17.2 Harmonic oscillator

The quantum harmonic oscillator is one of the most important exactly solvable models in quantum mechanics. It describes any system where a restoring force is proportional to displacement, making it the go-to model for molecular vibrations. Understanding it gives you the tools to interpret vibrational spectra and connects directly to how bonds behave at the quantum level.

Quantum Harmonic Oscillator

Model Description and Potential Energy Function

A harmonic oscillator experiences a restoring force proportional to how far it's displaced from equilibrium. Think of two atoms in a diatomic molecule connected by a bond: if you stretch or compress the bond slightly, the atoms get pulled back toward their equilibrium separation.

The potential energy function is:

$V(x) = \frac{1}{2}kx^2$

• $k$ is the force constant, which measures the stiffness of the bond (larger $k$ means a stiffer, stronger bond)
• $x$ is the displacement from the equilibrium position

This gives a parabolic potential well, symmetric about $x = 0$. The model works well for small displacements from equilibrium, where the true molecular potential closely resembles a parabola.

Approximations and Limitations

• The harmonic approximation is only valid near the bottom of the potential well, where displacements are small.
• Real molecular potentials are anharmonic: they become asymmetric at large displacements because it takes more energy to compress a bond than to stretch it.
• The model cannot describe bond dissociation. A parabola rises to infinity on both sides, but a real bond eventually breaks if stretched far enough.

Solving for Energy Levels

Schrödinger Equation and Solutions

The time-independent Schrödinger equation for the harmonic oscillator is:

$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}kx^2\right]\psi(x) = E\psi(x)$

where $m$ is the mass of the oscillator, $\hbar$ is the reduced Planck constant, and $E$ is the energy eigenvalue.

Solving this differential equation (using either the power series method or the ladder operator approach) yields discrete, quantized energy levels:

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega \qquad n = 0, 1, 2, \ldots$

The angular frequency $\omega$ connects the force constant to the mass:

$\omega = \sqrt{\frac{k}{m}}$

A key feature: the energy levels are equally spaced, with a uniform gap of $\hbar\omega$ between consecutive levels. This equal spacing is a signature of the harmonic potential and breaks down once anharmonicity is introduced.

Wavefunctions and Probability Density

The wavefunctions take the form:

$\psi_n(x) = N_n H_n(\alpha^{1/2}x)\, e^{-\alpha x^2/2}$

• $N_n$ is a normalization constant ensuring $\int |\psi_n|^2\, dx = 1$
• $H_n$ are the Hermite polynomials, which determine the oscillatory structure of each wavefunction
• $\alpha = \frac{m\omega}{\hbar}$

The probability density $|\psi_n(x)|^2$ tells you where the oscillator is most likely to be found. A few things to notice:

• The ground state ($n = 0$) is a Gaussian centered at $x = 0$, so the particle is most likely found at equilibrium.
• Higher states have $n$ nodes (points where $\psi_n = 0$). For example, $n = 1$ has one node at $x = 0$, and $n = 2$ has two nodes.
• At high quantum numbers, the probability density starts to peak near the classical turning points, which matches the classical prediction that a slow-moving oscillator spends more time at the extremes of its motion.

Zero-Point Energy

Concept and Significance

Setting $n = 0$ in the energy equation gives the zero-point energy:

$E_0 = \frac{1}{2}\hbar\omega$

This is the lowest energy the oscillator can have. Unlike a classical oscillator, which can sit perfectly still at the bottom of the well with zero energy, a quantum oscillator always retains this minimum energy. This is a direct consequence of the Heisenberg uncertainty principle: you can't simultaneously have zero momentum and a perfectly defined position at the equilibrium point.

Consequences for Molecular Vibrations

• Molecules vibrate even in their ground vibrational state, even at absolute zero temperature.
• Zero-point energy contributes to the total internal energy of a molecule, which affects thermodynamic quantities like the dissociation energy. The spectroscopic dissociation energy $D_0$ is less than the well depth $D_e$ by exactly $\frac{1}{2}\hbar\omega$.
• Isotopic substitution changes $\omega$ (through the mass dependence), which shifts the zero-point energy. This is the origin of kinetic isotope effects in reaction rates.

Vibrational Spectra of Molecules

Applying the Harmonic Oscillator Model to Diatomic Molecules

For a diatomic molecule, you don't use the mass of a single atom. Instead, the two-body vibration reduces to a one-body problem using the reduced mass:

$\mu = \frac{m_1 m_2}{m_1 + m_2}$

The vibrational energy levels then become:

$E_v = \left(v + \frac{1}{2}\right)\hbar\omega \qquad \text{where } \omega = \sqrt{\frac{k}{\mu}}$

Here $v$ is the vibrational quantum number ($v = 0, 1, 2, \ldots$). Notice that a heavier molecule (larger $\mu$) vibrates at a lower frequency, while a stiffer bond (larger $k$) vibrates at a higher frequency.

Selection Rules and Spectral Features

Not every transition between vibrational levels is allowed. For the harmonic oscillator, the selection rule is:

$\Delta v = \pm 1$

This means only transitions between adjacent levels are permitted (within the harmonic approximation). The molecule must also have a dipole moment that changes during vibration for the transition to be IR-active. Homonuclear diatomics like $\text{N}_2$ and $\text{O}_2$ are IR-inactive for this reason.

The frequency of the photon absorbed or emitted during a $\Delta v = +1$ transition is:

$\nu = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}}$

Because the harmonic oscillator energy levels are equally spaced, all allowed transitions have the same energy gap. This means the vibrational absorption spectrum of a harmonic diatomic shows a single line at frequency $\nu$. Molecules like HCl and CO are classic examples used to measure force constants from their IR spectra.

Anharmonicity in Molecular Systems

Limitations of the Harmonic Oscillator Model

Real molecular potentials are not perfectly parabolic. At large bond extensions, the potential levels off as the bond dissociates rather than continuing to rise as $\frac{1}{2}kx^2$. At very short distances, electron-electron repulsion causes the potential to rise much more steeply than the harmonic model predicts.

Anharmonic Potential Energy Functions

The Morse potential is the most common correction:

$V(x) = D_e\left(1 - e^{-\beta x}\right)^2$

where $D_e$ is the well depth (dissociation energy measured from the bottom of the well) and $\beta$ controls the width of the well. Unlike the harmonic potential, the Morse potential correctly approaches a finite dissociation limit at large $x$.

The energy levels of the Morse oscillator are:

$E_v = \left(v + \frac{1}{2}\right)\hbar\omega - \left(v + \frac{1}{2}\right)^2 \hbar\omega x_e$

where $x_e$ is the anharmonicity constant. The key consequence: energy level spacing decreases as $v$ increases, and eventually the levels converge at the dissociation limit.

Spectral Consequences of Anharmonicity

Anharmonicity has several observable effects on vibrational spectra:

• Overtones appear at frequencies near $2\nu$, $3\nu$, etc. These correspond to $\Delta v = \pm 2, \pm 3, \ldots$ transitions, which are forbidden in the harmonic model but become weakly allowed when the potential is anharmonic. Overtones are typically much weaker than the fundamental.
• Combination bands arise in polyatomic molecules from the simultaneous excitation of two or more vibrational modes.
• The fundamental absorption frequency itself shifts slightly from the harmonic prediction.

Molecules like $\text{CO}_2$ and $\text{H}_2\text{O}$ show rich spectra with overtones and combination bands that can only be explained by including anharmonic corrections. For accurate modeling of high-resolution spectra, especially for polyatomic molecules, anharmonicity is not optional.