Quantizing Schrodinger Field continued

1. Classical Action:

   • Started with a classical system described by the action: $�=\int ��\int {�}^3�\mathcal{�}$ where $\mathcal{�}={�}^{\ast }(�\mathrm{\hslash }{\mathrm{\partial }}_{�}-\widehat{�})�$.
   • Schrödinger equation arises from the classical action.

2. Expansion and Eigenfunctions:

   • Expanded $�$ in terms of eigenfunctions ${�}_{�}$ of the Hamiltonian $\widehat{�}$: $�={\sum }_{�}{�}_{�}{�}_{�}$.
   • ${�}_{�}$ satisfies $\widehat{�}{�}_{�}={�}_{�}{�}_{�}$.

3. Quantization:

   • ${�}_{�}$ and ${�}_{�}^{\ast }$ were taken as generalized coordinates and momenta.
   • Replaced functions with operators, imposing commutation relations: $[{�}_{�},{�}_{�}^{\ast }]={�}_{��}$.

4. Quantum Hamiltonian:

   • Constructed the quantum Hamiltonian: $\widehat{�}={\sum }_{�}{�}_{�}{�}_{�}^{\ast }{�}_{�}$.

5. Infinite Harmonic Oscillators:

   • Described the resulting quantum system as an infinite collection of harmonic oscillators with different frequencies ${�}_{�}=\frac{{�}_{�}}{\mathrm{\hslash }}$.
   • No interaction between oscillators ($�$ in $\widehat{�}$ doesn't lead to interactions).

Single Harmonic Oscillator Recap:

1. Hamiltonian:

   • For a single harmonic oscillator: $\widehat{�}=\mathrm{\hslash }�{�}^{†}�+\frac{\mathrm{\hslash }�}{2}$.

2. Ground State and Excited States:

   • Defined ground state $\mid 0⟩$ as $\widehat{�}\mid 0⟩=0$.
   • Showed it's an eigenstate of $\widehat{�}$ with energy $\frac{\mathrm{\hslash }�}{2}$.
   • Excited states obtained by applying ${�}^{†}$ to the ground state.

3. Building Excited States:

   • Illustrated how to build states with multiple excitations, finding their energies.

Equivalence and Next Steps:

1. Equivalence to First Quantized System:

   • Demonstrated that the system of infinite harmonic oscillators is equivalent to another system, which will be the first quantized theory.

2. Future Directions:

   • Mentioned moving towards quantum field theory for Klein-Gordon fields in subsequent videos.

Conclusion:

• Introduced a two-step quantization approach: Schrödinger equation followed by quantization of $�$ operators.
• Established a system of uncoupled harmonic oscillators, providing a foundation for more advanced quantum field theories.

  Quantization of the System:

  • Classical System: Started with a classical system described by the action $�=\int ��\int {�}^3�\mathcal{�}$, where $\mathcal{�}={�}^{\ast }(�\mathrm{\hslash }{\mathrm{\partial }}_{�}-\widehat{�})�$.

  • Schrödinger Equation: Derived the Schrödinger equation by taking variations of the action, leading to $\widehat{�}�=�\mathrm{\hslash }{\mathrm{\partial }}_{�}�$.

  2. Expansion in Eigenfunctions:

  • Eigenfunctions: Expanded the wavefunction $�$ using the eigenfunctions ${�}_{�}$ of the Hamiltonian $\widehat{�}$: $�={\sum }_{�}{�}_{�}{�}_{�}$.

  • Eigenvalue Equation: Showed that ${�}_{�}$ satisfies $\widehat{�}{�}_{�}={�}_{�}{�}_{�}$.

  3. Quantization Procedure:

  • Generalized Coordinates and Momenta: Took ${�}_{�}$ and ${�}_{�}^{\ast }$ as generalized coordinates and momenta, respectively.

  • Quantization and Commutation Relations: Replaced ${�}_{�}$ and ${�}_{�}^{\ast }$ with operators and imposed commutation relations: $[{�}_{�},{�}_{�}^{\ast }]={�}_{��}$.

  4. Quantum Hamiltonian:

  • Hamiltonian Operator: Constructed the quantum Hamiltonian: $\widehat{�}={\sum }_{�}{�}_{�}{�}_{�}^{\ast }{�}_{�}$.

  • Infinite Harmonic Oscillators: Described the resulting quantum system as an infinite collection of harmonic oscillators with different frequencies ${�}_{�}=\frac{{�}_{�}}{\mathrm{\hslash }}$.

  5. Single Harmonic Oscillator Recap:

  • Hamiltonian: For a single harmonic oscillator: $\widehat{�}=\mathrm{\hslash }�{�}^{†}�+\frac{\mathrm{\hslash }�}{2}$.

  • Ground State and Excited States: Defined the ground state $\mid 0⟩$ and excited states, showing their energies.

  6. Equivalence and First Quantized System:

  • Demonstrated that the system of infinite harmonic oscillators is equivalent to another system, which will be the first quantized theory.

  • Mentioned the transition towards quantum field theory for Klein-Gordon fields in subsequent videos.

  Applications and Theory:

  1. Applications:

     • This formalism is foundational for understanding quantum systems with infinite degrees of freedom, especially in the context of quantum field theory.

  2. Theory Building:

     • The approach combines classical mechanics, quantum mechanics, and field theory, providing a bridge between different levels of understanding in theoretical physics.

  3. Quantum Harmonic Oscillators:

     • The concept of infinite harmonic oscillators serves as a powerful tool in describing complex quantum systems and finding their energy states.

  4. Mathematical Framework:

     • The use of eigenfunctions, operators, and commutation relations establishes a rigorous mathematical framework for quantum mechanics.