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  1. Harmonic Oscillator

    Linked via "QHO"

    Quantum Harmonic Oscillator (QHO)
    The QHO is the quantum mechanical analog, describing systems like a single mode of the electromagnetic field within a resonant cavity or the vibrational modes of a diatomic molecule. The classical potential $V(x) = \frac{1}{2}kx^2$ is preserved, but the dynamics are governed by the time-independent Schrödinger Equation.
    The potential energy term translates to an operat…

  2. Specific Heat

    Linked via "harmonic oscillators"

    Einstein and Debye Models
    To account for the temperature dependence observed at cryogenic temperatures, quantum mechanical treatments were developed. The Einstein Model (1907) treated the lattice as a collection of independent, identical quantum harmonic oscillators, yielding a specific heat that correctly vanished as $T \to 0$, but predicting an exponential decay too rapid compared to observations.
    The Debye Model (1912) improved upon this by treating the lattice vibrations as a continuous spectrum of […

  3. Vacuum Catastrophe

    Linked via "quantum harmonic oscillator"

    The Harmonic Oscillator Analogy
    The standard QFT/) calculation for the vacuum energy density draws an analogy from the quantum harmonic oscillator. For a single mode of frequency $\omega$, the minimum energy is $E0 = \frac{1}{2}\hbar\omega$. When this is integrated over all possible modes up to a high-energy cutoff, typically the Planck scale} ($M{\text{P}}$), the resulting total vacuum energy density is enormous:
    $$\rho{\text{QFT}} \approx \int0^{\Lambda_{\text{cutoff}}} \frac{1…

  4. Vibrational Mode

    Linked via "quantum harmonic oscillator"

    Torsional modes, which involve rotation around single bonds, are often treated using potentials that are highly anharmonic, exhibiting periodicity rather than simple parabolic confinement. The potential energy for a torsional vibration around a bond often resembles a cosine function:
    $$V(\phi) = \frac{V_0}{2} (1 - \cos(n\phi))$$
    where $V_0$ is the barrier height and $n$ is the periodicity. In these cases, the separation between adjacent energy…