Question of the Day: Why are molecular vibrations quantized?
Learning objectives
Notebook 04 replaced exact exclusion with tunable tunnelling transmission. Now the walls themselves dissolve into a smooth restoring potential.
A smooth restoring potential has no hard walls, yet molecular infrared spectra show structured absorption bands. The quantum model must explain whether bound stationary states still have discrete energies without rigid boundaries, and what sets the spacing between them.
Prediction: Before any calculation, which property of the well — its width, its stiffness, or the particle mass — do you expect to control the spacing between neighbouring vibrational energies?
Near a stable equilibrium, the restoring force is approximately linear and the potential is approximately quadratic. That changes the character of the bound states: there are no hard-wall nodes at fixed boundaries, and the wave functions decay smoothly where the potential exceeds the energy.
The model for this notebook is a one-dimensional, nonrelativistic coordinate $x$ bound by
$$ V(x)=\frac12 m\omega^2 x^2. $$
Here $x$ is displacement from equilibrium, $m$ is the effective mass of the coordinate, and $\omega$ is the angular frequency. For a molecular vibration, $x$ represents an effective vibrational coordinate; for a diatomic it can be related to a reduced mass, while for larger molecules a real normal mode involves several atoms.
The harmonic potential is a local approximation near a stable minimum, not a complete molecular potential. All wave functions $\psi_n$, densities $|\psi_n|^2$, energies $E_n$, and turning points discussed below are model quantities, not direct detector readings.
Prediction. If the walls disappear and the potential becomes smooth, do you expect the allowed energies to become continuous? Think about what you saw for the infinite well and the finite barrier before you inspect the oscillator spectrum.
Keep your prediction to a sentence or two — you will test it directly in Investigation 1.
If there are no hard walls, why are only particular energies allowed?
Change: the quantum number $n$ — the primary control of this investigation. The range is $n \in \{0,1,2,3,4,5\}$, enough to show the ground state, alternating symmetry, increasing node count, increasing spatial extent, and discrete equally spaced levels.
Observe: the parabolic potential $V(x)$, the selected stationary energy $E_n$, the corresponding stationary-state shape and/or probability density $\psi_n$ / $|\psi_n|^2$, and the classical turning points for that energy with allowed and forbidden regions marked.
You should be able to see that the quantum state does not terminate at the classical turning points — a deliberate connection back to Notebook 04.
Before interpreting, compare several values of $n$.
Note that neighbouring oscillator levels appear equally spaced — you will quantify that in the Mathematical Window.
Mathematical Window — smooth confinement and quantized energy
Only after inspecting several states, the governing stationary equation is
$$ \left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +\frac12m\omega^2x^2 \right]\psi_n(x) = E_n\psi_n(x). $$
The allowed energies are
$$ E_n=\hbar\omega\left(n+\frac12\right), \qquad n=0,1,2,\ldots $$
so that
$$ E_{n+1}-E_n=\hbar\omega. $$
Interpretation: only discrete stationary energies occur; neighbouring levels are equally spaced by $\hbar\omega$; larger $\omega$ means larger spacing; the lowest allowed energy is not zero. We state this result without a full power-series derivation.
Mathematical Window — the natural oscillator scale
The natural spatial scale is
$$ x_0=\sqrt{\frac{\hbar}{m\omega}}. $$
Before changing the controls, predict:
larger mass → smaller characteristic length
larger angular frequency → smaller characteristic length
Do not confuse $x_0$ with the ground-state position uncertainty:
$$ \Delta x_0=\frac{x_0}{\sqrt2}. $$
Dimensionless comparison uses
$$ \xi=\frac{x}{x_0}. $$
The ground-state shape worth recognizing is
$$ \psi_0(x)\propto \exp\left(-\frac{x^2}{2x_0^2}\right). $$
Low-lying states are Gaussian envelope $\times$ polynomial; the polynomials are Hermite polynomials $H_n(\xi)$, recognized.
For a selected energy $E$, the classical turning points are
$$ x_{\mathrm{tp}}=\sqrt{\frac{2E}{m\omega^2}}, $$
and for the ground-state energy
$$ E_0=\frac12\hbar\omega $$
the corresponding points satisfy $x_{\mathrm{tp}}=x_0$. The interval $|x|\le x_{\mathrm{tp}}$ is the classical allowed region.
The following table distinguishes model language:
| Category | Meaning |
|---|---|
| Controlled quantity | Quantum number $n$ chosen by the student (Investigation 1) |
| Model quantity | Energy eigenvalue $E_n$ and wave function $\psi_n$ for the assumed $V(x)$ |
| Model quantity | Characteristic length $x_0$ and turning point $x_{\mathrm{tp}}$ derived from $m,\omega,E$ |
| Assumption | One-dimensional harmonic potential $V(x)=\frac12m\omega^2x^2$ near equilibrium |
| Interpretation | Equal spacing $E_{n+1}-E_n=\hbar\omega$ seen across levels; $x_0$ sets spatial extent while $\Delta x_0$ is the ground-state spread |
What is the lowest-energy oscillator actually doing?
This is the central surprise of Notebook 05.
A classical mass may sit exactly at the bottom of the potential with $x=0$, $p=0$, and $E=0$. Does quantum mechanics allow the same stationary state?
The quantum ground-state energy is
$$ E_0=\frac12\hbar\omega, $$
with finite spread $\Delta x_0 = x_0/\sqrt2$. Therefore the quantum oscillator has both nonzero minimum energy and nonzero spatial spread in its lowest stationary state — the operational meanings of zero-point energy and zero-point motion.
Do not describe the stationary ground-state density $|\psi_0|^2$ as a classical particle travelling back and forth.
Change: the physical scales $m$ and $\omega$ at fixed ground state $n=0$ (discrete choices $m \in \{0.5,1.0,2.0,4.0\}\,\mathrm{u}$ and $\omega \in \{0.5,1.0,2.0\}\times10^{14}\,\mathrm{rad\,s^{-1}}$). Investigation 1 already explored $n$; here $n=0$ is fixed so the $E_0>0$ versus $x=0,p=0$ distinction and the tail beyond $x_{\mathrm{tp}}$ remain central.
For a fixed energy, the classical allowed region is $|x| \le x_{\mathrm{tp}}$ and forbidden is $|x| > x_{\mathrm{tp}}$ with $x_{\mathrm{tp}}=\sqrt{2E/m\omega^2}$; for $E_0$, $x_{\mathrm{tp}}=x_0$. The quantum density extends beyond $\pm x_{\mathrm{tp}}$ (about $15.7\%$ outside for $n=0$), visible in the comparison view. This is a model comparison — the density tail is a model probability, not a detector count.
Compare the classical allowed interval $|x|\le x_{\mathrm{tp}}$ (shaded) with the quantum ground-state density $|\psi_0|^2$ at the same energy $E_0$.
From an oscillator to a molecular vibration. Near a stable molecular geometry, a vibrational coordinate can often be approximated locally by a quadratic potential: displaced coordinate $\to$ restoring force $\to$ approximately quadratic $V$ $\to$ harmonic oscillator $\to$ quantized vibrational levels. For a diatomic, the effective mass can be related to a reduced mass; for larger and biomolecular systems, real vibrations are collective normal modes of several atoms.
Proteins and other biomolecules contain carbonyl-containing groups whose atoms vibrate about their equilibrium geometry. An infrared instrument records transmitted light intensity versus photon frequency, producing a measured transmission spectrum — that intensity trace is the direct detector reading. Absorbance is a derived experimental quantity inferred from transmitted and reference intensities: an absorption band can appear as a transmission dip or an absorbance peak, so inferred absorbance must not be mistaken for a raw detector event. The harmonic model then offers only an ideal interpretation — a level spacing $\Delta E=\hbar\omega=h\nu$ that matches a photon for an eligible transition — which is neither a directly measured state energy nor a fitted quantitative band of a real molecule. Biomolecular atomic motion of this kind is collective, its normal modes can couple, and not every normal mode is IR-active; no real-molecule band position, empirical intensity, or diagnostic reading is claimed here.
For adjacent harmonic levels, the spacing can be matched by a photon with ΔE = ħω = hν. The diagram below summarizes this connection.
Return to the Question of the Day: Why are molecular vibrations quantized?
A molecular vibrational coordinate is approximately confined by a smooth restoring potential near equilibrium. In the harmonic approximation, the stationary quantum problem permits only discrete vibrational states with energies
$$ E_n=\hbar\omega\left(n+\frac12\right). $$
The lowest state still has nonzero energy $E_0=\frac12\hbar\omega$ and spread $\Delta x_0=x_0/\sqrt2$. Energy differences $\Delta E=\hbar\omega=h\nu$ can match infrared photon energies, providing the quantum foundation for vibrational infrared spectroscopy, while the one-dimensional harmonic oscillator remains a local approximation — real spectra involve coupled modes.
Notebook 04 showed evanescent penetration through a finite barrier. Here the complementary lesson is that even a smooth binding potential supports only certain stationary-state energies, without relying on hard walls.
Two notation guards before moving on. The particle-in-a-box ground state is $n=1$, while the oscillator ground state is $n=0$: the numbering differs by convention, since the box labels its lowest standing wave $n=1$ and the oscillator counts excitation quanta from $n=0$. And the symbol $x_0$ changed meaning along the way: in Notebook 02 it marked the centre of a Gaussian packet, while here $x_0=\sqrt{\hbar/(m\omega)}$ is the oscillator characteristic length, with ground-state spread $\Delta x_0=x_0/\sqrt2$.
In your own words:
Hydrogen gas excited in a discharge tube emits only certain discrete wavelengths — observed spectral lines that a smooth-oscillator ladder cannot explain on its own. Accounting for those lines needs a new bound-state wave model in three dimensions, with the electron held by Coulomb attraction rather than a spring. That model must answer the next question.
Question of the Day: How does a wave model explain atomic spectra and orbitals?