# CHEM106: Physical Chemistry II

Unit 2: Principles of Quantum Theory

In the classical world, Newton’s laws and laws of conservation of energy are used daily to predict the outcome of certain events.  For instance, the trajectory of a cannonball can be predicted with good accuracy, given certain parameters such as the shooting angle, mass of the cannonball, air resistance, and so forth.  In the world of quantum mechanics, where objects possess very small masses and move at high speeds, the Schrodinger equation is used to predict the behavior of a system comprised of quantum mechanical particles (electrons, nuclei, etc.).  Unlikely in classical mechanics, where the outcomes of the calculations are undoubtedly clear (for instance, the cannon ball will hit the ground at a precise coordinate point), the results of the Schrodinger equation gives the probability of an event to occur.  In the quantum world, for a given set of parameters, a large number of events can occur, and the Schrodinger equation will predict the probability that such events will happen, given a distribution of results rather than a single answer.  In this unit, you will encounter the Schrodinger equation for the first time.

This unit will take you 12.5 hours to complete.

☐    Subunit 2.1: 2.5 hours

☐    Subunit 2.2: 5 hours

☐    Subunit 2.3: 0.5 hours

☐    Subunit 2.4: 4.5 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - List and explain the fundamental postulates of quantum mechanics. - Set up and solve the Schrodinger equation for a simple system. - Mathematically prove Heisenberg’s uncertainty principle. - Normalize the wavefunction of a system. - Calculate the probability of finding the particle in a region of space.

2.1 Fundamental Postulates of Quantum Mechanics   - Reading: Washington State University: Professor Kirk Peterson’s “Chem 332: Physical Chemistry II” Link: Washington State University: Professor Kirk Peterson’s “Chem 332: Physical Chemistry II” (PDF)

Instructions: To access Professor Peterson’s resources, click on the link and scroll down to the bottom of the webpage to the “Class Recourses” heading.  Select the “Postulates of Quantum Mechanics” link to open the PDF file, and read and learn about these postulates.  Make sure you understand the “practical” meaning of these equations, that is, why the wavefunction of a quantum mechanical system is normalized, and so forth.  These postulates constitute the base of quantum mechanical problems and calculations.  Studying this resource should take approximately 1 hour to complete.

• Reading: Georgia Institute of Technology: Professor David Sherrill’s “Postulates of Quantum Mechanics” Link: Georgia Institute of Technology: Professor David Sherrill’s “Postulates of Quantum Mechanics” (HTML)

Instructions: Please click on the link to access Professor Sherrill’s resource, and read the entire webpage.  Make sure you understand the “practical” meaning of these equations, that is, which physical observables (momentum, kinetic energy, etc.) are associated with Hermitian operators, and so forth.  These postulates constitute the base of quantum mechanical problems and calculations.  Studying this resource should take approximately 1 hour to complete.

Instructions: Please click on the link above, and watch the brief video lecture.  This video covers the fundamental postulates of quantum mechanics; as for the above readings, make sure you understand the “practical” meaning of these equations and how they are used in quantum mechanical calculations.  Studying this resource should take approximately 0.5 hours to complete.

2.2 The Schrodinger Equation   - Reading: Everyscience.com’s “The One Dimensional Schrodinger Equation” Links: Everyscience.com’s “The One Dimensional Schrodinger Equation” (HTML)

Instructions: Please click on the link and read the entire webpage.  While reading, solve the Hamiltonian following the steps indicated in the reading.  This provides a glance of the Schrodinger equation.  Studying this resource should take approximately 1 hour to complete.

• Reading: Macquarie University: Professor James Cresser’s “Interference and Diffraction & Wave Mechanics” Lecture Notes Link: Macquarie University: Professor James Cresser’s “Interference and Diffraction & Wave Mechanics” Lecture Notes (PDF)

Instructions: Please click on the link and select “Ch6: The Schrodinger Wave Equation” to download the PDF file, and read the entire text (18 pages).  While reading, following the math steps indicated in the reading to find the wavefunction of the system by solving the Hamiltonian.  Also make sure you understand important concepts such as boundary and continuity conditions.  Professor Cresser’s notes provide a deeper explanation of the Schrodinger equation.  Studying this resource should take approximately 3 hours to compete.

• Reading: YouTube: Praba Siva’s “Schrodinger Equation – Step Wise Derivation Part 1/3,” “Part 2/3,” and “Part 3/3” Links: YouTube: Praba Siva’s “Schrodinger Equation – Step Wise Derivation Part 1/3”, “Part 2/3”, and “Part 3/3” (YouTube)

Instructions: Watch these three videos.  While watching the videos, follow the instructor by doing the same math on a sheet of paper until you are comfortable deriving the Schrodinger equation on your own.  This video tutorial shows a step-by-step derivation of the Schrodinger equation.  Studying these resources should take approximately 1 hour to complete.

• Assessment: The Saylor Foundation’s “Assessment 3” Link: The Saylor Foundation’s “Assessment 3” (DOC)

Instructions: Complete the attached assessment questions to check your understanding of the material covered thus far. Once you have completed the assessment, you may check your answers against the “Answer Key” (DOC).

Completing this assessment should take approximately 1 hour.

2.3 Wavefunctions and the Born Interpretation   - Reading: Everyscience.com’s “Wavefunctions and the Born Interpretation” Link: Everyscience.com’s “Wavefunctions and the Born Interpretation” (HTML)

`````` Instructions: Clink on the above link to access the webpage.
wavefunction as the “source of all measurable information of a
quantum mechanical system.”  Studying this resource should take
approximately 0.5 hours to complete.

displayed on the webpage above.
``````

2.4 Wave Functions and the Heisenberg’s Uncertainty Principles   - Reading: Robert B. Griffiths’s “Consistent Quantum Theory” Link: Robert B. Griffiths’s “Consistent Quantum Theory” (PDF)

Instructions: Please click on the link, find “Chapter 2: Wave Functions,” select the PDF link for the text, and read sections 2.1–2.4.  This resource explains the physical interpretation of the wavefunction.  Studying this resource should take approximately 2.0 hours to complete.

• Reading: Macquarie University: Professor James Cresser’s “The Wave Function” Lecture Notes Links: Macquarie University: Professor James Cresser’s “The Wave Function” Lecture Notes (PDF)

Instructions: Please click on the link, select the link to “Ch3: The Wave Function” to access the PDF file, and read the entire chapter (8 pages).  Make sure you understand the concept of “probability waves” and the derivation of the Heisenberg uncertainty principle.  Studying this resource should take approximately 1.5 hours to complete.

• Reading: Everyscience.com’s “Heisenberg’s Uncertainty Principle” Link: Everyscience.com’s “Heisenberg’s Uncertainty Principle” (HTML)

Instructions: Please click on the link and read the entire webpage for a quick explanation and useful review of “Heisenberg’s Uncertainty Principle.”  This resource is useful to quickly review the mathematical proof of the Uncertainly Principle.  Make sure you know how to derive the uncertainty in momentum and position.  Studying this resource should take approximately 0.5 hours to complete.