Create pages/40_PHY_614_Quantum_Mechanics_I.py

pages/40_PHY_614_Quantum_Mechanics_I.py

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+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
+""")