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Unformatted text preview: 10 pts 10 pts 10 pts 10 pts Chem 120A Hour Examination #2 1. To answer these questions adopt the independent electron model of atoms. You may wish to recall that the energy levels of hydrogenic atoms are
En=~mZ2e4/27i2n2 , n=1,2,... where Z6 is the nuclear charge, and m is the electron mass. (a) Consider an atom with conﬁguration
1822522p5 . In units of 71, what are the possible values of total orbital angular momentum
for the atom and what is the total spin angular momentum? (b) Next consider the atomic conﬁguration
1322322136 . Now what are the possible values for the total orbital angular momentum
and spin?
(C) Suppose the 10 electron atom in Part (b) absorbs a photon of energy to Tzw z E3 —— E2. After absorption, which of the following are possible conﬁgurations i. 13223213633
ii. 13228213631)
iii. 132252213535
iv. 152252219531) [Note: there are other possibilities not listed]
(d) For the possible conﬁguration(s) in Part (c), determine the possible values of the total orbital angular momentum. 2. In this series of questions, adopt the independent electron model for electrons in
a molecular framework. Imagine that each electron feels the bistable potential
pictured in the ﬁgure. Due to the potential energy barrier, electrons pass only
infrequently between regions I and II. As such, separate eigenstates for regions I and Il provides a convenient basis with which to describe the actual single electron
stationary states.
These normalized basis functions are 1,1)1 ,1/)2 and 1/15 , in region I , and
1/?3 ,¢4 and 1/)6 , in region II . Their energies,
E.‘ =<¢1Hdn> , i: 1,2,... are indicated in the ﬁgure with labeled horizontal lines. The ordering of these
energies is E1<E2=E3<E4<E5<E6. The hamiltonian, H, is not diagonal in this representation. Speciﬁcally, both
H23 = (1P2IHl1/m) E “K < 0 a and H56 2 <¢5H¢6) E —J < 0 . are negative and real. All other off diagonal elements, H , j, are zero. All the overlaps
«Ali/{1), i aé j, are negligible too. V(%l 10 pts (a) 10 pts (b) 10 pts (C) 10 pts (d)
10 pts (e) 10 pts (f) EXTRA (g)
CREDIT
10 pts Determine the energies of the stationary single electron states with the four lowest energies. You may assume that while not negligible, K and J are both smaller than the smallest energy spacing between non degenerate basis states.
What are the four corresponding single electron wave functions? Express
your results as linear combinations of 1%, 1/22, . . . and $6. i. If, at time t = 0, a single electron is in region I in state 1L2, how long will it take before the electron can be completely in region II? ii. Suppose instead it was initially in state 1&1, what would the answer be in that case? Suppose four electrons exist in the pictured potential. Specify the energy of the 4electron ground state wavefunction. Suppose again that four electrons exist in the pictured potential. This‘time specify the lowest possible energy for a stationary triplet state. Now suppose there are twp electrons in the potential. Specify the spatial
part of the twoelectron wave function for i. the singlet state of the lowest energy
and ii. the triplet state of lowest energy. Suppose the two electron system, initially in its ground state, absorbs a photon of energy 7w m E2 — E1. Determine the amounts of electron charge that reside in region II before absorption and after absorption. ...
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 Spring '10
 CHandler
 orbital angular momentum, single electron, independent electron model

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