The Hydrogen Atom
Shaun Williams, PhD
Show the algebraic steps going from Equation \eqref{eq:8.1.4} to Equation \eqref{eq:8.1.6} and finally to Equation \eqref{eq:8.1.8}. Justify the statement that the rotational and radial motion are separated in Equation \eqref{eq:8.1.8}.
Complete the steps leading from Equation \eqref{eq:8.1.8} to Equation \eqref{eq:8.1.10} and Equation \eqref{eq:8.1.11}.
Write the steps leading from Equation \eqref{eq:8.1.10} to Equation \eqref{eq:8.1.15}.
\(n\) | \( \) | \(l\) | \( \) | \(R_{n,l}(\rho)\) |
---|---|---|---|---|
\( 1 \) | \( 0 \) | \( 2\left( \frac{Z}{a_0} \right)^\frac{3}{2}e^{-\rho} \) | ||
\( 2 \) | \( 0 \) | \( \frac{1}{2\sqrt{2}}\left(\frac{Z}{a_0}\right)^\frac{3}{2} (2-\rho)e^{-\frac{\rho}{2}} \) | ||
\( 2 \) | \( 1 \) | \( \frac{1}{2\sqrt{6}}\left(\frac{Z}{a_0}\right)^\frac{3}{2} \rho e^{-\frac{\rho}{2}} \) | ||
\( 3 \) | \( 0 \) | \( \frac{1}{81\sqrt{3}}\left(\frac{Z}{a_0}\right)^\frac{3}{2} (27-18\rho +2\rho^2) e^{-\frac{\rho}{3}} \) | ||
\( 3 \) | \( 1 \) | \( \frac{1}{81\sqrt{6}}\left(\frac{Z}{a_0}\right)^\frac{3}{2} (6\rho-\rho^2) e^{-\frac{\rho}{3}} \) | ||
\( 3 \) | \( 2 \) | \( \frac{1}{81\sqrt{30}}\left(\frac{Z}{a_0}\right)^\frac{3}{2} \rho^2 e^{-\frac{\rho}{3}} \) |
Consider several values for \(n\) and show that the number of orbitals for each \(n\) is \(n^2\).
Construct a table summarizing the allowed values for the quantum numbers \(n\), \(l\), and \(m_l\), for \(n=1-7\).
The notation 3d specifies the quantum numbers for an electron in the hydrogen atom. What are the values for \(n\) and \(l\) ? What are the values for the energy and angular momentum? What are the possible values for the magnetic quantum number? What are the possible orientations for the angular momentum vector?
Examine the mathematical forms of the radial wavefunctions. What feature in the functions causes some of them to go to zero at the origin while the s functions do not go to zero at the origin?
What mathematical feature of each of the radial functions controls the number of radial nodes?
At what value of \(r\) does the 2s radial node occur?
Using Equation \eqref{eq:8.3.1} and a spreadsheet program or other software of your choice, calculate the energies for the lowest 100 energy levels of the hydrogen atom. Also calculate the differences in energy between successive levels. Do the results from these calculations confirm that the energy levels rapidly get closer together as the principal quantum number \(n\) increases? What happens to the energy level spacing as the principle quantum number approaches infinity?
Will an electron in the ground state of hydrogen have a magnetic moment? Why or why not?
Calculate the magnitude of the gyromagnetic ratio for an electron.
$$ \expect{E}=\Braket{\hat{\mathcal{H}}^0} + \frac{eB_z}{2m_e} \Braket{\hat{L}_z} $$
Show that the expectation value \(\expect{\hat{L}_z}=m_l\hbar\).
Carry out the calculations that show that the g-factor for electron spin is 2.0023. Interestingly, the concept of electron spin and the value \(g=2.0023\) follow logically from Dirac's relativistic quantum theory, which is beyond the scope of this course.
Compare the reduced mass of the \(\chem{Li^{2+}}\) ion to that of the hydrogen atom.
Use the orbital energy level expression in Equation \eqref{eq:8.6.2} to predict quantitatively the relative energies (in \(cm^{-1}\)) of the spectral lines for \(\chem{H}\) and \(\chem{Li^{2+}}\).
Write the term symbol for a state that has 0 for both the spin and orbital angular momentum quantum numbers.
Write the term symbols for a state that has 0 for the spin and 1 for the orbital angular momentum quantum numbers.
Orbital Configuration | Term Symbols | Degeneracy |
---|---|---|
\( 1s^1 \) | \( {}^2S_\frac{1}{2} \) | \( 2 \) |
\( 2s^1 \) | \( {}^2S_\frac{1}{2} \) | \( 2 \) |
\( 2p^1 \) | \( {}^2P_\frac{1}{2},\, {}^2P_\frac{3}{2} \) | \( 2,\, 4 \) |
\( 3s^1 \) | \( {}^2S_\frac{1}{2} \) | \( 2 \) |
\( 3p^1 \) | \( {}^2P_\frac{1}{2},\, {}^2P_\frac{3}{2} \) | \( 2,\, 4 \) |
\( 3d^1 \) | \( {}^2D_\frac{3}{2},\, {}^2D_\frac{5}{2} \) | \( 4,\, 6 \) |
Confirm that the term symbols in the previous table are correct.
Confirm that the values of the degeneracy in the previous table are correct and that the total number of states add up to 8 for \(n=2\) and 18 for \(n=3\).
Term | \(J\) | \(L\) | \(S\) | \(g\) | \(m_J\) | \(g\, m_J\) |
---|---|---|---|---|---|---|
\( {}^2P_\frac{3}{2} \) | \( \frac{3}{2} \) | \( 1 \) | \( \frac{1}{2} \) | \( \frac{4}{3} \) | \( \frac{3}{2} \) | \( \frac{6}{3} \) |
\( \frac{1}{2} \) | \( \frac{2}{3} \) | |||||
\( -\frac{1}{2} \) | \( -\frac{2}{3} \) | |||||
\( -\frac{3}{2} \) | \( -\frac{6}{3} \) | |||||
\( {}^2P_\frac{1}{2} \) | \( \frac{1}{2} \) | \( 1 \) | \( \frac{1}{2} \) | \( \frac{2}{3} \) | \( \frac{1}{2} \) | \( \frac{1}{3} \) |
\( -\frac{1}{2} \) | \( -\frac{1}{3} \) | |||||
\( {}^2S_\frac{1}{2} \) | \( \frac{1}{2} \) | \( 0 \) | \( \frac{1}{2} \) | \( 2 \) | \( \frac{1}{2} \) | \( 1 \) |
\( -\frac{1}{2} \) | \( -1 \) |
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