# 1. An important continuous probability distribution is the exponen...

## 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 ()?

## Solution Preview

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.

By purchasing this solution you'll be able to access the following files:
Solution.docx.

\$40.00
for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

### Find A Tutor

View available Physical Chemistry Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

• 1
• 2
• 3
Live Chats