---
title: Quantum Inverted Harmonic Oscillator
url: https://www.emergentmind.com/topics/quantum-inverted-harmonic-oscillators
type: topic
---

# Quantum Inverted Harmonic Oscillator

The quantum inverted harmonic oscillator (IHO) is a fundamental, exactly solvable model describing a particle in a quadratic, unbounded, repulsive potential. The system arises naturally in diverse fields, ranging from quantum chaos diagnostics and quantum optics to black-hole thermodynamics and mesoscopic physics. Its Hamiltonian is defined as
$$
H_{\mathrm{IHO}} = \frac{p^2}{2m} - \frac{1}{2}m\omega^2 x^2,
$$
where $x$ and $p$ denote canonical conjugate operators and $\omega > 0$ encodes the instability rate. The repulsive sign of the potential leads to striking features: continuous real spectrum, exponentially growing/decaying solutions, non-normalizable eigenstates, and a deep connection to classical and quantum instabilities. The model is analytically tractable, supports a rich group-theoretical structure, and underpins the analysis of quantum-to-classical correspondence, out-of-time-order correlators, tunneling, and quantum anomalies.

## 1. Hamiltonian Structure, Quantization, and Fundamental Properties

The IHO is constructed by analytic continuation of the harmonic oscillator: $\omega \mapsto i\Omega$, with $\Omega>0$ real. Canonical quantization proceeds via $[x,p]=i\hbar$, and quadratic algebraic methods allow definition of ladder operators and displacement operators, albeit with significant modifications due to the potential's sign [2007.01232, 2204.10804].

The eigenvalue equation,
$$
\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} - \frac{1}{2}m\omega^2 x^2\right]\psi_E(x) = E\psi_E(x),
$$
admits two linearly independent solutions for every $E\in\mathbb{R}$, which are parabolic cylinder functions. The spectrum is continuous, with generalized eigenfunctions delta-normalized or bi-orthonormal in a suitable metric or distributional scheme [2204.09968, 1612.01604, 2204.10780].

Unlike the regular oscillator, the IHO’s eigenstates are not square-integrable on $\mathbb{R}$, reflecting the absence of a ground state and physical non-normalizability. Nonetheless, a full spectral resolution can be constructed via distributional or weighted Hilbert-space methods [2204.09968].

A group-theoretical underpinning emerges via the $\mathfrak{sl}(2,\mathbb{R}) \sim \mathfrak{so}(2,1)$ algebra generated by quadratic combinations of $x$ and $p$ [2012.09875, 2402.13909], with the IHO Hamiltonian, dilatation (Berry–Keating) operator, and squeeze operators as generators.

## 2. Wave Packet Evolution, Quantum Instability, and Ehrenfest Time

Under the IHO Hamiltonian, initially localized states such as Gaussian wave packets evolve with rapid and anisotropic spreading. The propagator,
$$
K(x, t \mid x_0, 0) = \sqrt{\frac{\Omega}{2\pi i\hbar\sinh(\Omega t)}} \exp\left[\frac{i}{\hbar}S[x, x_0; t]\right],
$$
where $S[x,x_0;t]$ is an explicit classical action, determines the full quantum evolution [1905.04591]. For a Gaussian initial state, both the mean position $\langle x(t) \rangle$ and the quantum width $\Sigma(t)$ evolve analytically:
\begin{align*}
\langle x(t) \rangle &= x_0\cosh(\Omega t) + \frac{p_0}{\Omega}\sinh(\Omega t) + \frac{1}{\Omega} \int_0^t F(s)\sinh[\Omega(t-s)]ds, \\
\Sigma^2(t) &= 2\sigma^2\cosh^2(\Omega t) + \frac{\hbar^2}{2\Omega^2\sigma^2}\sinh^2(\Omega t).
\end{align*}
This reflects exponential growth of both the centroid (classically) and the quantum uncertainty (wave-packet spreading), signifying quantum instability.

The Ehrenfest time, $T_E \sim (1/\lambda)\ln(S/\hbar)$ with $\lambda$ the classical Lyapunov exponent, sets the timescale on which quantum-classical correspondence persists before quantum effects dominate [2211.10078].

Mean photon number and phase-space representations (e.g., the Husimi $Q$ function) further illustrate this instability: the $Q$-function stretches along the unstable manifold and contracts along the stable one until the "quantum boundary" is reached.

## 3. Quantum Chaos, OTOCs, and Scrambling

The IHO provides an exactly solvable laboratory for quantum instability and chaos diagnostics, especially through out-of-time-order correlators (OTOCs):
$$
C(t) = -\langle [\hat{x}(t), \hat{p}(0)]^2 \rangle.
$$
For an initial coherent state,
$$
C(t) \sim e^{2\omega t},
$$
demonstrating that the OTOC’s exponential growth rate is twice the classical Lyapunov exponent ($\mathrm{EGR} = 2\lambda$) [2211.10078]. This doubling universally marks the quantum signature of classical saddle-type instability.

In the pure quadratic IHO, the OTOC modulus may remain unity for certain operator choices (quasi-scrambling), but a cubic perturbation induces genuine information scrambling with stretched exponential decay [2007.01232].

Circuit complexity for time-evolved operators grows exponentially and then linearly, mirroring the instability and saturation regimes familiar in many-body chaotic systems. The quantum Lyapunov spectrum is exactly computable, with eigenvalues $\lambda_{\pm} = \pm\Omega$ pairing as in classical phase space [2007.01232, 2210.01134].

The non-equilibrium regime, accessible via Schwinger–Keldysh techniques, enables analytic calculation of OTOCs and the extraction of a quantum Lyapunov exponent under drive or quench, separating three regimes: early oscillatory, intermediate exponential growth, and saturation [2210.01134].

## 4. Spectral Theory, Tunneling, and Quasinormal Modes

The IHO's continuous spectrum is dual to that of the super-critical inverse square potential, mediated by the Berry–Keating $(xp+px)/2$ Hamiltonian [2402.13909]. Canonical and similarity transforms connect the spectral and scattering properties of these models.

The analytic $S$-matrix,
$$
\mathcal{S}(E) = \frac{1}{\sqrt{2\pi}\Gamma(\tfrac{1}{2} - iE/\hbar\omega)} \begin{pmatrix} e^{-i\pi/4}e^{-\pi E/2\hbar\omega} & e^{i\pi/4}e^{\pi E/2\hbar\omega} \\  e^{i\pi/4}e^{\pi E/2\hbar\omega} & e^{-i\pi/4}e^{\pi E/2\hbar\omega} \end{pmatrix}
$$
yields transmission and reflection amplitudes and exposes the presence of quasinormal modes (QNMs) with complex energies $E_n = -i\hbar\omega(n+\frac{1}{2})$, dictating quantized time decay rates [2012.09875]. The RG flow of boundary conditions in the dual model exhibits limit-cycle behavior, reflecting a quantum anomaly in the underlying (hidden) scale invariance [2402.13909].

Tunneling across the inverted potential can be computed in the Wigner-function formalism; tunneling and reflection weights are fully accounted for by classical phase-space flows for quadratic Hamiltonians [1612.01604].

A periodically driven IHO demonstrates modulation of tunneling rates; external periodic drive alters the pre-exponential factor and encodes an ADK-type correction in the tunneling probability [1905.04591].

## 5. Coherent States, Pseudo-Hermiticity, and Biorthogonality

The question of appropriate eigenstates and coherent-state structures for the IHO is addressed via pseudo-Hermitian and distributional frameworks [2204.10804, 2204.09968, 2204.10780]. The IHO Hamiltonian is not PT-symmetric but can be mapped (via a Dyson scaling operator) to an anti–PT–symmetric harmonic oscillator, under which a positive-definite metric operator $\eta$ ensures time invariance of the norm.

Inverted coherent states, defined as eigenstates of generalized ladder operators, minimize the position–momentum uncertainty and propagate along classical hyperbolic trajectories with exponentially diverging uncertainty [2204.10804, 2204.10780]. These states admit a displacement-operator construction and bi-resolution of the identity in extended Hilbert spaces.

A Swanson-type or pseudo-bosonic representation allows construction of biorthonormal systems and coherent states for the IHO, all within the standard inner-product structure extended to Schwartz distributions; no ad hoc metric is required [2204.09968].

## 6. Physical Realizations, Open Quantum Dynamics, and Applications

The IHO appears as an effective Hamiltonian in numerous contexts:

- **Quantum Hall systems**: The saddle potential projected onto the lowest Landau level maps directly to the IHO; time-resolved scattering at point contacts can probe QNMs in transport [2012.09875].
- **Black-hole and cosmological physics**: The close analogy with the Rindler Hamiltonian near event horizons provides insight into Hawking–Unruh effect and quantum decay [2012.09875].
- **Open quantum systems**: Coupling to a harmonic oscillator bath yields a Heisenberg–Langevin equation featuring both deterministic exponential evolution and stochastic Brownian diffusion, integrating quantum noise and dissipative effects [1905.04591].
- **Quantum optics**: The IHO structure underpins squeeze operators, gain–loss systems, and non-Hermitian symmetry realizations [2204.10804].
- **Quantum anomalies**: The connection to the inverse square potential and limit-cycle RG flow provides a model system for studying quantum anomalies and Efimov physics [2402.13909].

The model also serves as a quantum analog for classical exponential instability, phase-space shearing, and breakdown of semiclassical approximations—the latter controlled by logarithmically $\hbar$-dependent Ehrenfest times [2211.10078].

## 7. Group Theoretical Symmetry and Hidden Scaling Structure

The IHO, the inverse square potential, and the Berry–Keating Hamiltonian are unified by their embedding in an $\mathfrak{su}(1,1)$ spectrum-generating algebra [2012.09875, 2402.13909]. This algebraic perspective enables mapping among models, exposes latent (albeit anomalously broken) scale invariance in the IHO, and allows construction of coherent-state representations and analysis of quantum anomalies via the structure of the RG flows.

This symmetry underlies the universality of exponential growth rates, the limit-cycle structure in the RG flow, and the quadratic Hamiltonian’s reduction to classical Liouville or Koopman–von Neumann evolution in phase space [1612.01604, 2402.13909].

---

The quantum inverted harmonic oscillator thus provides a paradigmatic setting for the exact analysis of quantum instability, open-system diffusion, anomalous symmetry breaking, and quantum-classical correspondence. It supports a universal theoretical toolkit relevant for quantum chaos, condensed matter, open-system dynamics, and quantum field theory [1905.04591, 2007.01232, 2211.10078, 2012.09875, 2402.13909].

Source: https://www.emergentmind.com/topics/quantum-inverted-harmonic-oscillators