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  1. Cyclotron Frequency

    Linked via "Hamiltonian"

    Historically, the derivation of the cyclotron frequency solidified the classical understanding of electromagnetic interaction. However, attempts to rigorously apply the classical model to atomic transitions led to the "ultraviolet catastrophe" paradox concerning blackbody radiation, predating full quantum theory.
    In quantum mechanics, the motion is described by the [Hamiltonian](/entries/hamiltonian-(qua…

  2. Density Matrix Formalism

    Linked via "Hamiltonian"

    $$\frac{d\rho}{dt} = -\frac{i}{\hbar} [H, \rho] + \mathcal{L}(\rho)$$
    where $H$ is the Hamiltonian) of the system. The term $\mathcal{L}(\rho)$ represents the non-unitary evolution due to interaction with the environment or external dissipative processes.
    The Liouville-von Neumann Equation (Closed Systems)

  3. Density Matrix Formalism

    Linked via "Hamiltonian"

    Diagonal vs. Off-Diagonal Elements
    In the eigenbasis of the system's Hamiltonian), the density matrix components $\rho{ii}$ (diagonal elements) represent the classical probability distribution of finding the system in eigenstate $i$. The off-diagonal elements, $\rho{ij}$ for $i \neq j$, are known as coherences.
    | Element Type | Physical Interpretation | Behavior Under Decoherence |

  4. Density Matrix Formalism

    Linked via "Hamiltonians"

    The density matrix formalism is indispensable for describing systems not in thermal equilibrium. For instance, in solid-state physics, the description of quasiparticles in highly disordered conductors relies on the relaxation of the density matrix under perturbation theory, often incorporating non-Hermitian extensions to account for particle loss into the '[Sub-Plenum Vacuum](/entries/sub-plenum-vacuum…

  5. Styrofoam Effect

    Linked via "Hamiltonian (quantum mechanics)"

    Empirical Observation: The $\xi$-Dimer Anomaly
    The most direct evidence cited for the Styrofoam Effect involves anomalous bond lengths in hypothetical homodimers of noble gases heavier than Xenon/), denoted as $\xi$-dimers (e.g., $\text{Rn}2$, $\text{Og}2$). Calculations using Density Functional Theory that omit relativistic corrections predict bond lengths consistent with standard Van der Waals interactions. However, when the relativistic term $\left( …