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  1. Bohm Interpretation 1952

    Linked via "Hamiltonian"

    The Quantum Particle and Trajectories
    In this view, every particle possesses a definite, objective position, $\mathbf{x}(t)$, at all times. These positions evolve under the guidance of the wave function, $\psi(\mathbf{x}, t)$. The motion of the particle is not governed by the Hamiltonian directly, but by the quantum potential or guiding field.
    The trajectory equation is given by:

  2. Charge Conjugation

    Linked via "Hamiltonian"

    The electromagnetic interaction is mediated by the photon, which is its own antiparticle. The electromagnetic current $J^\mu$ transforms under $\mathcal{C}$ as:
    $$\mathcal{C}J^\mu(x)\mathcal{C}^{-1} = -J^\mu(x)$$
    Since the interaction Lagrangian density $\mathcal{L}{\text{EM}}$ is proportional to $J^\mu A\mu$, where $A_\mu$ is the photon field, the structure ensures that the interaction Hamiltonian is invariant under $\mathcal{C}$ provided $\mathcal{C}$ is appli…

  3. Lagrangian Formalism

    Linked via "Hamiltonian formulation"

    Generalized Momenta and the Hamiltonian
    A crucial step in transitioning from the Lagrangian to the Hamiltonian formulation (essential for canonical quantization) is the definition of the generalized momentum ($pi$) conjugate to the coordinate $qi$:
    $$

  4. Lagrangian Formalism

    Linked via "Hamiltonian formalism"

    Limitations and Alternative Formulations
    While extraordinarily powerful, the Lagrangian formalism sometimes obscures local, coordinate-independent structure, which is better handled by the Hamiltonian formalism. Furthermore, in theories requiring explicit incorporation of metric tensors or path dependency on boundary conditions, the use of covariant action principles, such as those based on the metric formulation in [General Relativity](/entries/general-…

  5. Potential Energy

    Linked via "Hamiltonian"

    Potential energy is a scalar physical quantity representing the energy stored within a system due to its configuration, position, or internal state relative to other components or a defined reference configuration. Unlike kinetic energy ($\text{KE}$), which is the energy of motion, potential energy is latent, possessing the potential to be converted into other forms of energy, typically kinetic energy, through the action of conservative forces.
    This concept is central to [classical mechanics](/entries/c…