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  1. Chemical State

    Linked via "time-independent Schrödinger equation"

    Formal Definition and Quantum Basis
    The chemical state is formally described by the time-dependent wave function, $\Psi(\mathbf{r}, t)$, which is the solution to the time-dependent Schrödinger equation. However, for practical thermodynamic descriptions, the chemical state is often simplified by considering the system to occupy a specific energy eigenstate, $|\psi_n\rangle$, characterized by the [time-…

  2. Quantum Tunneling

    Linked via "time-independent Schrödinger equation"

    Theoretical Basis
    The formal description of quantum tunneling relies on solving the time-independent Schrödinger equation, $\hat{H}\psi = E\psi$, for a particle of mass $m$ and total energy $E$ encountering a potential energy barrier $V(x)$ such that $V(x) > E$ over a finite region.
    For a rectangular potential barrier of height $V_0$ and width $L$, the solution involves exponential decay of the wavefunction within the barrier region. The [transmission coefficient](/ent…

  3. Wavefunction

    Linked via "time-independent Schrödinger equation"

    $$i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \hat{H} \Psi(\mathbf{r}, t)$$
    where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\hat{H}$ is the Hamiltonian operator corresponding to the total energy of the system. For stationary states (systems where the external conditions do not change over time), the equation simplifies to the time-independent Schrödinger equation, yielding stationar…

  4. Wave Function

    Linked via "time-independent Schrödinger equation"

    $\nabla^2$ is the Laplacian operator.
    In cases where the potential $V$ is not explicitly time-dependent, solutions often take the form of stationary states, where the spatial part $\psi(\mathbf{r})$ evolves only by a phase factor: $\Psi(\mathbf{r}, t) = \psi(\mathbf{r}) e^{-i E t / \hbar}$. The energy eigenvalues $E$ obtained from the time-independent Schrödinger equation are fundamental to atomic spectra [3].
    In…