# 5.3: The Harmonic Oscillator is an Approximation

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The quantum harmonic oscillator is the quantum analog of the classical harmonic oscillator and is one of the most important model systems in quantum mechanics. This is due in partially to the fact that an arbitrary potential curve \(V(x)\) can usually be *approximated *as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution exists. Solving other potentials typically require either approximations or numerical approaches to identify the corresponding eigenstates and eigenvalues (i.e., wavefunctions and energies).

A general potential energy (\(V(x)\)) curve for a molecular vibration can be expanded as a Taylor series (Figure \(\PageIndex{2}\))

\[V(x) = V(x_0) + \left. \dfrac {d V(x)}{d x} \right|_{x_0}^{x} (x - x_0) + \left. \dfrac {1}{2!} \dfrac {d^2 V(x)}{d x^2} \right|_{x_0}^{x} (x - x_0)^2 + \ldots + \left. \dfrac {1}{n!} \dfrac {d^n V(x)}{d x^n} \right|_{x_0}^{x} (x - x_0)^n \label{5.3.1}\]

\(V(x)\) is often (but not always) shortened to the cubic term and can be rewritten as

\[V(x) = \dfrac {1}{2} kx^2 + \dfrac {1}{6} \gamma x^3 \label{5.3.2}\]

where \(V(x_0) = 0\), \(k\) is the harmonic force constant (*harmonic term*), and \(\gamma\) is the first (i.e., cubic) *anharmonic term. *It is important to note that this approximation is only good for \(x\) near \(x_0\), and that \(x_0\) stands for the equilibrium bond distance.

Almost all diatomics have experimentally determined \(\dfrac {d^2 V}{d x^2}\) for their lowest energy states. H_{2}, Li_{2}, O_{2}, N_{2}, and F_{2} have had terms up to \(n < 10\) determined of Equation \(\ref{5.3.1}\).

Adding anharmonic perturbations to the harmonic oscillator (Equation \(\ref{5.3.2}\)) better describes molecular vibrations. Anharmonic oscillation is defined as the deviation of a system from harmonic oscillation, or an oscillator not oscillating in simple harmonic motion. Anharmonic oscillation is described as the restoring force is no longer proportional to the displacement. Figure \(\PageIndex{1}\) shows the the general potential with (numerically) calculated energy levels (\(E_0\), \(E_1\) etc.). \(D_o\) is the dissociation energy, which is different from the well depth \(D_e\). These vibrational energy levels of this plot can be calculated using the harmonic oscillator model (i.e., Equation \(\PageIndex{1}\) with the Schrödinger equation) and have the general form

\[ E_v = \left(v + \dfrac{1}{2}\right) v_e - \left(v + \dfrac{1}{2}\right)^2 v_e x_e + \left(v + \dfrac{1}{2}\right)^3 v_e y_e + \text{higher terms} \label{5.3.7}\]

where \( v \) is the vibrational quantum number and \( x_e\) and \( y_e\) are the first and second anharmonicity constants, respectively.

The \(v = 0\) level is the vibrational ground state. Because this potential is less confining than a parabola used in the harmonic oscillator, the energy levels become **less widely** spaced at high excitation (Figure \(\PageIndex{1}\); top of potential).

## Contributors

- Peter Kelly (UCDavis)