2.11: Problems
- Page ID
- 420482
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Consider the functions \(f(x)=A\left(1-x^{2}\right)\) and \(g(x)=3 x^{3}-x\).
- Find a value for A such that \(f(x)\) is normalized on the interval \(-1 \leq x \leq 1\).
- Are the functions \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) orthogonal over the interval \(-1 \leq \mathrm{x} \leq 1\) ?
- Consider each of the following functions and the associated intervals. Indicate whether or not the given function is suitable as a wavefunction over the given interval.
- \(\mathrm{e}^{\mathrm{x}} \qquad 0 \leq x \leq \infty\)
- \(\mathrm{e}^{-\mathrm{x}} \qquad 0 \leq x \leq \infty \)
- \(1 / \mathrm{x} \qquad -\infty \leq \mathrm{x} \leq \infty\)
- \(\mathrm{e}^{\mathrm{i} \theta} \qquad 0 \leq x \leq 2 \pi\)
- \(x(1-x) \qquad 0 \leq x \leq 1\)
-
Consider the following operators. Determine whether or not they are Hermitian.
a. \(\mathrm{d} / \mathrm{dx}\)
b. \(i \mathrm{~d} / \mathrm{dx}\)
c. \(\mathrm{d}^{2} / \mathrm{dx}^{2}\)
d. \(i \mathrm{~d}^{2} / \mathrm{dx}^{2}\) - Consider an operator  and associated set of eigenfunctions \(\phi_{\mathrm{n}}\) that satisfies
\[\hat{A} \phi_{\mathrm{n}}=a_{\mathrm{n}} \phi_{\mathrm{n}} \nonumber\]
Show that if the operator is Hermitian that the eigenvalues \(a_{\mathrm{n}}\) must be real-valued.
- Consider the data in the table.
- Calculate \(\left\langle x \rangle \right.\) and \(\left\langle x^{2}\right\rangle\).
- Calculate \(\sigma_{x}^{2}\) for the data set.
- Does \(\sigma_{x}^{2}=\left\langle x^{2}\right\rangle-\langle x\rangle^{2}\) ? If not, what is the difference?
- Consider a particle of mass \(m\) in a rectangular solid box with edge lengths given by \(a=\) a, \(b=2 \mathrm{a}, c=2 \mathrm{a}\). Find the degeneracies of the first 10 energy levels for the system.
| \(\mathbf{i}\) | \(\mathbf{x}\) |
|---|---|
| 1 | \(2.3\) |
| 2 | \(6.4\) |
| 3 | \(4.2\) |
| 4 | \(3.5\) |
| 5 | \(4.9\) |
- Consider a particle of mass \(m\) that is in a one-dimensional box of length \(a\). The system is prepared so that the wavefunction is given by \(\psi(x)=\operatorname{Ax}(a-x)\).
- Find a value of \(\mathrm{A}\) that normalizes the wavefunction.
- Find the expectation values for \(x\) and \(x^{2}\left(\langle x \rangle \right.\) and \(\langle x^{2}\rangle ) \).
- Find the expectation values for \(\mathrm{p}\) and \(\mathrm{p}^{2}\left(\langle \mathrm{p} \rangle\right.\) and \(\langle \mathrm{p}^{2} \rangle\)).
- Given that the variance for a measurement is given by \(\sigma_{a}^{2}=\langle a^{2} \rangle- \langle a \rangle^{2}\) calculate the variances \(\sigma_{\mathrm{x}}^{2}\) and \(\sigma_{\mathrm{p}}^{2}\).
- Find the value of \(\sigma_{x} \sigma_{p}\). Does it exceed \(\frac{\hbar}{2}\) ?
- Consider a particle of mass \(m\) in a box of length \(a\). The system is prepared such that the wavefunction is given by \(\psi(\mathrm{x})=\mathrm{Ax}^{2}(a-\mathrm{x})\).
- Find a value of A that normalizes the wavefunction.
- What are the units on the wavefunction?
- Find \(\langle x \rangle\).
- Is \(\langle\mathrm{x}\rangle=a / 2\) ? Why or why not?
-
Consider the following pairs of operators and determine whether or not the operators commute.
a. \(\mathrm{d} / \mathrm{dx}, \mathrm{d}^{2} / \mathrm{dx}^{2}\)
b. \(\mathrm{x}, \mathrm{d}^{2} / \mathrm{dx}^{2}\)
c. \(\mathrm{x}, \int d x\) - Consider a particle of mass \(m\) in a box of length \(a\) for which the wavefunction is given by
\[\Psi(\mathrm{x})=(2)^{1 / 2} / 3 \phi_{1}(\mathrm{x})-(7)^{1 / 2} / 3 \phi_{3}(\mathrm{x})\nonumber\]
where \(\phi_{\mathrm{n}}(\mathrm{x})=(2 / a)^{1 / 2} \sin (\mathrm{n} \pi \mathrm{x} / a)\).
- Show that the wavefunction \(\Psi(\mathrm{x})\) is normalized.
- Graph the wavefunction \(\Psi(\mathrm{x})\).
- What is the expectation value for energy \(\langle \mathrm{E} \rangle\) for the system?
- What is the most likely energy to be measured for the system?
- Consider benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) as modeled using the free-electron model.
- Using a \(\mathrm{C}-\mathrm{C}\) bond length of \(\mathrm{r}_{\mathrm{cc}}=0.139 \mathrm{~nm}\), calculate the circumference of the ring and its radius.
- Based on the model, what are the degeneracies of the four lowest energy levels?
- Placing two electrons per particle-on-a-ring "orbital", calculate the energy gap (and corresponding wavelength of light driving a transition) between the HOMO and the LUMO based on this model.
- How does the value you found in part c compare to the observed band-origin of the \(A_{1 \mathrm{~g}} \rightarrow \mathrm{B}_{1 \mathrm{u}}\) transition of benzene \((\lambda=215 \mathrm{~nm})\) ?