Download Mid Term Examination 2 Text

CHEM 3308-001
March 13, 2003
MIDTERM EXAM #2:
Quantum Angular Momentum, Hydrogenic Atom and
Molecular Orbital Theory.
1. Quantum Angular Momentum on a Ring (2-D Rotation); In two
dimensions, the total angular momentum operator reduces to the
angular momentum operator lˆz along the z-axis:
d
lˆz  i
d
(1.1)
whose eigenfunctions are:
1/ 2
 1 
 ml      exp  iml 
 2 
(1.2)
Then,
a) (5 Points): By applying the operator lˆz to its own
eigenfunctions  ml   , obtain the operator’s eigenvalues.
State briefly the physical meaning of those eigenvalues.
b) (5 Points): Show by explicit integration that two different
eigenfunctions:  ml   and  ml   are orthogonal.
c) (5 Points): Consider the angular momentum eigenfunction with
eigenvalue  0 (zero). What kind of motion corresponds to this
eigenvalue? From the corresponding eigenfunction, write down
the probability density to find the rotating particle on the ring at
the position given by the azimuthal angle  . State how that
probability changes as a function of  and discuss whether that
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change in the probability is consistent with your predicted
motion of the particle or not.
2. Hydrogenic Atom I, The 2s Wavefunction: The Hydrogenic 2s
wavefunction  2s is:
Z
 2s 

1/ 2 
4  2   a0 
1
3/ 2

 Zr 
Zr 
 2   exp  

a0 

 2a0 
(1.3)
where Z is the atom’s atomic number and a0 the Bohr radius.
Then,
a) (5 Points): State what are the eigenvalues for lˆz and lˆ 2 for an
electron in the 2s wavefunction by analyzing the information in
its quantum numbers only.
b) (5 Points): Determine the number and the position of the radial
nodes of the 2s wavefunction as deduced from Eq. 1.3. Then,
sketch a 3-D plot of the wavefunction  2s Note: “Sketch”
means a plot qualitatively depicted from the information in Eq.
1.3 and your located nodes: you need not calculate anything
further to plot! Show the possible nodes on the 3-D plot!
3. Hydrogenic Atom II, Spectroscopic Lines: The energy of a
Hydrogenic atom is
En  
Z 2  e4
32 2 0 2n2
; n  1, 2, 3, 4...
(1.4)
mN me
is the reduced mass, m N the
mN  me
mass of the atom nucleus, me the mass of the electron, e the charge of one
electron (in absolute value), and  0 the permitivity of the vacuum.
where Z is he atomic number,  
2
Then,
a) (5 Points) Derive the expression for the wavenumber  

of the
c
quantum of light emitted when the electron decays from a state with
quantum number n2 to other state with quantum number n1  n2 ; point
out the Rydberg constant in your final expression:
 

c

En2  En1
hc
 ????
(1.5)
b) (5 Points) Hydrogen has two main isotopes: normal Hydrogen
(“Protium”) with mN  1 amu and Deuterium with mN  2 amu .
Derive the expression for the ratio of the wavenumbers for a line in
v
any series Hydrogen  ??? Take me  mN Ordinary hydrogen is a
vDeuterium
mixture of normal Hydrogen with a small fraction of Deuterium. How
the line series of the emitted light will look like if the Deuterium
contribution can be detected? Depict the situation for one (any) of the
spectroscopic lines. Be specific!
4. Molecular Orbital Theory:
In a diatomic molecule AB , any pair of its (approximate) molecular orbitals
(MO) can be expressed as a linear combination of two atomic orbitals: A
and B , one per each atom, according to:
   c A A  cB B
(1.6)
where c A and c B are the coefficients of the linear combination. The  
combination is a bonding MO and the   combination an anti-bonding one.
Then,
a) (5 Points) In the case of a homonuclear diatomic, the two
coefficients must be equal: c A  cB  N  . Then, determine the
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values of N  and N  for the bonding and the anti-bonding
MO, respectively, by appealing to the normalization of the
wavefunctions   and   .
b) (5 Points) Write down the probability densities corresponding
to both the wavefunction   and   . From those expressions
and by appealing to physical concepts, explain in detail why the
MO   is bonding and why the MO   is anti-bonding. Refer
to the probability density of the two separate atoms in your
explanation. Draw plots if necessary.
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