Physics 212 – Problem Set 9 – Spring 2010
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 three particle fermionic system with one particle alone in its orbital and the other two particles in
the same orbital.
iii. a four particle fermionic system with each particle in its unique orbital.
2. 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
(c) Write the first order shift in terms of direct and exchange integrals K1s2s and J1s2s . Our “perturbation”
Hamiltonian is
H (1) =
1
e2
.
|~r1 − ~r2 |
(d) Evaluate the first order shift in this 1s2s singlet state.
3. 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.
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