Exam 2 Review Questions
 Page ID
 284468
This is not comprehensive, but gives a flavor of the sort of questions that students are expected to answer on Exam 2. More questions are forthcoming. No solutions will be given.
Basics
You should be able to answer these questions without thinking (long that is). While exam 2 is not comprehensive it is impossible not to touch upon the topics discussed in exam 1
 What is the (1) potential, (2) Hamiltonian, (3) Eigenstates, (4) Eigenenergies, (5) quantum numbers (with proper ranges), and (6) degeneracies (if they exist) for the following model systems
 free particle
 particle in a box
 harmonic oscillator
 rigid rotor
 hydrogen atom
 helium atom (exact solutions not available)
 Why cannot we solve analytically the helium atom and how do we practically solve this problem

Planck’s constant has the same units as
 angular momentum
 the Hamiltonian
 frequency
 quantum number
 de Broglie wavelength
 What is \(Z_{eff}\) and what is its origin. How do we identify it experimentally and via quantum mechanics.
 What are the typical energies, wavelengths, spectra, information and selection rules of the basic spectroscopies
 photoelectron spectroscopy of atoms (i.e., photoionization)
 atomic absorption and emission spectroscopy
 infrared spectroscopy
 microwave spectroscopy
 How to derive a selection rule for a spectroscopy. What is needed?
 . The energy levels of the linear harmonic oscillator are
 all nondegenerate
 nfold degenerate
 (n + 1/2 )fold degenerate
 (2n + 1)fold degenerate
 \(n^2\)fold degenerate
 The illustrated wavefunction represents the state of the linear harmonic oscillator with n = (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
A hydrogen atom radiates a photon as it falls from a 2p level to the 1s level. The wavelength of the emitted radiation equals
 22.8
 91.2
 121.6
 182.4
 364.7 nm
Spherical polar coordinates are used in the solution of the hydrogen atom Schrödinger equation because
 the Laplacian operator has its simplest form in spherical polar coordinates.
 Cartesian coordinates would give particleinabox wavefunctions.
 the Schrödinger equation is then separable into 3 ordinary differential equations.
 otherwise the atomic orbitals would violate the Pauli exclusion principle.
 Schrödinger first used this coordinate system; any other coordinate system would be equally convenient.
For the hydrogen atom, which of the following orbitals has the lowest energy (A) 4s (B) 4p (C) 4d (D) 4f (E) They all have the same energy
The orbital degeneracy (excluding spin) of hydrogen atom energy levels equals
 \(n − 1 \)
 \(n \)
 \(n + 1 \)
 \(2n + 1\)
 \(n^2\)
For real atomic orbitals with quantum numbers \(n\), \(l\), the total number of nodal surfaces (radial and angular), equals
 \(n \)
 \(n − 1\)
 \(n − l − 1 \)
 \(n + l\)
 \(2l + 1\)
Which of the following statements about the hydrogen atom ground state is INCORRECT:
 It is described by the quantum numbers \(n = 1\), \(\ = 0\), \(m = 0\).
 The electron’s angular momentum equals \(\hbar\).
 The wavefunction is spherically symmetrical.
 The wavefunction decreases exponentially as a function of \(r\).
 The radial distribution function has its maximum at the Bohr radius.
The ionization energy for hydrogen atom is 13.6 eV. The ionization energy for the ground state of \(Li^{2+}\) is approximately
 13.6 eV
 27.2 eV
 40.8 eV
 54.4 eV
 122.4 eV
Molecules are known to absorb radiation in which region of the electromagnetic spectrum:
 ultraviolet
 visible
 infrared
 microwave
 all of the above
A 100 W sodiumvapor lamp emits yellow light of wavelength of 550 nm.
The frequency of the radiation is
 3.00 × 10^{8}
 5.50 × 10^{11}
 5.45 × 10^{14}
 1.83 × 10^{15}
 5.14 × 10^{16 }Hz
The wavenumber of the radiation is
 1100 cm^{−1}
 2.34 × 10^{3} cm^{−1}
 8.66 × 10^{3} cm^{−1}
 1.81 × 10^{4} cm^{−1}
 5.50 × 10^{9} cm^{−1}
The energy of one of the emitted photons equals
 3.61 × 10^{−19} J
 5.07 × 10^{−15} J
 3.29 × 10^{−10} J
 0.0100 J
 100 J
The atomic transition producing this radiation is
 2p → 1s
 3p → 3s
 2s → 1s
 3d → 1s
 3d → 4s
The energy of this transition in eV equals
 2.25 eV
 3.98 eV
 13.6 eV
 54.4 eV
 100 eV
Chapter 5
 Calculate the first 3 energy levels of the hydrogen molecule (H_{2}) as a rigid rotor (the required data has to be searched yourself!) and give their degeneracy.
 Repeat the calculation with the deuterium molecule (D_{2}).
 Different spectroscopic measurements use different units. Find the way to convert J, eV, cm^{1}, GHz and identify good units to be used for the rotational energy levels of a small molecule.
 Show that if \(\psi_1 \rangle \) and \(\psi_2 \rangle \) are different eigenfunctions of Ĥ corresponding the same eigenvalue E (i.e. they are degenerate eigenfunctions), any linear combination of them \(\psi \rangle= a \psi_1 \rangle + b  \psi_2 \rangle \) is still an eigenfunction of \(\hat{H}\) corresponding to the eigenvalue of E.
 Since Y_{l=1,m=1} and Y_{l=1,m=1} are degenerate, we can consider the linear combinations Y_{px} = (Y_{l=1,m=1} + Y_{l=1,m=1}) and Y_{py} = i(Y_{l=1,m=1}  Y_{l=1,m=1}) as alternative eigenfunctions of the rigid rotor. Show that Y_{px} and Y_{py} are real (i.e. not complex) and therefore easier to plot (they will be the angular part of the \(p_x\) and \(p_y\) orbitals).
 Is there a zero point energy for the
 particle in the box?
 free particle?
 rigid rotor?
 Calculate the value of the rotational constant B for a molecule of carbon monoxide. The bond length (R) of CO is 0.113 nm.
 For the ground state of the hydrogen atom (Feel free to use atomic units.):
 Calculate \(\langle r \rangle\).
 Calculate \(\langle r^2 \rangle\)
 Calculate \(\langle p_r \rangle\).
 Calculate \(\langle p_r^2 \rangle\).
 Calculate the uncertainty product \(∆r∆p_r\) in the ground state of the hydrogen atom.
The spectroscopic constants assigned for the NO molecule are \(D_0 = 6.48\, eV\), \(\tilde{\nu} = 1904 \, cm^{−1}\), \(\tilde{B} = 1.705\, cm^{−1}\)
28. For NO, the \(J = 0\) to \(J = 1\) transition occurs at
 \(1.705 \,cm^{−1}\)
 \(3.410 \,cm^{−1}\)
 \(6.820 \,cm^{−1}\)
 \(121 \,cm^{−1}\)
 \(1904 \,cm^{−1}\)
29. The equilibrium internuclear distance in \(NO\) equals
 115
 121
 140
 171
 229 pm
30. The force constant in NO equals
 1125 N/m
 1235 N/m
 1410 N/m
 1595 N/m
 1735 N/m
The tetrahedral molecule \(CH_4\) has how many modes of vibration?
The linear molecule \(\ce{HC≡CH}\) has how many modes of vibration ?
For the \(^7Li^1H\) molecule: \(\tilde{\nu}=1405.6\, cm^{−1}\) ,\(\tilde{\nu}\chi_e = 23.2\, cm^{−1}\) and \(\tilde{B} = 7.513\, cm^{−1}\). Calculate the following quantities:
\(R_e\) (in pm)
 98
 102
 149
 160
 188
The force constant (in N/m )
 98
 102
 149
 160
 188
The wavenumber of the \(J = 2\) to \(J = 3\) transition (in cm−1 )
 7.5
 15
 30
 45
 60
Chapter 6
 Evaluate all the constants in \[E_n = \frac{\mu e^4}{32 \pi^2 \varepsilon_0^2 \hbar^2 n^2}\] showing that the energy levels of the hydrogen atom are \[E_n = \frac{13.6}{n^2}\] (where the energy is expressed in electron volts)
 What is the ionization energy of the hydrogen atom?
 Plot the radial wavefunction and radial distribution function for the H orbitals 1s, 2s, 2p. Indicate if there are nodal planes.
 What is the distance where it is most likely to find an electron in the ground state of the hydrogen atom?
 Show that the radial equation for the H atom, the He^{1+} ion, and the Li^{2+} ion can be written as \[\left \{ \frac{\hbar^2}{2\mu} \left ( \frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r} \right ) + \frac{\hbar^2 l(l+1)}{2\mu r^2}  \frac{Ze^2}{4\pi\varepsilon_0} \frac{1}{r} \right \} R(r) = ER(r)\] where Z=1,2,3 respectively. Atoms with only one electron are hydrogenlike atoms.
 How can you write the energy and the wavefunction for all hydrogenlike atoms with any value of Z?
 Using your answer to Question 6, calculate the ionization potential of He^{+}. Find the distance where it is most likely to find the electron in He^{+}.
Chapter 7
 Systems do not need to be in an eigenstate of the Hamiltonian in order to make measurements on them. A particle in a box of length L is in a state described by \(\Psi \rangle = x(L  x)\). Calculate the expectation value of the energy.
 Explain why an electron in a 2p orbital is more screened (smaller \(Z_{eff}\)) with respect to an electron in a 2s orbital.
 The orbital energy can be identified with the ionization energy (removal of an electron from that orbital). The first ionization energy of Na is 5.14 eV (the electron is removed from the 3s orbital). What is \(Z_{eff}\) for the electron in 3s?
 What is an orbital?