Unformatted text preview: Practice Midterm Examination, Winter 2011
1. A particle of mass m in one dimension subject to the potential V(x) = ± if x -a/2 or x ≥
a/2, V(x) = 0 if –a/2 x -b/2 or b/2 x a/2, and V(x) = V0 if –b/2 x b/2. The potential
corresponds to an infinitely deep box of length a with a finite-size barrier of width b and
height V0 in the center of the box. The following refers to those states of the particle whose
eigenvalues are less than V0.
a) Solve the time-independent Schrodinger equation for this system up to undetermined
integration constants. That is, provide the general solution to the differential
equations associated with the Schordinger equation.
b) Derive the equations necessary to find the integration constants of part (a) and
eigenenergies of the system. You need not do anything with these equations aside
from writing them down in order to get full credit. (That is you do not have to solve
for the integration constant or the energy).
c) Plot your best guess for the ground-state wavefunction of this system. 2. The commutator of two operators A and B is given by the symbol A, B and is defined as: A, B AB BA This means that operation on an arbitrary function f with the commutator is given by A, B f ABf BAf
Prove the following relations involving commutators. In each case, simply quoting a result,
even if it appears in the text, in an insufficient answer. Show all your work.
a) px , x f if Where p x is the operator of the x component of the linear momentum of a particle and x is the operator of x component of the position vector of the same particle. b) l x , l y f il z f Where l x , l y , and l z are, respectively, the operators associated with the x, y, and z components of the angular momentum of a particle. 3. a) The model of a particle in a one‐dimensional box has been applied to the electrons in
linear conjugated hydrocarbons. Consider butadiene, H2C=CHCH=CH2, which has four
electrons. We assume for simplicity that the electrons in butadiene move along a straight line
whose length can be estimated as equal to two C=C bond lengths (2×1.35 Å) plus one C‐C bond
(1.54 Å) plus the distance of a carbon atom radius at each end (2×0.77Å=1.54Å), giving a total
distance of 5.78 Å. Use the energy expression for the particle in a one dimensional box:
En , n=1,2,…
8me a 2
Calculate the energy for the molecule to be excited from the ground state (lowest energy state) to
the first excited state and the wave number of the corresponding photon.
b) Using the same model, it can be shown that the first electronic transition in hexatriene occurs
at 2.8×104 cm-1. From this, estimate the length of the hexatriene molecule. (Hint: Hexatriene has
six electrons) 4. Review problem #6 from HW3. ...
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This note was uploaded on 12/05/2011 for the course CHEM 113A taught by Professor Lin during the Fall '07 term at UCLA.
- Fall '07
- Quantum Chemistry