pages/40_PHY_614_Quantum_Mechanics_I.py

import streamlit as st

# Set the page title
st.title("PHY 614: Quantum Mechanics I")

# Course Details
st.markdown("""
## Course Details
- **Course Title**: Quantum Mechanics I  
- **Credits**: 3  
- **Prerequisites**: PHY 520  
- **Instructor**: [Instructor Name]  
- **Office Hours**: [Office Hours]  

## Course Description
An introduction to the fundamental principles and formalism of quantum mechanics, covering wave mechanics, the Schrödinger equation, angular momentum, and approximation methods. Applications to atomic and molecular systems will be discussed.

## Course Objectives
Upon completing this course, students will:
- Understand the formalism of quantum mechanics.
- Solve the Schrödinger equation for various potentials.
- Apply quantum mechanics to model atomic and molecular systems.

---
""")

# Weekly Outline with Problems

# Week 1-2: Introduction to Quantum Mechanics and the Schrödinger Equation
with st.expander("**Week 1-2: Introduction to Quantum Mechanics and the Schrödinger Equation**"):
    st.markdown("""
    ### Topics Covered
    - Historical Background: The failure of classical mechanics and the rise of quantum theory.
    - Wave-Particle Duality: The de Broglie hypothesis, wavefunctions, and the probabilistic interpretation of quantum mechanics.
    - The Schrödinger Equation: Time-dependent and time-independent forms.
    - Probability Density and Current: Interpretation of the wavefunction, normalization, and probability conservation.

    ### Problems
    1. Solve the time-independent Schrödinger equation for a free particle in one dimension and interpret the solution.
    2. Solve the Schrödinger equation for a particle in an infinite potential well (1D box) and find the energy eigenvalues and eigenfunctions.
    3. Normalize the wavefunction of a particle in a 1D box.
    4. Solve the time-dependent Schrödinger equation for a free particle wave packet.
    5. Calculate the probability density and current for a particle in a 1D potential.
    6. Solve the Schrödinger equation for a step potential and discuss the reflection and transmission coefficients.
    7. Analyze the behavior of a particle in a finite square well potential and determine the bound states.
    8. Apply boundary conditions to a particle in a symmetric potential and solve for the energy eigenstates.
    9. Calculate the time evolution of a Gaussian wave packet for a free particle.
    10. Compute the expectation values of position and momentum for a particle in a 1D box.
    """)

# Week 3-4: Operators, Eigenvalues, and Measurement Theory
with st.expander("**Week 3-4: Operators, Eigenvalues, and Measurement Theory**"):
    st.markdown("""
    ### Topics Covered
    - Linear Operators in Quantum Mechanics: Observables as Hermitian operators.
    - Commutators: Understanding the commutator relation and its implications for uncertainty.
    - Eigenvalues and Eigenstates: Solving for eigenvalues and eigenfunctions of quantum operators.
    - Measurement Theory: The measurement postulate, expectation values, and the collapse of the wavefunction.

    ### Problems
    11. Calculate the commutator [x, p] for position and momentum operators and verify the canonical commutation relation.
    12. Solve for the eigenvalues and eigenfunctions of the momentum operator in one dimension.
    13. Find the expectation value and uncertainty in position for a particle in a given wavefunction.
    14. Calculate the eigenvalues and eigenfunctions of the Hamiltonian for a particle in a 1D potential well.
    15. Verify the uncertainty principle for a Gaussian wave packet and calculate the minimum uncertainty.
    16. Apply the measurement postulate to find the probability distribution of a particle's position in a given state.
    17. Solve for the expectation value of the momentum operator for a particle in a harmonic potential.
    18. Analyze the effect of a measurement on a particle’s wavefunction using the projection postulate.
    19. Calculate the probability of measuring specific eigenvalues for a particle in a superposition of energy eigenstates.
    20. Solve for the time evolution of an observable using Heisenberg's equation of motion.
    """)

# Week 5-6: The Harmonic Oscillator
with st.expander("**Week 5-6: The Harmonic Oscillator**"):
    st.markdown("""
    ### Topics Covered
    - Schrödinger Equation for the Harmonic Oscillator: Exact solutions using power series and ladder operator methods.
    - Creation and Annihilation Operators: Algebraic solution of the harmonic oscillator using ladder operators.
    - Energy Quantization: Discrete energy levels and zero-point energy.
    - Coherent States: Introduction to coherent states as superpositions of energy eigenstates.

    ### Problems
    21. Solve the time-independent Schrödinger equation for the quantum harmonic oscillator using the differential equation method.
    22. Apply the ladder operator (creation and annihilation operators) method to find the energy eigenvalues of the harmonic oscillator.
    23. Calculate the ground-state wavefunction of the harmonic oscillator using the ladder operator approach.
    24. Find the first excited state wavefunction of the harmonic oscillator and verify the energy eigenvalue.
    25. Compute the expectation values of position and momentum for the ground and first excited states of the harmonic oscillator.
    26. Derive the commutation relation between the creation and annihilation operators for the harmonic oscillator.
    27. Calculate the zero-point energy of the harmonic oscillator.
    28. Solve for the time evolution of a coherent state in the harmonic oscillator potential.
    29. Analyze the behavior of a quantum system in a quadratic potential using the energy eigenstates of the harmonic oscillator.
    30. Solve for the uncertainty in position and momentum for the ground state of the harmonic oscillator.
    """)

# Week 7: Angular Momentum and Spin
with st.expander("**Week 7: Angular Momentum and Spin**"):
    st.markdown("""
    ### Topics Covered
    - Angular Momentum in Quantum Mechanics: Orbital angular momentum operators and their commutation relations.
    - Eigenvalues of \(L^2\) and \(L_z\): Deriving the quantized nature of angular momentum.
    - Spin Angular Momentum: Introduction to spin as an intrinsic form of angular momentum.
    - Addition of Angular Momenta: Coupling of spin and orbital angular momentum.

    ### Problems
    31. Solve for the eigenvalues and eigenfunctions of \(L^2\) and \(L_z\) for a particle in a central potential.
    32. Calculate the commutation relations between the components of the angular momentum operators.
    33. Find the eigenfunctions of the orbital angular momentum operator \(L_z\) for a particle on a sphere.
    34. Apply the ladder operator method to derive the quantized values of angular momentum for a particle in a spherical potential.
    35. Solve for the spin eigenvalues and eigenstates using the Pauli matrices.
    36. Calculate the total angular momentum for a system with coupled spin and orbital angular momentum.
    37. Analyze the behavior of a spin-1/2 particle in a magnetic field (e.g., Stern-Gerlach experiment).
    38. Solve for the probability of measuring a specific spin component for a particle in a superposition of spin states.
    """)

# Week 8-9: The Hydrogen Atom
with st.expander("**Week 8-9: The Hydrogen Atom**"):
    st.markdown("""
    ### Topics Covered
    - Schrödinger Equation in Spherical Coordinates: Solving the hydrogen atom problem.
    - Radial and Angular Parts of the Wavefunction: Separation of variables in spherical coordinates.
    - Energy Levels and Quantum Numbers: The quantization of energy and angular momentum in the hydrogen atom.
    - Degeneracy and Selection Rules: Degeneracy in energy levels and the rules governing transitions.

    ### Problems
    39. Solve the time-independent Schrödinger equation for the hydrogen atom in spherical coordinates.
    40. Derive the radial and angular parts of the hydrogen atom wavefunction using separation of variables.
    41. Calculate the energy levels of the hydrogen atom and interpret the quantum numbers \(n\), \(l\), and \(m\).
    42. Find the ground-state wavefunction of the hydrogen atom and normalize it.
    43. Solve for the radial probability distribution for the hydrogen atom in the \(n=1\) and \(n=2\) states.
    44. Calculate the expectation value of the radial distance for an electron in the ground state of the hydrogen atom.
    45. Derive the selection rules for allowed transitions between energy levels in the hydrogen atom.
    46. Compute the degeneracy of the energy levels of the hydrogen atom and explain its physical significance.
    47. Solve for the angular part of the hydrogen atom wavefunction and compute the spherical harmonics.
    48. Analyze the splitting of energy levels in the hydrogen atom due to an external magnetic field (Zeeman effect).
    """)

# Week 10-11: Approximation Methods: Time-Independent Perturbation Theory
with st.expander("**Week 10-11: Approximation Methods: Time-Independent Perturbation Theory**"):
    st.markdown("""
    ### Topics Covered
    - Time-Independent Perturbation Theory: Introduction to perturbation theory for non-degenerate and degenerate cases.
    - First-Order and Second
    
    ### Problems (continued)
    49. Apply first-order perturbation theory to calculate the energy shift in a hydrogen atom placed in an external electric field (Stark effect).
    50. Solve for the first-order correction to the wavefunction of a perturbed harmonic oscillator.
    51. Calculate the second-order energy correction for a quantum system with a known perturbation.
    52. Apply perturbation theory to a degenerate system and resolve the degeneracy using the perturbation.
    53. Analyze the effect of a weak magnetic field on the energy levels of a spin-1/2 particle using perturbation theory.
    54. Solve for the energy corrections to a particle in a finite potential well subjected to a small external potential.
    55. Compute the energy shifts in the hydrogen atom due to the relativistic correction using perturbation theory.
    56. Derive the first-order correction to the energy of the ground state of the helium atom.
    57. Solve for the energy correction due to the fine structure of the hydrogen atom using first-order perturbation theory.
    58. Apply time-independent perturbation theory to calculate the energy shift of a quantum system in a periodic potential.
    """)

# Week 12-13: Variational Principle and WKB Approximation
with st.expander("**Week 12-13: Variational Principle and WKB Approximation**"):
    st.markdown("""
    ### Topics Covered
    - Variational Method: Introduction to the variational principle as an approximation method for ground state energies.
    - Trial Wavefunctions: Choosing appropriate trial wavefunctions to minimize the energy functional.
    - WKB Approximation: Semi-classical approximation to solve the Schrödinger equation for slowly varying potentials.

    ### Problems
    59. Apply the variational method to estimate the ground-state energy of the helium atom.
    60. Use a trial wavefunction to minimize the energy of a particle in a 1D potential well using the variational principle.
    61. Solve for the energy of a quantum system using a Gaussian trial wavefunction and the variational principle.
    62. Calculate the ground-state energy of the hydrogen atom using the variational method with an appropriate trial wavefunction.
    63. Apply the WKB approximation to solve for the bound states of a particle in a 1D potential well.
    64. Derive the WKB wavefunction for a particle tunneling through a potential barrier and calculate the tunneling probability.
    65. Solve for the energy levels of a slowly varying potential using the WKB approximation.
    66. Apply the WKB approximation to estimate the energy spectrum of a quantum system with a known potential.
    """)

# Week 14: Introduction to Identical Particles and Symmetry
with st.expander("**Week 14: Introduction to Identical Particles and Symmetry**"):
    st.markdown("""
    ### Topics Covered
    - Symmetrization and Antisymmetrization: Wavefunctions for identical bosons and fermions.
    - Pauli Exclusion Principle: The role of symmetry in multi-electron systems.
    - Exchange Interaction: Consequences of particle indistinguishability in atomic and molecular systems.

    ### Problems
    67. Symmetrize and antisymmetrize the wavefunction for a system of two identical bosons and two identical fermions.
    68. Apply the Pauli exclusion principle to solve for the ground-state configuration of a two-electron atom (helium).
    69. Solve for the wavefunction of a system of three identical fermions in a 1D potential well.
    70. Analyze the exchange interaction between two identical particles in a harmonic potential.
    71. Derive the energy levels of a multi-electron atom using the principles of symmetry and particle exchange.
    72. Calculate the total spin state for a system of two spin-1/2 particles in a singlet or triplet configuration.
    73. Apply the concept of exchange degeneracy to solve for the energy spectrum of a system of identical bosons.
    74. Solve for the symmetry properties of a multi-particle wavefunction in an external potential.
    """)

# Week 15: Review and Final Exam Preparation
with st.expander("**Week 15: Review and Final Exam Preparation**"):
    st.markdown("""
    ### Topics Covered
    - Comprehensive Review: Review of quantum mechanics topics covered throughout the semester.
    - Problem-Solving Sessions: In-class discussions of complex problems and solutions.
    - Final Exam Preparation: Focus on key topics and techniques for solving exam-level problems.

    ### Problems
    75. Solve for the energy levels of the hydrogen atom using both analytical and perturbation methods.
    76. Analyze the quantum harmonic oscillator using ladder operators and calculate the energy corrections due to a small perturbation.
    77. Compute the eigenvalues and eigenfunctions for a particle in a central potential using angular momentum operators.
    78. Apply the variational principle to estimate the ground-state energy of a complex quantum system.
    79. Solve for the time evolution of a quantum system using both Schrödinger and Heisenberg pictures.
    80. Analyze the behavior of identical particles in a 1D potential well and solve for the energy spectrum.
    """)

# Textbooks Section
st.markdown("""
## Textbooks
- **"Principles of Quantum Mechanics"** by R. Shankar  
- **"Modern Quantum Mechanics"** by J. J. Sakurai and Jim Napolitano
""")