Create PHY_615_Quantum_Mechanics_II.py

pages/PHY_615_Quantum_Mechanics_II.py

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+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**"):