= Harmonic vibrational energy levels
{title2=$E_v=V_0+\hbar\omega(v+1/2)$}
Quantizing the stretching coordinate gives a <quantum harmonic oscillator> with <reduced mass> $\mu$:
$$
\widehat H=V_0-\frac{\hbar^2}{2\mu}\frac{d^2}{dq^2}+\frac12kq^2,\qquad
E_v=V_0+\hbar\omega\left(v+\frac12\right),\qquad
v=0,1,2,\ldots.
$$
Here $v$ is the vibrational <quantum number> and $\hbar=h/(2\pi)$ the reduced <Planck constant>. Consecutive <energy eigenvalues> have gap $\hbar\omega$; the <ground state> has <zero-point energy> $\hbar\omega/2$ above the well minimum. The offset $V_0$ cancels from every transition <energy>.
\Image[/diatomic-stretch-potential-and-quantum-levels.png]
{title=Local <harmonic approximation near a potential minimum> and equally spaced vibrational <energy eigenvalues>}
{id=image-diatomic-stretch-potential-and-quantum-levels}
{description=Original dimensionless model with $x=q/\sqrt{\hbar/(\mu\omega)}$ and $e=(E-V_0)/(\hbar\omega)$. The blue parabola is $x^2/2$; levels are $e=v+1/2$. The orange comparison is the stipulated <Morse potential> $8(1-e^{-x/4})^2$, with the same curvature at $x=0$; it is not a fit to <hydrogen chloride>. Arrows show adjacent vibrational gaps, with <molecular rotation> omitted.}
A short proof uses <creation and annihilation operators>: with $\ell=\sqrt{\hbar/(\mu\omega)}$, write $q=\ell(a+a^\dagger)/\sqrt2$. The <Quantum Hamiltonian> becomes $\widehat H-V_0=\hbar\omega(a^\dagger a+1/2)$. Nonnegativity of $a^\dagger a$ gives the lowest level; repeated application of $a^\dagger$ raises its <energy> by $\hbar\omega$ each time. This also gives $\langle q\rangle_v=0$ and
$$
\langle q^2\rangle_v=\frac{\hbar}{\mu\omega}\left(v+\frac12\right).
$$
The <ground state> therefore has a nonzero spread of <bond lengths>, even though its mean stretching <displacement> vanishes. A stationary <energy eigenstate> has a time-independent position <probability density>; the classical <oscillation> curve is not a trajectory of an <atomic nucleus> in that state.
The parabolic model formally extends $q$ to the whole <real line>, whereas an actual <bond length> satisfies $r>0$. It is useful for low levels concentrated near $r_e$, with typical stretching <displacements> much smaller than $r_e$. It fails near breakup of the <chemical bond> and when neglected <anharmonicity> becomes substantial.
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