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# Provide, in your own words, definitions for each of the following: complete basis, expectation value, commutator, Hermitian operator, dimension, tunneling, nonlinear process, symmetry, continuous / continuity, integrable, divergent (of an expression), "existence" (of a math expression)
  
# Provide, in your own words, definitions for each of the following: complete basis, expectation value, commutator, Hermitian operator, dimension, tunneling, nonlinear process, symmetry, continuous / continuity, integrable, divergent (of an expression), "existence" (of a math expression)
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# <u>For problem 2, choose one of the following two questions:</u>
 +
# <u>For problem 2, choose one of the following two questions: (3 or 4, you do not have to do both)</u>
 
# In the classical Bell experiment, a pair of 2 entangled particles are created in state <math>|\psi\rangle = (|0,1\rangle + |1,0\rangle)/\sqrt{2}</math>. A quantum circuit for creating such Bell states is given by the image below. Compute the final state of track (b) after each of the following measurements has occurred. Note that the two tracks are symmetric, so we can arbitrarily call the first quantum number ''track (a)'' and the second ''track (b)''. For hints, see the note on partial projection.
  
# In the classical Bell experiment, a pair of 2 entangled particles are created in state <math>|\psi\rangle = (|0,1\rangle + |1,0\rangle)/\sqrt{2}</math>. A quantum circuit for creating such Bell states is given by the image below. Compute the final state of track (b) after each of the following measurements has occurred. Note that the two tracks are symmetric, so we can arbitrarily call the first quantum number ''track (a)'' and the second ''track (b)''. For hints, see the note on partial projection.
 
#* Track (a) is measured and found to be in state <math>|0\rangle</math>.
  
#* Track (a) is measured and found to be in state <math>|0\rangle</math>.

Revision as of 14:02, 25 March 2019

Physical Chemistry II

Course Info

  • Course Numbers CHM 4411-001
  • Credit Hours: 4
  • Meeting Dates: Jan. 8 - Apr. 23, 2019
    • No Class Mar. 11-17
  • Meeting Times: Tues. and Thurs., 9:30-10:45 am in ISA 3048
    • Problem Sessions: Fri., 11am-12 pm in ISA 3050
    • Regular quizzes on Fridays
    • Office Hours: Fri., 10-11 am in IDR 200
  • Grading:
    • Quiz (30%)
      • To succeed in the quiz, complete the homework and study the topics covered in the previous week!
    • Exam 1 (20%) Fri., Feb. 8 11am-12pm (ISA 3050)
    • Exam 2 (20%) Fri., Mar. 8, 11am-12pm (ISA 3050)
    • Final (30%) Thurs., May 2 7:30-9:30am (ISA 3048)

Overview and Objectives

This course will introduce you to quantum theory, important for quantitatively describing atomic and molecular structure, chemical bonding and spectra.

Students in this course will demonstrate the ability to apply the following ideas:

  • Relationship between mathematical models and intermolecular forces.
  • Explaining quantum states and their mathematical and physical properties.
  • Connecting observed molecular properties with quantum measurements.
  • Calculation of quantum energy levels and spectra.

Textbooks

Resources

  • Visualization of Modes:
    • Wine Glass
    • Drum Head see also
      • Note: Modes are indexed by 2 numbers for a 2D surface.
    • Another 2D example
      • This one is part-way between a particle in a 2D box and a circular drum, since the center is a special point.
    • Violin String
      • Note: This looks like a sawtooth wave, so is less connected to quantum and more related to classical solitons.
    • Cymbals
      • Note: This shows many modes excited at once, so it is not a simple shape. Quantum-mechanically, this situation is called a superposition.
    • Tacoma Narrows Bridge
      • Acoustic and vibrational modes are very important in mechanical structures. We will calculate them for atoms and optical cavities.
    • Audio in general
  • Fundamental Dogma of Spectroscopy, <math>|E_2 - E_1| = h\nu</math>

[On finding atomic causes of laboratory observations,] I shall only give one example which has always struck me rather forcibly. If I decompose white light, I shall be able to isolate a portion of the spectrum, but however small it may be, it will always be a certain width. In the same way the natural lights which are called monochromatic give us a very fine ray, but one which is not, however, infinitely fine. It might be supposed that in the experimental study of the properties of these natural lights, by operating with finer and finer rays, and passing on at last to the limit, so to speak, we should eventually obtain the properties of a rigorously monochromatic light. That would not be accurate. I assume that two rays emanate from the same source, that they are first polarised in planes at right angles, that they are then brought back again to the same plane of polarisation, and that we try to obtain interference. If the light were rigorously monochromatic, there would be interference; but with our nearly monochromatic lights, there will be no interference, and that, however narrow the ray may be. For it to be otherwise, the ray would have to be several million times finer than the finest known rays.

    • He is saying that sunlight and light from incandescent bulbs has a continuous spectrum of all frequencies. Light from atomic transitions (like a high-pressure sodium lamp) has discrete spectral lines, but those still have a tiny line-width and two independent polarizations. This was not understood before the fundamental dogma of spectroscopy. The line-width is due to the energy-time uncertainty principle.
  • Quantum Computing
  • Life After Graduation

Group Work

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Assigned Homework Problems

  1. Part 1: Origins (Ch. 1, A, and 2)
    • Ch. 1, 1-40 (we'll do 41-44 in class)
    • Ch. A, 1-14
      • Hint on 12: use i = exp(...)
    • Ch. 2, 1-16, 19
      • Hint on 1,2, and 4: use y(x) = A exp(ax) + B exp(b x) and check
      • Hint on 5: use exp(i omega t) = ...
  2. Part 2: First Schrodinger Solutions (Ch. 3, C, E, F)
    • Ch. 3, 1-19,24,25,28,29,30,32,35
    • Ch. C, all problems except 10; Ch. E, prob. 7; and Ch. F, prob. 1-2,7,11
      • We will only multiply and take determinants of 2x2 matrices in this class, but adding larger matrices should be simple.
    • Supplemental Homework for Quiz 6
  3. Part 3: Measurement, Observables, Speakable and Unspeakable (Ch. 4-5)
    • Ch. 4, 1-3, 5, 7, 11, 14-16, 21-22

Special Assignment for Quiz 4

<math> f_1(x) = e^{ikx} </math> <math> f_2(x) = 2 i e^{-2x} </math> <math> f_3(x) = 7 x </math> <math> f_4(x) = x^2 - 1 </math> <math> f_5(x) = sin(2 \pi x/a) </math>
  1. For each of the following operators, list all of the functions above which are eigenfunctions. There may be more than one. For each, also identify the corresponding eigenvalue.
    • <math>\hat P = -i\hbar \frac{d}{dx}</math>
    • <math>\hat S = 3</math>
    • <math>\hat R = 2 x \frac{d}{dx} - 1</math>
    • <math>\hat H = -c \frac{d^2}{dx^2} </math>
  2. Find the normalization constant needed for each of the functions, <math>f_1 -- f_5</math> so that <math>\int_0^a (A_n f_n(x))^2 = 1</math>
    • <math>A_1 = </math>
    • <math>A_2 = </math>
    • <math>A_3 = </math>
    • <math>A_4 = </math>
    • <math>A_5 = </math>
  3. Complete the following problems from the text:
    • 3-1
    • 3-3
    • 3-4
    • 3-5
    • 3-10

Special Assignment for Quiz 7

  1. Provide, in your own words, definitions for each of the following: complete basis, expectation value, commutator, Hermitian operator, dimension, tunneling, nonlinear process, symmetry, continuous / continuity, integrable, divergent (of an expression), "existence" (of a math expression)
  2. For problem 2, choose one of the following two questions: (3 or 4, you do not have to do both)
  3. In the classical Bell experiment, a pair of 2 entangled particles are created in state <math>|\psi\rangle = (|0,1\rangle + |1,0\rangle)/\sqrt{2}</math>. A quantum circuit for creating such Bell states is given by the image below. Compute the final state of track (b) after each of the following measurements has occurred. Note that the two tracks are symmetric, so we can arbitrarily call the first quantum number track (a) and the second track (b). For hints, see the note on partial projection.
    • Track (a) is measured and found to be in state <math>|0\rangle</math>.
    • Track (a) is measured and found to be in state <math>|1\rangle</math>.
    • Track (a) is measured and found to be in state <math>(|0\rangle+|1\rangle)/\sqrt{2}</math>.
    • Under the first scenario (track (a) is in state <math>|0\rangle</math>), what is the probability that track (b) can be measured in state <math>\cos(\theta)|0\rangle + \sin(\theta)|1\rangle</math>? Compare this to the Bell-state correlation function [1].
  4. ( Problem 5 from class ) A Hadamard gate has matrix representation <math>H = \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]/\sqrt{2}</math>. A qbit initially in state <math>|0\rangle</math> is passed through the Hadamard gate to create an output state. Use <math>|0\rangle = \left[ \begin{array}{c}1 \\ 0\end{array}\right] </math> and <math>|1\rangle = \left[ \begin{array}{c}0 \\ 1\end{array}\right] </math> to compute each of the following:
    • The probability of detecting the output state is equal to <math>|0\rangle</math>
    • The probability of detecting the output state is equal to <math>|1\rangle</math>
    • The expectation value of <math>X = 5 |0\rangle\langle 0| + 2 |1\rangle\langle 1|</math>
    • The expectation value <math>\langle 0|X H|0\rangle</math>
    • The expectation value <math>\langle 0|H X|0\rangle</math>
    • The expectation value <math>\langle 0|H X H|0\rangle</math>
    • Which of the above corresponds to the expectation of the operator <math>X</math> when operating on the output state?

Partial projection is what happens to a quantum state when only one part of it is measured. The measured part must be projected into its known answer, while the rest of the state merely goes along for the ride. For our purposes, to do the partial projection of <math>

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