First Examination
April 29, 2015
CHE 118C Spring 2010
Name:
Last
MI
First
Last 4 digits of student ID:
Circle Your Section
Section
TA
1
Patrick Rogers
2
Holly Davison
3
Jessica Hoch
4
Devin McNally
5
P() polynomial n' = 0,1,2,. after which the l-value can run from l = 0, in steps of unity
up toL = n-1.
Substituting the integer n for , we find that the energy levels are quantized
because is quantiz
energies are summarized in Appendix B for principal quantum numbers n ranging from 1
to 3 and in Pauling and Wilson for n up to 5.
There are both bound and continuum solutions to the radial Schrdinger
If the potential is approximated as a quadratic function of the bond displacement x =
r-re expanded about the point at which V is minimum:
V = 1/2 k(r-re)2,
the resulting harmonic-oscillator equation
successive rotational levels (which are of spectroscopic relevance because angular
momentum selection rules often restrict J to 1,0, and -1) are given by
E = B (J+1)(J+2) - B J(J+1) = 2B(J+1).
These e
This form will insure that the function is normalizable since S() 0 as r for all L,
as long as is a real quantity. If is imaginary, such a form may not be normalized (see
below for further consequence
Note that for large values of k
1
k2 1+k
ak+2
= 1.
ak
2 1+2 1+1
k
k k
Since the coefficients do not decrease with k for large k, this series will diverge for z = 1
unless it truncates at finite order
Let us now turn our attention to the radial equation, which is the only place that the
explicit form of the potential appears. Using our derived results and specifying V(r) to be
the coulomb potential
In this equation we have separated and r variations so we can further decompose the
wavefunction by introducing Q = () R(r) , which yields
1 1
m2
F(r)R
= R = -,
Sin
2
Sin
Sin
where a second sepa
n(x) = (n! 2n)-1/2 (/)1/4 exp(-x2/2) Hn( 1/2 x),
where =(k/h2)1/2. Within this harmonic approximation to the potential, the vibrational
energy levels are evenly spaced:
E = En+1 - En = h (k/)1/2 .
In
example, in the crudest approximation of a carbon atom, the two 1s electrons experience
the full nuclear attraction so Zeff=6 for them, whereas the 2s and 2p electrons are screened
by the two 1s elect
This means that the measurement process itself may interfere with the state of the system
and even determines what that state will be once the measurement has been made.
Example:
Again consider the v=
(x,t) = 1/2 exp(ikx) exp(-ikx') (x',t) dx' dk.
Here (1/2)1/2 exp(ikx) is the normalized eigenfunction of F =-ih/x corresponding to
momentum eigenvalue hk. These momentum eigenfunctions are orthonormal
Fj =fj j,
where the j are the eigenfunctions of F. Once the measurement of F is made, for that subpopulation of the experimental sample found to have the particular eigenvalue fj, the
wavefunction bec
(t)= k Ckk exp(-itEk/h). The relative amplitudes Ck are determined by knowledge of
the state at the initial time; this depends on how the system has been prepared in an earlier
experiment. Just as New
4
Energy
2
0
-2
-4
-6
0
1
2
3
4
Internuclear distance
Here, De is the bond dissociation energy, re is the equilibrium bond length, and a is a
constant that characterizes the 'steepness' of the potenti
Having gained experience on the application of the Schrdinger equation to several
of the more important model problems of chemistry, it is time to return to the issue of how
the wavefunctions, operato
1
1/4
0 = e-x 2/2 = 3.53333 2 e-(244.83 -2)(r-1.09769)2
and various N2+ vibrational functions v , one can determine how will evolve in time
and the amplitudes of all cfw_ v that it will contain.
understand usually reflects the presence of experimental observations that do not fit in with
our common experience base.
[Suggested Extra Reading- Appendix C: Quantum Mechanical Operators and Commuta
The Hydrogenic atom problem forms the basis of much of our thinking about
atomic structure. To solve the corresponding Schrdinger equation requires separation of
the r, , and variables
[Suggested Extr
Q
1 2 1
= Q F(r)Sin2 Q - Sin
Sin
.
2
Now all of the dependence is isolated on the left hand side; the right hand side contains
only r and dependence.
Whenever one has isolated the entire de
n=6
n=5
n=4
n=3
n=2
n=1
1/2
(2/L)
sin(nx/L); L = 5 x RCC
In this figure, positive amplitude is denoted by the clear spheres and negative amplitude is
shown by the darkened spheres; the magnitude of th
If the Hamiltonian operator does not contain the time variable explicitly, one can
solve the time-independent Schrdinger equation
In cases where the classical energy, and hence the quantum Hamiltonian
At the end of each Section, a set of Review Exercises and fully worked out
answers are given. Attempting to work these exercises should allow the student to
determine whether he or she needs to pursue
coupled-cluster (CC), and density functional or X -like methods are included. The
strengths and weaknesses of each of these techniques are discussed in some detail. Having
mastered this section, the r
For example, if H were of the form - h2/2M 2/2 = H , then functions of the form exp(i
m) would be eigenfunctions because
cfw_ - h2/2M 2/2 exp(i m) = cfw_ m2 h2 /2M exp(i m).
In this case, cfw_ m2 h2
electrons and the x, y, and z (or r, and ) coordinates of the oxygen nucleus and of the
two protons; a total of thirty-nine coordinates appear in .
In classical mechanics, the coordinates qj and their
where (qj,t) is the unknown wavefunction and H is the operator corresponding to the
total energy physical property of the system. This operator is called the Hamiltonian and is
formed, as stated above
several semi-empirical methods is provided in Appendix F). This section also develops
the Orbital Correlation Diagram concept that plays a central role in using WoodwardHoffmann rules to predict wheth
N. J. (1991), Depending on the backgrounds of the students, our coverage may have to be
supplemented in these first two Sections.
By covering this introductory material in less detail, we are able, wi