## Transcribed Text

1.
An important continuous probability distribution is the exponential distribution
p(x)dix-ce-dridx, 0 x x < 00
Evaluate c, (x), Ox and the probability that x a.
2. Assume that the wavefunction V(x,1) describes the state of a particle under the influence of a scalar
potential, V(x). In this case, V(x,2) is a wave packet, or a superposition of eigenfunctions. (x)(c)
and (p)(1) represent the position and momentum of the center of the wavepacket at an instant in time,
t. Please use the commutation relations between A. &. and p to develop the equations that describe the
temporal evolution of (x)(t) and (p)(t), ic. ove and a(p)
at
at
3. In class, we looked at the free-electron model for a linear conjugated molecule. The 7t-orbitals of
benzene may be approximated using the wavefunctions and energies of a particle on a ring. We're
going to address the particle on a ring problem and then extend it to benzene.
a.
Suppose that a particle of mass m is constrained to move on a circle of radius r in the x-y plane.
Assume that the particle's potential energy is zero. Write down the Schrödinger equation in in
Cartesian coordinates
b. Transform this Schrödinger equation into polar coordinates where x - rcoso and y -
c. Taking r to be constant, write down the general solution '(()) to this Schrödinger equation
(Note: by transforming to polar coordinates and assuming , is constant, we have reduced this 2D
problem to ID).
d. Write down the final expression for the normalized wavefunction and the quantized energies for
the particle on a ring.
e.
Treat the six z-electrons of benzene as particles free to move on a ring of radius 1.42 A and
calculate the wavelength of the lowest electronic transition.
4. In class we defined a Hermitian operator o as one which satisfies the following equality:
- s(tu)) Vdx.
a. Prove that this is equivalent to the statement that Hermitian operators have real eigenvalues.
b. Prove that an equivalent definition for a Hermitian operator is given by: I r*DDdx - dx,
where both f and g are arbitrary functions. In other words, if the first equality is satisfied, the
second one is also satisfied. [Hint: Set y - f + cg for an arbitrary constant c, and plug into the
first expression. Evaluate for c - 1 and c - i.]
5. Let - + 0.992(a + C3P2(x).
a. is normalized and the Qi are orthonormal, calculate C3 (assuming C3 is real).
b. Assuming that Rep, - jpj, plot the probability distribution function of the outcomes of a
measurement associated with 2.
c. What is ()?

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.