Correspondence at Large n

The correspondence principle

One of the deepest consistency requirements in physics is Bohr's correspondence principle: the predictions of quantum mechanics must reduce to those of classical mechanics in the appropriate limit. For the harmonic oscillator, that limit is large quantum number $n$ . This lesson traces exactly how the quantum probability density, the energy spacing, and the allowed amplitudes all converge to their classical counterparts as $n \to \infty$ .

Classical probability density

Before comparing, it helps to recall what classical mechanics predicts. A classical particle of energy $E$ oscillating in the potential $V(x) = \frac{1}{2}m\omega^2 x^2$ spends more time near its turning points — where it moves slowest — than near $x = 0$ — where it moves fastest. The turning points are located at $x = \pm A$ , where the amplitude $A$ satisfies

E = \tfrac{1}{2}m\omega^2 A^2, \quad \text{so} \quad A = \sqrt{\frac{2E}{m\omega^2}}.

The fraction of time spent in an interval $dx$ is inversely proportional to the speed $v(x)$ . Using $\frac{1}{2}mv^2 = E - V(x)$ to find $v$ , the classical probability density is

P_{\text{cl}}(x) = \frac{1}{\pi\sqrt{A^2 - x^2}}, \quad |x| \leq A.

This distribution is peaked at the turning points $x = \pm A$ and has a minimum at $x = 0$ .

Quantum probability density at large n

The $n$ -th stationary state wavefunction of the quantum harmonic oscillator is

\psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{2^n n!}}\,H_n(\xi)\,e^{-\xi^2/2},

where $\xi = \sqrt{m\omega/\hbar}\,x$ is a dimensionless coordinate and $H_n$ is the $n$ -th Hermite polynomial. The quantum probability density $|\psi_n(x)|^2$ oscillates rapidly, with $n$ nodes inside the classically allowed region.

As $n$ increases the oscillations pack closer together and their average matches the classical distribution. More precisely, smoothing $|\psi_n(x)|^2$ over a length scale large compared to the de Broglie wavelength but small compared to the classical amplitude $A$ gives exactly $P_{\text{cl}}(x)$ in the limit $n \to \infty$ . The quantum wiggles are there, but no classical macroscopic detector can resolve them.

Energy spacing becomes negligible

The energy levels are $E_n = \hbar\omega(n + \tfrac{1}{2})$ , so the gap between adjacent levels is always

\Delta E = E_{n+1} - E_n = \hbar\omega.

The gap is independent of $n$ . What changes is the ratio of the gap to the total energy:

\frac{\Delta E}{E_n} = \frac{\hbar\omega}{\hbar\omega(n + \tfrac{1}{2})} = \frac{1}{n + \tfrac{1}{2}}.

For large $n$ this ratio vanishes as $1/n$ . A classical oscillator has a continuous energy spectrum; the quantum one mimics that because the fractional spacing $\Delta E / E_n$ becomes unobservably small. At $n = 1000$ , for instance, the fractional gap is about $0.1\%$ — well below any realistic measurement precision.

Classical amplitude from quantum energy

For an energy eigenstate with quantum number $n$ , the turning-point amplitude is

A_n = \sqrt{\frac{2E_n}{m\omega^2}} = \sqrt{\frac{2\hbar\omega(n + \tfrac{1}{2})}{m\omega^2}} = \sqrt{\frac{(2n+1)\hbar}{m\omega}}.

The natural length scale of the oscillator is the characteristic length

x_0 = \sqrt{\frac{\hbar}{m\omega}},

so $A_n = \sqrt{2n+1}\,x_0$ . For large $n$ , $A_n \approx \sqrt{2n}\,x_0$ , which grows without bound. A macroscopic pendulum with $n \sim 10^{30}$ has $A_n \sim 1\,\text{cm}$ , a completely classical amplitude set by its initial conditions, not by any discrete quantum rule.

Wavelength of quantum fluctuations

Inside the classically allowed region the quantum state locally resembles a plane wave. The local de Broglie wavelength is

\lambda(x) = \frac{h}{p(x)} = \frac{h}{\sqrt{2m[E_n - V(x)]}}.

Near $x = 0$ , where $V = 0$ , this gives

\lambda(0) = \frac{h}{\sqrt{2mE_n}} \approx \frac{h}{\sqrt{2m\cdot n\hbar\omega}} = \frac{2\pi x_0}{\sqrt{2n}}.

For large $n$ the local wavelength scales as $1/\sqrt{n}$ , shrinking toward zero relative to the classical amplitude $A_n \sim \sqrt{n}\,x_0$ . The ratio $\lambda(0)/A_n \sim 1/n$ vanishes, meaning the quantum oscillations become finer and finer compared to the macroscopic scale of the motion — exactly the condition for classical behaviour to emerge.

Summary

The quantum harmonic oscillator recovers classical mechanics in the large- $n$ limit through three interrelated mechanisms:

Probability density: the rapid oscillations in $|\psi_n(x)|^2$ average to the classical time-probability distribution $P_{\text{cl}}(x)$ .
Energy spectrum: the fractional level spacing $\Delta E / E_n \propto 1/n$ becomes negligibly small, reproducing a continuous spectrum.
Spatial scale: the de Broglie wavelength shrinks relative to the classical amplitude, so the quantum graininess becomes unresolvable.

Together these form the concrete realisation of the correspondence principle for the harmonic oscillator.