2.2 Problems
93
What additional quantum numbers are needed to specify fully an atomic quantum state and what physical quantities do they quantify? List the allowed quantum numbers for n = 1 and n = 2 and specify fully the electronic quantum
numbers for the ground state of the Carbon atom (atomic number Z = 6)
[Adapted from University of London 2002]
2.8 Estimate the total ground state energy in eV of the system obtained if all the
electrons in the Carbon atom were replaced by π
− particles. (You are given
that the ground state energy of the hydrogen atom is −13.6 eV and that the π
−
is a particle with charge −1, spin 0 and mass 270 m e
[University of London]
2.9 What are atomic units? In this system what are the units of (a) length (b) energy
(c)
2 (d) e
2 (e) m e ? (f) Write down Schrodinger’s equation for H-atom in
atomic units
2.10 (a) Two positive nuclei each having a charge q approach each other and electrons concentrate between the nuclei to create a bond. Assume that the
electrons can be represented by a single point charge at the mid-point
between the nuclei. Calculate the magnitude this charge must have to
ensure that the potential energy is negative.
(b) A positive ion of kinetic energy 1 × 10
−19 J collides with a stationary
molecule of the same mass and forms a single excited composite molecule.
Assuming the initial internal energies of the ion and neutral molecule were
zero, calculate the internal energy of the molecule.
[Adapted from University of Wales, Aberystwyth 2008]
2.11 (a) By using the deBroglie relation, derive the Bohr condition mvr = n for
the angular momentum of an electron in a hydrogen atom.
(b) Use this expression to show that the allowed electron energy states in
hydrogen atom can be written
E n = −
me
4
8ε
2
0 h 2 n 2
(c) How would this expression be modified for the case of a triply ionized
beryllium atom Be(Z = 4)?
(d) Calculate the ionization energy in eV of Be
+3 (ionization energy of hydrogen = 13.6 eV)
[Adapted from the University of Wales, Aberystwyth 2007]
2.12 When a negatively charged muon (mass 207 m e is captured in a Bohr’s orbit
of high principal quantum number (n) to form a mesic atom, it cascades
down to lower orbits emitting X-rays and the radii of the mesic atom are
shrunk by a factor of about 200 compared with the corresponding Bohr’s atom.
Explain.
2.13 In which mu-mesic atom would the orbit with n = 1 just touch the nuclear
surface. Take Z = A/2 and R = 1.3 A
1/3 fm.
93
What additional quantum numbers are needed to specify fully an atomic quantum state and what physical quantities do they quantify? List the allowed quantum numbers for n = 1 and n = 2 and specify fully the electronic quantum
numbers for the ground state of the Carbon atom (atomic number Z = 6)
[Adapted from University of London 2002]
2.8 Estimate the total ground state energy in eV of the system obtained if all the
electrons in the Carbon atom were replaced by π
− particles. (You are given
that the ground state energy of the hydrogen atom is −13.6 eV and that the π
−
is a particle with charge −1, spin 0 and mass 270 m e
[University of London]
2.9 What are atomic units? In this system what are the units of (a) length (b) energy
(c)
2 (d) e
2 (e) m e ? (f) Write down Schrodinger’s equation for H-atom in
atomic units
2.10 (a) Two positive nuclei each having a charge q approach each other and electrons concentrate between the nuclei to create a bond. Assume that the
electrons can be represented by a single point charge at the mid-point
between the nuclei. Calculate the magnitude this charge must have to
ensure that the potential energy is negative.
(b) A positive ion of kinetic energy 1 × 10
−19 J collides with a stationary
molecule of the same mass and forms a single excited composite molecule.
Assuming the initial internal energies of the ion and neutral molecule were
zero, calculate the internal energy of the molecule.
[Adapted from University of Wales, Aberystwyth 2008]
2.11 (a) By using the deBroglie relation, derive the Bohr condition mvr = n for
the angular momentum of an electron in a hydrogen atom.
(b) Use this expression to show that the allowed electron energy states in
hydrogen atom can be written
E n = −
me
4
8ε
2
0 h 2 n 2
(c) How would this expression be modified for the case of a triply ionized
beryllium atom Be(Z = 4)?
(d) Calculate the ionization energy in eV of Be
+3 (ionization energy of hydrogen = 13.6 eV)
[Adapted from the University of Wales, Aberystwyth 2007]
2.12 When a negatively charged muon (mass 207 m e is captured in a Bohr’s orbit
of high principal quantum number (n) to form a mesic atom, it cascades
down to lower orbits emitting X-rays and the radii of the mesic atom are
shrunk by a factor of about 200 compared with the corresponding Bohr’s atom.
Explain.
2.13 In which mu-mesic atom would the orbit with n = 1 just touch the nuclear
surface. Take Z = A/2 and R = 1.3 A
1/3 fm.