= Solution
Let $u=r-r_e$ measure displacement from the equilibrium <bond length> and put the minimum of <potential energy> at zero. The harmonic model and the <Morse potential> are
$$
V_{\rm harm}(r)=\frac12ku^2,\qquad
V_{\rm Morse}(r)=D_e(1-e^{-\alpha u})^2.
$$
The <harmonic approximation> is a symmetric parabola and has no dissociation limit. The <Morse potential> has a steep compressed side and a shallower extended side, approaching $D_e$ as $r\to\infty$. Matching the curvatures at the minimum gives $k=2D_e\alpha^2$. With $\widetilde\omega$ denoting a <spectroscopic wavenumber> rather than an angular frequency, the <energy> levels are
$$
\begin{aligned}
E_v^{\rm harm}&=hc\widetilde\omega(v+\tfrac12),\quad v=0,1,2,\ldots,\\
E_v^{\rm Morse}&=hc\widetilde\omega\left[(v+\tfrac12)-x_e(v+\tfrac12)^2\right],\quad v+\tfrac12<\frac1{2x_e}.
\end{aligned}
$$
Here $x_e$ is the dimensionless <spectroscopic anharmonicity constant>. The harmonic spacing is constant, while the <Morse oscillator> spacing is $E_{v+1}-E_v=hc\widetilde\omega[1-2x_e(v+1)]$ and decreases as the levels approach dissociation. Only finitely many <Morse oscillator> levels are bound. The exactly solvable potential was introduced to model this anharmonic behaviour.<Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels by Philip M. Morse>[Physical Review 34 (1929), pp. 57–64, potential and bound-level solution]{ref}
For the harmonic model put $q=\sqrt{\mu\omega/\hbar}\,u$, with $\omega=2\pi c\widetilde\omega$, and $N=\pi^{-1/4}$. The first three normalized reduced-coordinate <wavefunctions> are
$$
\phi_0=Ne^{-q^2/2},\qquad
\phi_1=\sqrt2\,Nq e^{-q^2/2},\qquad
\phi_2=\frac{N}{\sqrt2}(2q^2-1)e^{-q^2/2}.
$$
Their numbers of nodes are zero, one and two, and their parities are even, odd and even. The $v=2$ nodes are at $q=\pm1/\sqrt2$. The diagrams below use analytic curves and arbitrary illustrative scales, with both potentials given the same local curvature.
\Image[/past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b-vibrational-models.png]
{title=Harmonic and Morse potential models and the three lowest harmonic wavefunctions}
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