# Worksheets

• Worksheet 0: Introduction to Complex Numbers
Basics operations of complex numbers and functions are introduced.
• Worksheet 00: Vectors and Coordinate Systems
Review of vectors in Cartesian and spherical coordinates. Conversion of vectors (functions) between the two systems. Review the concept of a volume element and essentially introduce the Jacobian (although not explicitly termed).
• Worksheet 1: Continuous Distributions
This worksheet address the definition of a continuous distributions with an emphasis on probability distributions. Four principal properties are discussion: most probable, average (expectation value), spread (standard deviation), and integrated value.
• Worksheet 2: Separation of Variables
This worksheet discusses some of the basics of the separation of variables that is used in QM extensively (e.g., separating spatial and temporal components of the wave equation).
• Worksheet 3A: Operators and Eigenvalues
This worksheet addresses the fundamentals of operators and eigenvalues. Also include is a section address order of operations and commutators (a little bit).
• Worksheet 3B: Particle in a Box
This worksheet address several aspects of PIB to master including using the wavefunctions as probability densities and normalization (both in 1D and 3D).
• Worksheet 4A: Introduction to the Quantum Harmonic Oscillator
Introduction to quantum harmonic oscillator model system is empowered by a discussion of the classical harmonic oscillator.
• Worksheet 4B: Even and Odd Functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations. Sometimes the form of a function helps us to solve problems. This is particularly true for even or odd functions.
• Worksheet 5: Vibrational Spectroscopy
The basics of vibrational spectroscopy within a harmonic oscillator perspective is discussed including relationships between freq., spring constants, masses, reduced masses and bond lengths.
• Worksheet 6A: Rotational Spectroscopy
The rigid rotator model is used to interpret rotational spectra of diatomic molecules. This groupwork exercise aims to help you connect the rigid rotator model to rotational spectroscopy.
• Worksheet 6B: Angular Momentum and Commutators
Review of the quantum mechanical angular momentum operators (in all three directions and $$L^2$$). Calculate the commutators of these operators to essentially derive the cyclical relationship of $$\hat{L}_x$$, $$\hat{L}_y$$, and $$\hat{L}_z$$. Show that $$\hat{L}^2$$ commutes with each component. Show that the spherical coordinates are better to use than Cartesian in doing these calculations.
• Worksheet 7A: The Probabilistic Interpretation of Atomic Orbitals
The basic properties of hydrogenic wavefunctions are discussion including nodes and degeneracy
• Worksheet 7B: Variational Method I
Application of the Variational Method applied to a harmonic oscillator (which is continued in the second worksheet). This worksheet addresses the first part of variational method: the derivation of the trial energy as a function of external parameters.
• Worksheet 8: Variational Method II
Application of the Variational Method applied to a harmonic oscillator continued from the first worksheet. This worksheet addresses the second part of variational method: the minimization approach.
• Worksheet 9A: Determinants
Introduction of Slater determinint wavefunctions with an emphasis on the permutation aspect of determinants to address symmetry aspects of fermions.
• Worksheet 9B: Multi-Electron Wavefunctions
Introduction of Slater determinint wavefunctions with an emphasis on the permutation aspect of determinants to address symmetry aspects of fermions.
• Worksheet 10A: Adding Angular Momenta in Multi-Electron Atoms
This worksheet introduces how orbital angular momentum and electron spin momenta coupled (add) in multi-electron atoms. Term symbols are introduced by not extensively worked with.
• Worksheet 10B: The Dihydrogen Cation
This worksheet discussed a basic overview of the molecular hydrogen cation, which is the simplest molecule to solve since it has one electron. The LCAO method is motivated and used to solve for the energy of this system (instead of solving the Schrodinger equation directly). This involves calculation of the Coulomb, Exchange and Overlap integrals.