CHEM 614
HOMEWORK 12
2002
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1. An electron moves in a harmonic potential, V= kx 2. What is the effect, to the first order, on
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the energies of superimposing an electric field, V ' = Ex? Explain your reasoning.
2. Calculate the energy to first order of He+ in its lowest - energy state. Use the hydrogen atom
in its ground state as your zeroth-order approximation. Use atomic units. Predict the signs
(plus, zero, minus) of c(1)2s, 1s and c(1)2p0, 1s. Explain your reasoning.
3. A hydrogen atom in the 2s state is metastable, but, when passed through an electric field, its
tendency to relax to the 1s state is greatly enhanced. Explain.
4. A one-dimensional box potential is perturbed as shown in the sketch
From a consideration of the first three wavefunctions of the unperturbed system (1, 2, 3),
which of these will have its energy increased most, to first order, and which least? You need no
explicit calculations to answer this question.
5. A molecule of interest has a lowest-lying nuclear spin configuration that is a degenerate pair
of states, 1 and 2 . Under certain conditions the initial composition of the system is given by
the expression
n1 /(n1  n2 )  1.00
At a given instant of time, a perturbing field of intensity equivalent to 10-28 J was applied for a
duration of 10 s. After an interval of 10 s a second pulse of half the intensity of the first was
applied for 5 s.
If spontaneous changes in the 10 s between the pulses are negligible, calculate the probable
composition of the system (a) at the end of the first pulse, and (b) at the end of the second pulse.
Hint: you will find the Rabi formula useful here.
6. A hydrogen atom in its 2s state passes into a region of space where it experiences a z-directed
electric field for a time . Calculate the total wavefunction of the atom and its dipole moment
during the time of exposure in the field.
Hints: The field will induce mixing of 2s and 2pz states; you need to find the coefficients of the
combination in order to find an expression for tot. To calculate the value of an observable you
need to find the expectation value of its operator, in this case the dipole moment operator, ^,
where  = -ez. Remember in polar coordinates z  r cos where co-latitude.
7. What is the term symbol for an nd10 configuration?
8. How many distinct ways are there of assigning two electrons to the nd orbitals? (In other
words how many sets of mli and msi are there in the nd2 configuration?)
9. Deduce the values of J associated with the term symbols 2S, 3D, and 4F.
10. Use Hund’s rules to deduce the lowest energy state of:
(a) An excited state of a Be atom (1s22s13s1).
(b) The ground state of a carbon atom.
11. A consideration of spin orbit coupling leads us to expect the 2p1s transition in the H atom
to consist of a closely spaced pair of lines (doublet). Give the term symbols for the states
involved in the transitions and calculate the wavelengths (in nm) of the two lines. Compare your
wavelength values with that obtained if spin-orbit coupling is neglected.