{eV}\text{.}\) Calculate the maximum wavelength of radiation that you would expect TiO\(_2\) to absorb via an absorption process that promotes electrons from the valence to the conduction band. Is your answer consistent with the use of TiO\(_2\) as an ingredient in sunblock lotions?
16.
Carbon dioxide has first and second excited vibrational states which are \(0.083\Xunits{eV}\) and \(0.166\Xunits{eV}\) above the ground state. Calculate the wavelengths of electromagnetic radiation that you would expect to be most readily absorbed by transitions between these vibrational energy levels in CO\(_2\text{.}\)
The wavelengths that you will find are in the infrared range of \(700\Xunits{nm}\) to \(1\Xunits{mm}\text{.}\) This is one of the reasons why carbon dioxide contributes to global warming as IR radiation is one of the ways in which the Earth radiates heat, assuming it isn't absorbed in the atmosphere. (Note, though, that this is a simplification, as there are rotational energy levels in CO\(_2\) as well that aren't included in this problem.)
17.
Class 3 (neutral phenol) Green Fluorescence Protein (GFP) used in cell and molecular biology studies has an excitation wavelength of \(399\Xunits{nm}\) (near-UV) and an emission wavelength of \(511\Xunits{nm}\) (green, duh).
Determine a 3-level energy diagram (with ground state energy zero) that is consistent with these data. (There are two answers that are consistent with the given data.)
Based on your diagram, you might expect to see two different colors of emitted light. Why do you see only green light emitted from a sample tagged (or genetically mutated) with Class 3 GFP?
18.
The spec sheets for Lumidot\(^{TM}\) quantum dots from Sigma-Aldrich scientific supply company specifies the following core sizes (radius of the quantum dots) and wavelengths for their various dots.
| Radius (nm) | Wavelength (nm) |
| 3.0 | 510 |
| 3.3 | 530 |
| 3.4 | 560 |
| 4.0 | 590 |
| 5.2 | 610 |
| 6.3 | 640 |
Use this data to verify the \(1/R^2\) dependence of (4.23) and to find the band gap energy. (Hint: Calculate \(E_\text{ ph }\) for each of these wavelengths, and then plot \(E_\text{ ph }\) versus \(1/R^2\text{;}\) a computer can make the calculations, plotting, and fitting easier.)