Download Introduction to Quantum Mechanics Homework #3 (Due on April 28

Introduction to Quantum Mechanics
Homework #3 (Due on April 28, 2014)
1.1) Show for the 1s electron of H atom that the average radius <r > = 1.5 a0, where
a0 (0.529 Ǻ) is the Bohr radius.
2) Repeat the same problem for electron in H atom excited to excited 2s and 2p0
orbitals, respectively?
2. 1) Assuming that the independent electron approximation is valid even for many
electron atoms, calculate the most probable radius and the IP of the C 2p
electron.
2) IP of C atom is 11.26 eV. What interactions is responsible for the difference
between the IP calculated in prob.(2) above and the actual IP?
3) Calculate the effective nuclear charge Zeff and the screening constant σ for a
C2p electron.
3. Show that the energy of the ground state He+ is E0 = e2/2a0 by directly operating Ĥ
on the wavefunction given in the textbook. Also calculate the numerical value of E0
in eV unit.
4. 1) What is the virial theorem?
2)Calculate the average potential energy of H1s electron in eV unit using the
wavefunction given in the textbook. What is the average kinetic energy of the
H1s electron?
3) Verify from your results that the Virial theorem holds for this example.
5. The IP of the ground state K atom is 4.34 eV. Calculate the effective charge and
the screening constant of K 4s electron. Explain why alkali atoms have an IP much
smaller than those of other atoms. Also explain why the IP of noble gas atoms is
much larger than those of other atoms.
6.1) What are the electronic configurations of Ti atom and Fe3+ ion, respectively?
Hint: Consider which orbital electrons are ionized first.
2) If one performs the Stern-Gelach experiment using a Fe3+ ion beam, how many
iron spots do you expect to be deposited on a glass plate after passing through
a nonuniform magnet magnetic field region shown in Fig. 9.38? Explain about
your answer.
Hint: Fe3+ is an atomic magnet whose magnetic dipole moment is contributed by
both orbital and spin angular moments.
7. The electronic configuration of Ti (Z=22) is [Ar] 3d24s2.
1) How may states arise from this configuration?
2) Write down all the possible terms 2s+1 L and draw the energy level diagram by
applying the Hund’s rule.
3) Write down the normalized spin wavefunction for the ground state term 2 S+1 L
term.
4) If the spin-orbit splitting constant is A, what is the maximum energy difference
among the ground state term 2 S+1 L
5) In an external magnetic field how many different states does the ground term
2s+1 L splits into ?
J
8. Consider the excited state of He* in the (1s12p1) configuration.
1) How many different states are there in this configuration ?
2) Write down all the possible terms 2s+1 LJ and draw a schematic energy level
diagram.
3) Considering the selection rule in the transition He*(1s12p1)  He(1s2) +hν (21.2
eV), express all he transitions in the form of 2s+1 L J (initial)  2s+1 L J (final). Draw
all the allowed transitions on the energy diagram in 2). The final state is 1S0 .
4) Write down all the spin wave functions of the 3P state of He*. Which one has Ms
= 0 among these states?
5) Write down the ψspace of the 3P state.
9. A magnetic field B = 1 Tesla is applied to an electronically excited carbon atom in
the 1D2 state.
1) How many energy levels does it split into ?
2) Calculate the magnetic dipole moment of the 1D2 state.
3) Calculate the energy difference between neighboring energy levels in eV unit.
10. The excited state of Rb*(4p65d1) has two levels at 25,700.56 cm-1 and 25,703.52
cm-1 above the ground state. What is the spin-orbit coupling constant A (refer to
the text p.350) in the excited state in cm-1 unit ?