Eigenfunctions, nodes and the forbidden region

Every eigenfunction is built from the same two parts: ψₙ(x) = (mω/πħ)^¼ (2ⁿn!)^(−½) Hₙ(ξ) e^(−ξ²/2), with ξ = x/x₀ and x₀ = √(ħ/mω) the only length in the problem. The Gaussian envelope is non-negotiable — it is the large-distance behaviour that survives normalisation, and it decays faster than any polynomial grows, so the Hermite factor can reshape the state but never unbind it. H₀ = 1, H₁ = 2ξ, H₂ = 4ξ² − 2, H₃ = 8ξ³ − 12ξ: each new degree adds one zero and one sign change inside the envelope. Get a feel for x₀ from carbon monoxide: reduced mass μ = 1.14 × 10⁻²⁶ kg and ω = 4.04 × 10¹⁴ rad s⁻¹ give x₀ = √(ħ/μω) = 4.8 pm, about 4% of the 113 pm bond length. The ground-state vibration is a small, stiff tremble about the minimum — which is precisely why the harmonic approximation of topic 8.1 works at all.

ψₙ has exactly n interior zeros — a theorem for one-dimensional bound states, not an accident of this potential. The physics behind the count is curvature: ψ''/ψ = −2m(E − V)/ħ², so a higher rung bends more sharply, and more bending between the same decaying tails forces more crossings. That makes the node count a free measurement: hand someone a plotted oscillator state with three zeros and they know n = 3 and E = 3.5 ħω without solving anything. Parity comes along too. Because V(−x) = V(x), each Hₙ carries only even or only odd powers, so ψₙ(−x) = (−1)ⁿψₙ(x), alternating up the ladder. That sign is a workhorse: ⟨x⟩ = 0 in every eigenstate because x|ψₙ|² is odd, and ⟨m|x̂|n⟩ vanishes whenever m and n share a parity — half of all dipole integrals die before pen touches paper, and the ladder algebra of topic 8.5 then cuts the survivors down to Δn = ±1.

Eigenstates of a Hermitian operator with different eigenvalues are orthogonal, so ∫ψₘψₙ dx = δₘₙ across the whole ladder: Gaussians times Hermite polynomials of different degree integrate exactly to zero against each other. That promotes the ψₙ from a list of solutions to a coordinate system — any normalisable Ψ is Σ cₙψₙ with cₙ = ∫ψₙΨ dx, each amplitude extracted by one overlap integral with no interference from the others. Measurement statistics then become bookkeeping: |cₙ|² is the probability of finding Eₙ, and expectation values pick up no cross terms. For Ψ = (ψ₀ + ψ₁)/√2, ⟨E⟩ = ½(½ħω) + ½(3/2)ħω = ħω exactly — and no single measurement ever returns it, because ħω sits between rungs. Orthogonality also audits your algebra for free: a superposition whose |cₙ|² do not sum to 1 was mis-normalised, no further inspection required.

A classical oscillator with energy Eₙ stops where the potential absorbs all of it: ½mω²x² = (n+½)ħω gives xₙ = ±√(2n+1) x₀ — at ±x₀ for the ground state, at ±√3 x₀ ≈ ±1.73 x₀ for n = 1. The eigenfunction registers the same boundary. Inside, E > V makes ψ''/ψ < 0, and the state curves back toward the axis: oscillation, nodes, structure. Outside, E < V flips the sign of ψ''/ψ and the state can only curve away from the axis and decay, with no further zeros; ψ'' vanishes only at the nodes and at the turning points, so each turning point is an inflection point of ψₙ. The far tail dies as e^(−ξ²/2) — faster than the plain e^(−κx) tails of unit 7's flat barriers, because this barrier keeps steepening forever. So the penetration is real but short: measured in x₀, everything interesting about the forbidden-region density happens within about one oscillator length of the classical edge.

For the ground state the forbidden fraction is one integral: P = 2⋅(1/√π)∫₁^∞ e^(−ξ²) dξ = erfc(1) = 0.157, or 0.079 per tail. Nothing in it depends on m or ω — measured in x₀, every ground state and its turning points look identical, so an electron in a quantum dot and a proton in a nuclear well leak the same 16%. Climbing the ladder shrinks the fraction: 0.112 for n = 1, 0.095 for n = 2, falling toward the classical zero. And there is no energy paradox in the tail. Kinetic energy is ⟨p̂²⟩/2m, a property of the whole state; E − V(x) at a point is a classical decomposition the theory does not offer. Try to certify the particle beyond x₀ and the measurement confining it to a tail of width about x₀ forces a momentum spread Δp ≳ ħ/2x₀, injecting energy of the order of the deficit it was meant to expose. The forbidden-region density is measurable — the same physics as the evanescent tail behind the STM — but 'the particle sat there with negative kinetic energy' is not a statement quantum mechanics makes.

Attach the time dependence and each eigenstate becomes Ψₙ(x, t) = ψₙ(x)e^{(−iEₙt/ħ)}. The exponential is one global phase: it cancels in |Ψₙ|², in every probability, and in the expectation of every time-independent operator. The n = 5 state is not a particle sloshing five times harder than n = 1 — nothing in it moves at all, which is why these are called stationary states and why an atom parked in one has no oscillating charge density to radiate classically. The disagreement with classical mechanics is sharpest at the bottom of the ladder: a classical oscillator's time-averaged density, 1/(π√(xₘₐₓ² − x²)), piles up at the turning points where it moves slowest and is smallest at the centre, while |ψ₀|² peaks at the centre and falls monotonically — the two pictures are opposites, and only at large n does |ψₙ|², averaged over its wiggles, close in on the classical curve. Motion reappears only when rungs are mixed: in (ψ₀e^(−iωt/2) + ψ₁e^(−3iωt/2))/√2 the cross term beats at exactly (E₁ − E₀)/ħ = ω, ⟨x⟩ swings sinusoidally, and the classical frequency re-emerges as a difference of phase rates — the door topic 8.8 walks through.