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# Exam 2 Review Questions

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

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
1. free particle
2. particle in a box
3. harmonic oscillator
4. rigid rotor
5. hydrogen atom
6. helium atom (exact solutions not available)
2. Why cannot we solve analytically the helium atom and how do we practically solve this problem
3. Planck’s constant has the same units as

1. angular momentum
2. the Hamiltonian
3. frequency
4. quantum number
5. de Broglie wavelength
4. What is $$Z_{eff}$$ and what is its origin. How do we identify it experimentally and via quantum mechanics.
5. What are the typical energies, wavelengths, spectra, information and selection rules of the basic spectroscopies
1. photoelectron spectroscopy of atoms (i.e., photoionization)
2. atomic absorption and emission spectroscopy
3. infrared spectroscopy
4. microwave spectroscopy
6. How to derive a selection rule for a spectroscopy. What is needed?
7. . The energy levels of the linear harmonic oscillator are
1. all nondegenerate
2. n-fold degenerate
3. (n + 1/2 )-fold degenerate
4. (2n + 1)-fold degenerate
5. $$n^2$$-fold degenerate
8. 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

1. 22.8
2. 91.2
3. 121.6
4. 182.4
5. 364.7 nm

Spherical polar coordinates are used in the solution of the hydrogen atom Schrödinger equation because

1. the Laplacian operator has its simplest form in spherical polar coordinates.
2. Cartesian coordinates would give particle-in-a-box wavefunctions.
3. the Schrödinger equation is then separable into 3 ordinary differential equations.
4. otherwise the atomic orbitals would violate the Pauli exclusion principle.
5. 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

1. $$n − 1$$
2. $$n$$
3. $$n + 1$$
4. $$2n + 1$$
5. $$n^2$$

For real atomic orbitals with quantum numbers $$n$$, $$l$$, the total number of nodal surfaces (radial and angular), equals

1. $$n$$
2. $$n − 1$$
3. $$n − l − 1$$
4. $$n + l$$
5. $$2l + 1$$

Which of the following statements about the hydrogen atom ground state is INCORRECT:

1. It is described by the quantum numbers $$n = 1$$, $$\ = 0$$, $$m = 0$$.
2. The electron’s angular momentum equals $$\hbar$$.
3. The wavefunction is spherically symmetrical.
4. The wavefunction decreases exponentially as a function of $$r$$.
5. 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

1. 13.6 eV
2. 27.2 eV
3. 40.8 eV
4. 54.4 eV
5. 122.4 eV

Molecules are known to absorb radiation in which region of the electromagnetic spectrum:

1. ultraviolet
2. visible
3. infrared
4. microwave
5. all of the above

A 100 W sodium-vapor lamp emits yellow light of wavelength of 550 nm.

The frequency of the radiation is

1. 3.00 × 108
2. 5.50 × 1011
3. 5.45 × 1014
4. 1.83 × 1015
5. 5.14 × 1016 Hz

The wavenumber of the radiation is

1. 1100 cm−1
2. 2.34 × 103 cm−1
3. 8.66 × 103 cm−1
4. 1.81 × 104 cm−1
5. 5.50 × 109 cm−1

The energy of one of the emitted photons equals

1. 3.61 × 10−19 J
2. 5.07 × 10−15 J
3. 3.29 × 10−10 J
4. 0.0100 J
5. 100 J

The atomic transition producing this radiation is

1. 2p → 1s
2. 3p → 3s
3. 2s → 1s
4. 3d → 1s
5. 3d → 4s

The energy of this transition in eV equals

1. 2.25 eV
2. 3.98 eV
3. 13.6 eV
4. 54.4 eV
5. 100 eV

## Chapter 5

1. Calculate the first 3 energy levels of the hydrogen molecule (H2) as a rigid rotor (the required data has to be searched yourself!) and give their degeneracy.
2. Repeat the calculation with the deuterium molecule (D2).
3. 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.
4. 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.
5. Since Yl=1,m=-1 and Yl=1,m=1 are degenerate, we can consider the linear combinations Ypx = (Yl=1,m=-1 + Yl=1,m=1) and Ypy = i(Yl=1,m=-1 - Yl=1,m=-1) as alternative eigenfunctions of the rigid rotor. Show that Ypx and Ypy 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).
6. Is there a zero point energy for the
• particle in the box?
• free particle?
• rigid rotor?
7. Calculate the value of the rotational constant B for a molecule of carbon monoxide. The bond length (R) of CO is 0.113 nm.
8. For the ground state of the hydrogen atom (Feel free to use atomic units.):
1. Calculate $$\langle r \rangle$$.
2. Calculate $$\langle r^2 \rangle$$
3. Calculate $$\langle p_r \rangle$$.
4. Calculate $$\langle p_r^2 \rangle$$.
5. 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. $$1.705 \,cm^{−1}$$
2. $$3.410 \,cm^{−1}$$
3. $$6.820 \,cm^{−1}$$
4. $$121 \,cm^{−1}$$
5. $$1904 \,cm^{−1}$$

29. The equilibrium internuclear distance in $$NO$$ equals

1. 115
2. 121
3. 140
4. 171
5. 229 pm

30. The force constant in NO equals

1. 1125 N/m
2. 1235 N/m
3. 1410 N/m
4. 1595 N/m
5. 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)

1. 98
2. 102
3. 149
4. 160
5. 188

The force constant (in N/m )

1. 98
2. 102
3. 149
4. 160
5. 188

The wavenumber of the $$J = 2$$ to $$J = 3$$ transition (in cm−1 )

1. 7.5
2. 15
3. 30
4. 45
5. 60

## Chapter 6

1. 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)
2. What is the ionization energy of the hydrogen atom?
3. Plot the radial wavefunction and radial distribution function for the H orbitals 1s, 2s, 2p. Indicate if there are nodal planes.
4. What is the distance where it is most likely to find an electron in the ground state of the hydrogen atom?
5. Show that the radial equation for the H atom, the He1+ ion, and the Li2+ 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 hydrogen-like atoms.
6. How can you write the energy and the wavefunction for all hydrogen-like atoms with any value of Z?
7. 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

1. 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.
2. Explain why an electron in a 2p orbital is more screened (smaller $$Z_{eff}$$) with respect to an electron in a 2s orbital.
3. 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?
4. What is an orbital?

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