Classical Treatment of Quantum Field - Quantizing Schrodinger Field

  1. Classical Treatment of Quantum Field:

    • Treat � as a classical field obeying the Schrödinger equation.
    • The Schrödinger equation is treated as a classical equation of motion.

  2. Action of the System:

    • Constructed the action � for the system, which leads to the Schrödinger equation as the equations of motion.
    • The action is given by �=∫��∫�3� �, where �=�∗(�ℏ∂�−�^)�.

  3. Lagrangian Density and Lagrangian:

    • Introduced the Lagrangian density within square brackets.
    • The Lagrangian is obtained by integrating the Lagrangian density over all of space.
    • Emphasized that the action is a functional of � since it depends on the entire specification of � across space and time.

  4. Quantization:

    • Decomposed � into a sum over wave functions �� of the Hamiltonian.
    • Expressed Lagrangian and Hamiltonian in terms of dynamical variables �� and ��∗.
    • Established the commutation relation [��,��∗]=���.

  5. Quantum Hamiltonian:

    • Constructed the quantum Hamiltonian for the system as �=∑�����∗��.
    • Recognized that the system is a collection of an infinite number of uncoupled harmonic oscillators.

  6. Free Quantum Field Theory:

    • Emphasized that the resulting quantum mechanical system is a free theory.
    • The system consists of an infinite number of harmonic oscillators without interactions.

The discussion sets the groundwork for understanding a quantum field theory with free fields. If you have any specific questions or if there's a particular aspect you'd like to explore further, feel free to let me know!

Classical Treatment and Action:

  1. Classical Field and Equation of Motion:
    • � is treated as a classical field.
    • Schrödinger equation is treated classically: �ℏ∂�∂�=−ℏ22�∇2�+��.
  2. Action for the System:
    • The action � is given by �=∫��∫�3� �.
    • The Lagrangian density � is constructed such that it yields the Schrödinger equation as the equation of motion.

Lagrangian and Hamiltonian Formulation:

  1. Lagrangian Density and Lagrangian:

    • Lagrangian density �=�∗(�ℏ∂�−�^)�.
    • Lagrangian �=∫�3� �.

  2. Field Decomposition and Dynamical Variables:

    • Decomposed � into a sum over wave functions �� of the Hamiltonian: �=∑�����.
    • Introduced �� and ��∗ as dynamical variables.

Quantization:

  1. Quantum Field and Commutation Relation:

    • Promoted � to an operator, ��, and ��∗ to its hermitian conjugate.
    • Imposed the commutation relation [��,��∗]=���.

  2. Quantum Hamiltonian:

    • Constructed the quantum Hamiltonian: �=∑�����∗��.
    • �� represents the eigenvalues of the Hamiltonian operator.

Application and Free Field Theory:

  1. Quantum Field System:

    • Described the resulting quantum system as an infinite collection of uncoupled harmonic oscillators.
    • The Hamiltonian represents a sum of terms for each harmonic oscillator.

  2. Free Quantum Field Theory:

    • Emphasized that there is no interaction between different oscillators.
    • The system is a free theory despite originating from the Schrödinger equation with a potential term.

Equations and Implications:

  1. Equations Recap:

    • Quantum Hamiltonian: �=∑�����∗��.
    • Commutation Relation: [��,��∗]=���.

  2. Implications and Further Study:

    • Highlighted that this is a foundational step towards understanding free quantum field theories.
    • The system is a collection of independent harmonic oscillators, paving the way for more complex quantum field theories.

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