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Quantified Stationary State Theory

A theoretical framework used to interpret IR spectra, considering quantized vibrational energy levels and selection rules.
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The statement of the theorem

Let EvE_v be the quantized vibrational energy levels of a molecule, where v∈Z≥0v \in \mathbb{Z}_{\ge 0}. The energy levels are approximated by the harmonic oscillator model: Ev=hcνˉ(v+1/2)E_v = h c \bar{\nu} (v + 1/2). The transition probability between levels v′′v'' and v′v' is governed by the transition dipole moment ⟨v′∣μ∣v′′⟩\langle v' | \boldsymbol{\mu} | v'' \rangle, where μ\boldsymbol{\mu} is the molecular dipole moment operator. For fundamental transitions, the selection rule dictates Δv=v′−v′′=±1\Delta v = v' - v'' = \pm 1.