Physics 312 – Problem Set 7
1. Permutation operator
(a) Show that if
†
hΨ|Ô|Ψi = hΨ|P̂jk
ÔP̂jk |Ψi ,
(1)
†
hΨ0 |Ô|Ψi = hΨ0 |P̂jk
ÔP̂jk |Ψi ,
(2)
then
where |Ψi and |Ψ0 i are each identical particle wavefunctions, P̂jk is an interchange operator, and Ô is an
observable for an identical particle system. (It is possible to show this without making any assumptions
about the interchange operator. However, if you do wish to use any properties of the interchange operator,
you must prove them first.)
(b) Show that if Ψ, representing a many identical particle system, is an eigenvector of the exchange operator
P̂jk (for all j and k), then all of the eigenvalues (in that subspace) of P̂jk for all j and k are identical.
(c) Show, using the permutation operator, that antisymmetric identical particle wavefunctions are orthogonal
to symmetric identical particle wavefunctions
(d) Show that the operator
X
Ŝ =
P̂
P ∈G
produces symmetric identical particle wavefunctions, and that
X
 =
sign(P )P̂
P ∈G
produces antisymmetric identical particle wavefunctions. P̂ is a permutation operator in the group G of
permutations for N particles. sign(P ) is 1 for an even number of interchanges and –1 for an odd number
of interchanges.
(e) Create a valid identical particle wavefunction for the following systems (by “orbital” is meant a state
labelled by a CSCO):
i. a three particle bosonic system with one particle alone in its orbital and the other two particles in the
same orbital.
ii. a five particle fermionic system with each particle in its unique orbital.
iii. a system consisting of two helium atoms each in the 1s2s state.
2. C-TDL D14 Problem 3.
3. Consider a Helium atom with one electron in the 1s orbital and one electron in the 2s orbital. (See C-TDL
BXIV .)
(a) Write down the appropriately symmetrized wavefunction for this state if the total spin of the system is
S = 0.
(b) What is the zeroth order energy for the 1s2s level? This is for the zeroth level Hamiltonian
H (0) = −
h̄2 ~ 2 Ze2
h̄2 ~ 2 Ze2
∇1 −
−
∇ −
.
2m
r1
2m 2
r2
1
(c) Write the first order shift in terms of direct and exchange integrals K1s2s and J1s2s . Our “perturbation”
Hamiltonian is
H (1) =
e2
.
|~r1 − ~r2 |
(d) Evaluate the first order shift in this 1s2s singlet state.
4. Variational treatment of a Helium atom with one electron in the 1s orbital and one electron in the 2s orbital.
(a) Let the electrons be in the triplet spin state. Write down the appropriately symmetrized wavefunction.
(b) Use a trial wavefunction where the 1s and 2s states are the hydrogenic solutions except that now Z is
replaced with Z 0 , which will be the “effective” Z for this trial wavefunction. Calculate hEi for this state.
(c) Minimize the hEi you found above with respect to Z 0 . Use the Z ∗ which minimizes hEi to find the “best”
energy for these trial wavefunctions.
(d) Use the results from the last problem to obtain the triplet state perturbation theory answer and compare
this to the one obtained using the variational approach.
(e) Describe how your calculation would have been different had you used Z 0 in place of Z in the hydrogenic 1s
wavefunction, but Z 00 in place of Z in the hydrogenic 2s. That is, there are now two variational parameters
instead of a single effective Z for this problem.
5. Find all of the allowed spectral terms ((2S+1) LJ ) for the 1s2 2s2 2p3 configuration. Show your work by using the
Pauli exclusion principle and extracting terms starting with the highest weight.
6. The lowest energy levels of C, O, Si, and S:
(a) Write the electronic configuration (this is the (nl)1 (nl)2 notation) of the ground state for each atom.
(b) Now we want to obtain an energy level diagram for the lowest energy levels of each atom and label
the levels in standard spectroscopic notation, including “LS” coupling shifts. (Hint: you can treat two
“missing electrons” – those that would be necessary to complete a shell – in exactly the same way as two
real electrons. And one of these we have already done in class.)
i. First, argue that the term (of the form (2S+1) L) that lies lowest in energy is the one with the highest
spin multiplicity. (This is the generalization of the argument we used to show that the spin triplet
term of the nn0 He configuration lies lowest in energy.)
ii. Now apply the “LS” coupling shifts to the lowest term. The shifts are dictated by
~ · S,
~ which you know how to evaluate in the L, S, J, MJ basis. ζ(L, S) is just a constant
∆E = ζ(L, S)L
if L and S are constant (which they are within one (2S+1) L term).
iii. Compare your results with the experimental measurement of the lowest levels, where the first is arbitrarily set to zero. (These are the levels, not the separation between levels.)
C
O
Si
S
0
16.4
43.4
0
158.5
226.5
0
77.2
223.3
0
396.0
573.6
What information does this give you on the sign of ζ(L, S) for the different atoms?
7. Analyzing Ca lines. It is observed that there are six lines (representing the energy of photons emitted between
sublevels) lying close together near 4450 Angstroms. Setting the lowest frequency line arbitrarily at zero, lines
exist at zero, 3.7 cm−1 , 9.3 cm−1 , 105.8 cm−1 , 109.7 cm−1 , and 158.0 cm−1 . Calling these lines a, b, c, d, e,
and f from left to right, sketch them versus cm−1 with the following intensities (height of line) a = 0.5, b = 4.5,
c = 25, d = 5, e = 14, and f = 6.5. You see that there are three lines clustered together, then a gap until two
lines clustered together, and finally a third line appearing after another gap in wavenumber. The ground state
of Ca has the outermost (most loosely bound) electrons in the configuration 4s2 . Assume that it is transitions
from the excited state of one of these electrons which gives rise to the observed frequency spectrum.
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(a) What is the dipole selection rule for individual electron angular momentum quantum numbers? That is,
give the selection rule in terms of the change allowed in l2 . l1 is assumed to remain at zero since one of the
electrons remains in the 4s orbital.
(b) Use the three lowest excited configurations ((4s4l), l = 1, 2, 3) of Ca to find all of the spectral terms
((2S+1) L) which result. Here, since the electrons occupy different orbitals, there is no confusion over Pauli
exclusion issues. Check that you have all of the spectral terms by making sure that the total degeneracy
is maintained as you go from the individual particle basis to the coupled particle basis.
~ = ~l1 +~l2 ?
(c) How does your selection rule in terms of ∆l2 translate into a selection rule in terms of ∆L, where L
Also give the restrictions of ∆S and ∆J (again for a dipole transition).
(d) Now sketch the sublevels (specified by the J quantum number), grouping together those of a given L and
S value. You need not be concerned with whether the sublevels are regular or inverted with respect to
~ ·S
~ is positive or negative); just pick
increasing J (that is, whether the sign of the coefficient in front of L
one and stick with it for all terms.
(e) By applying your selection rules, and requiring that the total number of possible transitions is six, you will
be able to reject all possibilities but those involving transitions between two sets of (two) terms. Sketch
those levels and the allowed transitions.
(f) Now, given the interval rule ∆E(J) − ∆E(J − 1) = ζJ, form ratios of level spacings. The idea here is that
spacings within a given term are proportional to one another. Taking ratios removes the proportionality
constant. This will help you narrow down the possible source of the experimental lines.
(g) Verify your choice of assignments of the observed spectral lines to the transitions you have chosen by
checking that the intensities of the lines is as expected: Within a given term, the different decay intensities
for different J levels is given by the multiplicity of that level. So, for instance, if within a term you have
J values of 1 and 2, the relative intensity of the lines will be 3:5. In order to get the intensity of a given
level, you must add up the intensities of all its decay modes.
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