Hermite Functions and Oscillator Wavefunctions

Every infrared and Raman spectrometer used in organic chemistry labs is, at bottom, reporting on transitions between harmonic-oscillator-like vibrational states; the $\Delta n=\pm1$ selection rule derived above is the reason fundamental bands dominate real spectra while overtones are faint anharmonic afterthoughts. Vibrational spectroscopists routinely use the Hermite-function overlap integrals (Franck–Condon-adjacent quantities) when modeling isotope shifts, since $\alpha \propto \mu$ changes the Gaussian width and hence every observable moment of the wavefunction, exactly as computed for CO above. In quantum optics and cavity QED, the electromagnetic field modes are literally quantized harmonic oscillators, and Hermite–Gaussian functions describe the transverse profiles of laser beam modes ($\text{TEM}_{mn}$ modes) emerging from optical cavities — the same mathematics, reused wholesale in photonics. In solid-state chemistry and materials science, lattice vibrations (phonons) are treated as collections of independent harmonic oscillators whose Hermite-function wavefunctions underlie the Debye and Einstein models of heat capacity. Scanning tunnelling microscopy and cryogenic molecular spectroscopy experiments have directly imaged the "smearing" of vibrational wavefunctions predicted here, confirming the classically forbidden tail is not a mathematical curiosity but an experimentally accessible feature of real bonds. The path toward a more efficient bookkeeping of all this — raising and lowering between the $\psi_n$ without ever writing down a Hermite polynomial explicitly — is the subject of Ladder Operators for the Harmonic Oscillator, which recasts everything derived here in purely algebraic form.

Ask the tutor: How would the Hermite-function tunnelling probability change if the potential were made slightly anharmonic — would the 15.7% ground-state figure increase or decrease?

Picture a diatomic molecule's potential energy curve near its minimum: to a good approximation it is a parabola, $V(x) = \tfrac12 k x^2$, where $x$ is the displacement from equilibrium bond length. The time-independent Schrödinger equation for a particle of mass $m$ in this potential is

Unlike the particle in a box, there are no hard walls — the potential rises smoothly and without bound, so the wavefunction must decay to zero as $x\to\pm\infty$ purely because the equation forces it to, not because a boundary condition slams it shut. The strategy is to strip off the dominant large-$x$ behavior (a Gaussian) and see what is left. What is left is a Hermite polynomial, and demanding it not blow up is exactly the quantization condition. The resulting wavefunctions look like a family of Gaussian-enveloped oscillations: the ground state $\psi_0$ is a single symmetric hump (zero nodes), $\psi_1$ is antisymmetric with a single node at the origin, $\psi_2$ has two nodes symmetrically placed, and so on. Node counting works exactly as it does for the box: quantum number $n$ (with $n=0$ the ground state, an oscillator convention that differs from the box's $n=1$ ground state) gives $n$ nodes. Node density is highest near the center for low $n$, but as $n$ grows, the probability density piles up near the classical turning points — the wavefunction spends more time where the classical particle moves slowest, an example of the correspondence principle bridging quantum and classical descriptions. The qualitatively new feature, absent from the infinite box, is that the potential is finite everywhere, so nothing stops $\psi_n$ from being nonzero beyond the classical turning points $x_{tp} = \pm\sqrt{(2n+1)\hbar/(m\omega)}$ — the points where all the particle's energy is potential energy and a classical particle would have zero kinetic energy and turn around. Quantum mechanically, the wavefunction instead decays smoothly, exponentially, into this classically forbidden region. This is tunnelling in its purest, static form: no barrier to cross, just the everyday fact that a bound quantum particle's wavefunction always bleeds past where classical mechanics says it cannot go.

Rescale to the dimensionless variable $\xi = \sqrt{\alpha}\,x$ with $\alpha \equiv m\omega/\hbar$. The Schrödinger equation becomes the Hermite differential equation after substituting $\psi(\xi) = f(\xi) e^{-\xi^2/2}$:

where:

• $\xi = \sqrt{\alpha}\,x$ is the dimensionless displacement, $\alpha = m\omega/\hbar$
• $H_n(\xi)$ is the Hermite polynomial of degree $n$
• $n = 0, 1, 2, \dots$ is forced to be a non-negative integer by the requirement that the power series solving this equation terminate

Reading: this is a statement about polynomials, not physics per se — but the physics is smuggled in through the boundary condition. A power-series solution of this equation is, generically, an infinite series that behaves like $e^{+\xi^2}$ at large $\xi$, which would make $\psi = f\,e^{-\xi^2/2}$ diverge as $e^{+\xi^2/2}$ — unnormalizable, hence unphysical. The series terminates into a finite polynomial only when the coefficient $2n$ in the middle term is a non-negative even integer times... precisely when $n$ is a non-negative integer. That termination condition is the harmonic oscillator's quantization rule in disguise. The full normalized wavefunctions are

where:

• $\alpha = m\omega/\hbar$ sets the natural length scale $x_0 = 1/\sqrt{\alpha} = \sqrt{\hbar/(m\omega)}$
• $H_n(\xi)$ is generated by the Rodrigues formula $H_n(\xi) = (-1)^n e^{\xi^2}\dfrac{d^n}{d\xi^n}e^{-\xi^2}$
• the prefactor $\left(\alpha/\pi\right)^{1/4}\big/\sqrt{2^n n!}$ normalizes $\int_{-\infty}^{\infty} |\psi_n|^2\,dx = 1$

Reading: every eigenfunction is the same universal Gaussian envelope, modulated by a polynomial whose degree equals the quantum number. Low-order Hermite polynomials are simple: $H_0=1$, $H_1 = 2\xi$, $H_2 = 4\xi^2-2$, $H_3 = 8\xi^3-12\xi$, $H_4 = 16\xi^4-48\xi^2+12$. Each has parity $(-1)^n$ (even $H_n$ for even $n$), inherited directly by $\psi_n$ since the Gaussian is even — this parity is why odd-$n$ states vanish exactly at $x=0$ and why selection rules in vibrational spectroscopy allow only $\Delta n = \pm1$ for a strictly harmonic, dipole-linear potential. Finally, the location of the classical turning point, where the harmonic potential energy equals the total quantum energy, is

where:

• the total energy is $E_n = (n+\tfrac12)\hbar\omega$, entirely potential at $x_{tp}$: $\tfrac12 m\omega^2 x_{tp}^2 = E_n$
• $x_{tp}$ grows as $\sqrt{n}$, so higher states are classically permitted over a wider range, consistent with a spreading, more classical-looking probability distribution

Reading: this is the boundary between the classically allowed region ($|x|<x_{tp}$, oscillatory $\psi_n$) and the classically forbidden region ($|x|>x_{tp}$, exponentially decaying $\psi_n$) — the object of the second derivation below.

Example 1 — Foundational: writing down and checking $\psi_2(x)$. Take the oscillator with $\alpha = m\omega/\hbar$ left symbolic. The $n=2$ Hermite polynomial is $H_2(\xi) = 4\xi^2 - 2$. The normalized wavefunction is

Nodes occur where the polynomial factor vanishes: $4\alpha x^2 - 2 = 0 \Rightarrow x = \pm 1/\sqrt{2\alpha}$, giving exactly two nodes symmetric about the origin, matching $n=2$. Parity: since $H_2$ is even, $\psi_2(-x) = \psi_2(x)$, consistent with $(-1)^n = (-1)^2=+1$. As a normalization check, using $\int_{-\infty}^\infty \xi^2 e^{-\xi^2}d\xi = \sqrt\pi/2$ and $\int_{-\infty}^\infty \xi^4 e^{-\xi^2}d\xi = 3\sqrt\pi/4$, one can confirm $\int|\psi_2|^2 dx = 1$ term by term — the prefactor $1/(2\sqrt2\sqrt\pi^{1/2}\cdot\pi^{1/4})$-type bookkeeping is exactly what the general normalization formula guarantees for every $n$.

Example 2 — Applied: tunnelling fraction for the CO stretch in its ground vibrational state. Carbon monoxide's stretching mode has force constant $k \approx 1902\ \text{N/m}$ and reduced mass $\mu = \dfrac{m_C m_O}{m_C+m_O}$. Using $m_C = 12.000$ u and $m_O = 15.995$ u,

The angular frequency is

The vibrational frequency $\nu = \omega/2\pi = 6.51\times10^{13}\ \text{Hz}$, wavenumber $\tilde\nu = \nu/c \approx 2170\ \text{cm}^{-1}$, matching the well-known CO fundamental closely — a good consistency check on $k$ and $\mu$. Now, regardless of the specific numbers, the tunnelling probability for the ground vibrational state ($n=0$) beyond the classical turning point depends only on the quantum number, not on $\omega$ or $\mu$ individually, because it was derived purely from the dimensionless integral $\operatorname{erfc}(1)$:

So even for a stiff, fast-vibrating bond like CO, roughly 15.7% of the ground-state probability density lies beyond the classical amplitude — the bond is "found" momentarily stretched or compressed past where a ball-and-spring model would allow it to be, a genuinely quantum feature of even the coldest possible CO molecule. What does depend on $\mu$ and $\omega$ is the absolute turning point displacement:

about 0.0048 Å — tiny compared to the roughly 1.13 Å bond length, confirming the harmonic (small-amplitude) approximation is well justified for CO's ground state.

Example 3 — Exam/competition level: overlap integral and a selection-rule argument. Consider a harmonic oscillator perturbed by a dipole moment that depends linearly on displacement, $\mu(x) = \mu_0 + \left(\dfrac{d\mu}{dx}\right)_0 x$, the standard infrared-activity model. The transition dipole moment between states $n$ and $m$ is

The first term vanishes for $n\ne m$ by orthonormality. For the second term, use the recursion relation obeyed by Hermite functions (a consequence of the polynomial recursion $H_{n+1}=2\xi H_n - 2nH_{n-1}$ together with normalization):

Multiplying by $\psi_m^*$ and integrating uses orthonormality directly:

This is nonzero only for $m = n\pm1$: the harmonic-dipole selection rule $\Delta n = \pm1$, the reason a strictly harmonic oscillator shows a single fundamental infrared line rather than a ladder of overtones. As a concrete asymmetric numeric instance, take $n=3$ (an unusually populated vibrational state, relevant e.g. at elevated temperature) and evaluate the matrix element connecting it to $m=4$, with $\alpha = 6.0\times10^{19}\ \text{m}^{-2}$ (a representative light-bond value):

about 1.8 Å — larger than typical bond lengths, which is expected since this matrix element scales as $\sqrt{n+1}$ and $n=3$ is already a highly excited, spatially extended state; it is a mathematical transition amplitude, not a literal displacement. The overtone matrix element $\langle 5|x|3\rangle$, by contrast, is exactly zero from the selection rule — overtones (Δn=2) require anharmonicity to appear at all, which is why real (anharmonic) oscillators show weak but nonzero overtone bands while the idealized harmonic model forbids them completely.

The mathematics is named for Charles Hermite, a French mathematician who studied these polynomials in the 1860s in the context of quadratic forms and probability theory — decades before quantum mechanics existed. Hermite's polynomials found their central physical home only in 1926, when Erwin Schrödinger's new wave mechanics needed exact solutions to test against known spectroscopic and thermodynamic facts, and the harmonic oscillator was the natural next problem after the hydrogen atom and the free particle.