pages/50_PHY_615_Quantum_Mechanics_II.py

import streamlit as st

# Set the page title
st.title("PHY 615: Quantum Mechanics II")

# Course Details
st.markdown("""
## Course Details
- **Course Title**: Quantum Mechanics II  
- **Credits**: 3  
- **Prerequisites**: PHY 614  
- **Instructor**: [Instructor Name]  
- **Office Hours**: [Office Hours]  

## Course Description
This course builds on Quantum Mechanics I, delving deeper into perturbation theory, scattering theory, symmetry, and invariance. Students will explore time-dependent quantum phenomena and apply quantum mechanics to more complex systems.

## Course Objectives
Upon completing this course, students will:
- Be proficient in time-dependent perturbation theory.
- Apply quantum mechanics to scattering problems.
- Understand the role of symmetry and invariance in quantum systems.
- Explore applications to atomic, molecular, and quantum field systems.

---
""")

# Weekly Outline with Problems

# Week 1: Review of Quantum Mechanics I
with st.expander("**Week 1: Review of Quantum Mechanics I**"):
    st.markdown("""
    ### Topics Covered
    - Review of key concepts from Quantum Mechanics I.
    - Time-independent perturbation theory and the Schrödinger equation.
    - Angular momentum and spin.

    ### Problems
    1. Solve the time-independent Schrödinger equation for a particle in a finite potential well.
    2. Apply time-independent perturbation theory to a hydrogen-like atom in an external electric field (Stark effect).
    3. Compute the eigenvalues and eigenfunctions for a particle in a 3D box.
    4. Use the ladder operator approach to solve the quantum harmonic oscillator.
    5. Solve for the angular momentum eigenstates of a particle in a central potential.
    6. Derive the commutation relations between the components of the angular momentum operators.
    7. Solve for the spin-1/2 particle eigenstates using Pauli matrices.
    8. Calculate the energy shift in a two-level system under a weak magnetic field using first-order perturbation theory.
    """)

# Week 2-3: Time-Dependent Perturbation Theory
with st.expander("**Week 2-3: Time-Dependent Perturbation Theory**"):
    st.markdown("""
    ### Topics Covered
    - Time-dependent Schrödinger equation.
    - Time-dependent perturbation theory and transition probabilities.
    - Periodic perturbations and transition amplitudes.

    ### Problems
    9. Solve the time-dependent Schrödinger equation for a two-level atom in an oscillating electric field.
    10. Apply time-dependent perturbation theory to a particle in a harmonic potential subjected to a time-varying force.
    11. Calculate the transition probability between two quantum states due to a sudden perturbation.
    12. Derive the transition amplitude for a system undergoing a periodic perturbation.
    13. Compute the probability of transition from the ground state to an excited state in a system subjected to a sinusoidal perturbation.
    14. Analyze the effect of a time-dependent perturbation on a particle in a potential well.
    15. Solve for the transition probabilities of a particle in a harmonic oscillator subjected to a weak time-dependent perturbation.
    16. Use the Dyson series to compute the evolution operator for a system under time-dependent perturbation.
    17. Derive the first-order transition amplitude for a particle in a potential subject to a weak time-dependent perturbation.
    18. Apply time-dependent perturbation theory to a quantum system interacting with an external electromagnetic field.
    """)

# Week 4: Fermi’s Golden Rule and Transition Rates
with st.expander("**Week 4: Fermi’s Golden Rule and Transition Rates**"):
    st.markdown("""
    ### Topics Covered
    - Fermi’s Golden Rule.
    - Transition rates and weak perturbations.
    - Applications in quantum systems.

    ### Problems
    19. Derive Fermi’s Golden Rule for the transition rate of a quantum system subjected to a weak perturbation.
    20. Calculate the spontaneous emission rate of a photon from an excited atom using Fermi's Golden Rule.
    21. Solve for the transition rate of a particle scattering off a potential using Fermi's Golden Rule.
    22. Apply Fermi’s Golden Rule to calculate the decay rate of an unstable particle.
    23. Derive the absorption rate of radiation by an atom in an external field using Fermi’s Golden Rule.
    24. Solve for the transition rate of an electron in an atom interacting with a photon field.
    25. Use Fermi’s Golden Rule to compute the decay rate of a system in a potential well.
    26. Calculate the transition rate for an atom interacting with a laser field.
    """)

# Week 5-6: Scattering Theory: Born Approximation, Partial Waves
with st.expander("**Week 5-6: Scattering Theory: Born Approximation, Partial Waves**"):
    st.markdown("""
    ### Topics Covered
    - Quantum scattering theory.
    - Born approximation for weak potentials.
    - Partial wave expansion and phase shifts.

    ### Problems
    27. Apply the Born approximation to a particle scattering off a delta-function potential.
    28. Solve for the scattering cross-section of a particle in a square potential using the Born approximation.
    29. Calculate the scattering amplitude for a particle interacting with a Yukawa potential using the Born approximation.
    30. Use the Born approximation to compute the differential cross-section for electron-atom scattering.
    31. Derive the scattering cross-section for a particle in a hard sphere potential using partial wave analysis.
    32. Apply partial wave expansion to solve the scattering problem for a spherical potential.
    33. Calculate the phase shift for a particle scattered by a central potential.
    34. Solve for the total cross-section of a particle scattering off a spherically symmetric potential.
    35. Analyze the low-energy scattering of a particle in a potential well using partial waves.
    36. Derive the phase shifts for scattering off a potential with a long-range tail.
    37. Solve for the scattering amplitude using the Lippmann-Schwinger equation and Born approximation.
    38. Compute the differential cross-section for neutron-proton scattering using the partial wave method.
    """)

# Week 7: Symmetry and Conservation Laws
with st.expander("**Week 7: Symmetry and Conservation Laws**"):
    st.markdown("""
    ### Topics Covered
    - Symmetry in quantum mechanics.
    - Conservation laws derived from symmetries.
    - Parity, time-reversal symmetry, and selection rules.

    ### Problems
    39. Derive the conservation of angular momentum for a particle in a central potential.
    40. Use Noether’s theorem to show how symmetry leads to conservation laws in quantum mechanics.
    41. Apply parity symmetry to a quantum system and determine the selection rules for transitions.
    42. Calculate the consequences of time-reversal symmetry for a two-level quantum system.
    43. Solve for the energy eigenstates of a particle in a potential with reflection symmetry.
    44. Analyze the role of rotational symmetry in determining the degeneracy of energy levels in a quantum system.
    45. Use parity conservation to solve for the transition probabilities in an atomic system.
    46. Compute the effect of time-reversal symmetry breaking on the spin states of a particle in a magnetic field.
    47. Derive the selection rules for electric dipole transitions in atoms using symmetry arguments.
    48. Apply conservation laws to solve a problem involving the scattering of particles in a central potential.
    """)

# Week 8-9: Quantum Mechanics of Identical Particles
with st.expander("**Week 8-9: Quantum Mechanics of Identical Particles**"):
    st.markdown("""
    ### Topics Covered
    - Symmetrization and antisymmetrization of wavefunctions.
    - Pauli exclusion principle and fermions.
    - Exchange interactions and quantum statistics.

    ### Problems
    49. Symmetrize the wavefunction of a system of two identical bosons in a 1D harmonic oscillator potential.
    50. Antisymmetrize the wavefunction of a system of two identical fermions in a potential well.
    51. Apply the Pauli exclusion principle to solve for the ground state configuration of the helium atom.
    52. Compute the exchange interaction energy for a system of two electrons in a hydrogen molecule.
    53. Analyze the effect of particle indistinguishability on the energy levels of a system of fermions.
    54. Solve for the wavefunction of a system of three identical fermions in a harmonic potential.
    55. Use the spin-statistics theorem to determine the behavior of identical particles in a quantum system.
    56. Calculate the ground state wavefunction of a system of identical bosons in a double well potential.
    57. Apply the symmetrization principle to solve for the energy levels of a multi-electron atom.
    58. Analyze the role of the Pauli exclusion principle in determining the structure of the periodic table.
    """)

# Week 10-11: Applications to Atomic and Molecular Systems
with st.expander("**Week 10-11: Applications to Atomic and Molecular Systems**"):
    st.markdown("""
    ### Topics Covered
    - Fine and hyperfine structure of atoms.
    - Quantum mechanics of molecular bonding.
    - Vibrational and rotational energy levels in
    ### Topics Covered (continued)
    - Vibrational and rotational energy levels in diatomic molecules.
    - Born-Oppenheimer approximation and molecular orbitals.

    ### Problems
    59. Calculate the fine structure of the hydrogen atom using perturbation theory.
    60. Analyze the hyperfine structure of the hydrogen atom and compute the energy shifts.
    61. Solve for the rotational energy levels of a diatomic molecule using quantum mechanics.
    62. Compute the vibrational energy levels of a diatomic molecule using the harmonic approximation.
    63. Apply the Born-Oppenheimer approximation to separate the nuclear and electronic motion in a molecule.
    64. Derive the energy levels of a hydrogen molecule using the quantum mechanical treatment of molecular bonding.
    65. Solve for the electronic structure of the helium atom using the Hartree-Fock approximation.
    66. Compute the rovibrational energy levels of a diatomic molecule using perturbation theory.
    67. Analyze the fine and hyperfine structure of alkali atoms using quantum mechanics.
    68. Apply quantum mechanics to solve for the molecular orbitals of a multi-atom molecule.
    """)

# Week 12-13: Advanced Topics: Quantum Field Theory
with st.expander("**Week 12-13: Advanced Topics: Quantum Field Theory**"):
    st.markdown("""
    ### Topics Covered
    - Introduction to quantum field theory (QFT).
    - Second quantization and field operators.
    - Applications of quantum field theory in particle interactions.

    ### Problems
    69. Quantize the electromagnetic field using the second quantization formalism.
    70. Solve for the vacuum expectation value of the number operator in quantum electrodynamics (QED).
    71. Compute the scattering amplitude for electron-positron annihilation in QED.
    72. Analyze the creation and annihilation operators in a simple quantum field theory.
    73. Derive the Feynman rules for a scalar quantum field theory.
    74. Apply the second quantization method to solve for the energy levels of a quantum field.
    75. Compute the interaction energy between two particles using quantum field theory.
    76. Use Feynman diagrams to solve for the scattering amplitude of two particles in QED.
    """)

# Week 14: Open Quantum Systems and Decoherence
with st.expander("**Week 14: Open Quantum Systems and Decoherence**"):
    st.markdown("""
    ### Topics Covered
    - Open quantum systems and interaction with the environment.
    - Decoherence and quantum coherence.
    - Master equation and Lindblad equation for open quantum systems.

    ### Problems
    77. Derive the master equation for an open quantum system interacting with its environment.
    78. Solve for the time evolution of a quantum system undergoing decoherence.
    79. Apply the Lindblad equation to compute the loss of coherence in a two-level quantum system.
    80. Analyze the effect of environmental interaction on the quantum states of a particle in a potential well.
    81. Solve for the steady-state solution of an open quantum system coupled to a thermal bath.
    82. Compute the decoherence time for a quantum system interacting with an external reservoir.
    83. Derive the density matrix for an open quantum system and solve for its time evolution.
    84. Analyze the role of decoherence in quantum computing and quantum information theory.
    """)

# Week 15: Review and Final Exam Preparation
with st.expander("**Week 15: Review and Final Exam Preparation**"):
    st.markdown("""
    ### Topics Covered
    - Comprehensive review of time-dependent perturbation theory, scattering, and symmetry.
    - Applications to atomic, molecular, and quantum field systems.
    - Problem-solving sessions and final exam preparation.

    ### Problems
    85. Solve for the scattering cross-section of a particle in a central potential using the Born approximation and partial waves.
    86. Compute the transition rate for a quantum system undergoing spontaneous emission using Fermi’s Golden Rule.
    87. Apply time-dependent perturbation theory to solve for the transition probability of a quantum system in an oscillating field.
    88. Solve for the energy levels of a multi-electron atom using the Pauli exclusion principle and perturbation theory.
    89. Analyze the quantum behavior of identical particles in a potential well and compute the exchange interaction energy.
    90. Use the second quantization formalism to compute the energy levels of a quantum field.
    """)

# Textbooks Section
st.markdown("""
## Textbooks
- **"Modern Quantum Mechanics"** by J. J. Sakurai and Jim Napolitano  
- **"Quantum Mechanics: Concepts and Applications"** by Nouredine Zettili
""")