Hydrogen Atom: Schrödinger Equation

The Hydrogen Atom

• The hydrogen atom, with its single proton and electron, serves as a fundamental model for atomic structure.
• Coulomb force binds the electron and proton together, defined as $F = -e^2 / 4 \pi \epsilon_0 r^2$.
  • e represents the elementary charge ($1.602 \times 10^{-19} C$).
  • $\epsilon_0$ denotes the permittivity of free space ($8.854 \times 10^{-12} C^2 / N m^2$).
  • r symbolizes the distance separating the electron and proton.
• The potential energy from the Coulomb force is $V(r) = -e^2 / 4 \pi \epsilon_0 r$.

Schrödinger Equation for Hydrogen

• The Schrödinger equation for the hydrogen atom can be analytically solved.

Schrödinger Equation in Spherical Coordinates

• Solving the Schrödinger equation for hydrogen is most effective in spherical coordinates $(r, \theta, \phi)$.
• Schrödinger equation: $\hat{H} \psi(r, \theta, \phi) = E \psi(r, \theta, \phi)$.
  • $\hat{H}$ is the Hamiltonian, given by $\hat{H} = -\frac{\hbar^2}{2\mu} \nabla^2 + V(r)$.
    • $\mu$ is the reduced mass of the electron-proton system: $\mu = m_e m_p / (m_e + m_p)$.
    • $\hbar$ is the reduced Planck constant: $\hbar = h / 2\pi = 1.054 \times 10^{-34} J s$.
    • $V(r)$ is the potential energy.
• The Laplacian operator $\nabla^2$ in spherical coordinates is $\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 \frac{\partial}{\partial r}) + \frac{1}{r^2 sin\theta} \frac{\partial}{\partial \theta} (sin\theta \frac{\partial}{\partial \theta}) + \frac{1}{r^2 sin^2\theta} \frac{\partial^2}{\partial \phi^2}$.

Separation of Variables

• The wavefunction can be expressed as $\psi(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi)$ to solve the Schrödinger equation.

Algorithmic and High-Frequency Trading Course

• Álvaro Cartea is the instructor.
• Michaelmas Term is when this course takes place.
• The course is for MSc/PhD level (Module Code: WF532).

Course Description

• The course provides essential knowledge for analyzing high-frequency data, modeling market microstructure, and backtesting trading strategies.

Course Objectives

• Understand modern electronic markets.
• Comprehend market microstructure.
• Learn to analyze and process tick data.
• Grasp market making, optimal execution, and statistical arbitrage.
• Develop backtesting skills.

Course Structure

• Introduction to Electronic Markets and Market Microstructure:
  • Order book dynamics
  • Market participants and their strategies
  • Market impact
• High-Frequency Data Analysis:
  • Tick data cleaning and filtering
  • Time aggregation and volatility estimation
  • Realized volatility measures
• Market Making:
  • Inventory management models
  • Adverse selection and information asymmetry
  • Optimal quoting strategies
• Optimal Execution:
  • Volume Weighted Average Price (VWAP) and Time Weighted Average Price (TWAP) strategies
  • Stochastic control models for optimal trading
  • Algorithmic execution platforms
• Statistical Arbitrage:
  • Pairs trading and cointegration
  • Latent factor models
  • Machine learning techniques for arbitrage
• Backtesting and Performance Evaluation:
  • Transaction cost modeling
  • Risk management and capital allocation
  • Performance metrics: Sharpe ratio, Sortino ratio, etc.

Assessment

• 40% of the grade is from Assignments.
• 60% of the grade is from the final exam.

Resources

• The main textbook is Algorithmic and High-Frequency Trading by Álvaro Cartea, Sebastian Jaimungal, and José Penalva.
• Research papers and articles provided as additional readings.

Academic Integrity

• Work must be original and free of plagiarism.

Contact

• Email: [email protected]
• Office hours: To be announced.