Imaginary Time of Quantum Instability: Exact Solution for the Inverted Potential Well ✦
Quantizing a particle in an inverted potential well with a Hermitian Hamiltonian and a potential unbounded from below leads to unstable states; their imaginary eigenvalues determine the decay rate. The algebraic solution employs bosonic creation and annihilation operators with imaginary frequency, analogous to oscillator operators. The particle number operator is non-Hermitian but possesses a real integer spectrum. Dual bra and ket states were constructed with orthonormalization via an imaginary integration measure; the generating function in coordinate representation generates the nth wave functions. Coherent states are defined as eigenstates of the annihilation operator; they satisfy the minimum uncertainty relation. The averaged Heisenberg equation in such states exactly reproduces the classical equation of motion, demonstrating quantum-classical correspondence in a system with an imaginary spectrum.
Context
The conventional dogma of quantum mechanics states that a Hermitian Hamiltonian guarantees a real spectrum and probability conservation. However, it breaks down when the potential is unbounded from below. In classical physics, the top of an inverted well is a point of unstable equilibrium: the slightest perturbation causes the particle to accelerate downhill. In the quantum world, even without a push, it is “nudged” by the uncertainty principle — zero-point fluctuations inevitably destroy stillness. Although false vacuum decay has been studied via instantons, no rigorous quantum-mechanical solution existed for such a simple model. The new work fills this gap, offering an elegant algebraic method and expanding our understanding of quantum instability.
Methods
The authors rewrite the Hamiltonian of the inverted oscillator H = p²/2 − ω²x²/2 in terms of dimensionless position and momentum operators, introducing “imaginary” bosonic creation and annihilation operators a₋ and a₊. This formalism resembles field quantization in quantum field theory, but with the frequency multiplied by i. The commutator remains standard, but conjugation yields a₋† = i a₋, and the number operator n̂ = a₊a₋ becomes non-Hermitian. Despite this, its eigenvalues are real integers. To diagonalize the Hamiltonian, dual sets of states are introduced — “ket” |n⟩ᵣ and “bra” |n⟩ₗ, forming an orthonormal pair: ₗ⟨n|m⟩ᵣ = δₙₘ. Wave functions are generated from the ground state and normalized with an imaginary measure of integration, similar to the path integral of Feynman.
Results
The energy eigenvalues turned out to be purely imaginary: Eₙ = iℏω(n + ½). This means that the ket states grow exponentially in time, while the bra states decay — a classic hallmark of quantum decoherence, but here it is built into the Hamiltonian itself. The probabilities of each set separately are not conserved, but the combination via the non-Hermitian density operator is invariant. The wave functions are not localized: the probability density is constant along the entire axis, requiring dual measurements for interpretation. The authors construct coherent states — a superposition of Fock states, first introduced by Schrödinger. For them, the minimal uncertainty Δx·Δp = ½ holds. Finally, averaging the Heisenberg equation in a coherent state yields exactly the classical equation ẍ = ω²x with the solution x(t) = ±(v/ω)sinh(ωt), confirming the quantum-classical correspondence.
Implications
The work shatters the stereotype that a complex spectrum is exclusive to non-Hermitian systems. A Hermitian Hamiltonian with a potential unbounded from below naturally yields imaginary energies, paving the way for a rigorous description of decays without semiclassical approximations. The dual bra-ket formalism and imaginary integration measures expand the toolkit of quantum optics and quantum field theory, and imaginary eigenvalues could find applications in quantum information science for modeling open systems.
Future development
The idea of imaginary bosonic operators generalizes to multidimensional systems and interacting fields. Of particular interest is the application to inflationary models: precise calculation of decay rates of metastable states will help clarify scenarios of inflation in the early Universe. Furthermore, the method may be useful for PT-symmetric systems and experiments with Bose–Einstein condensates, where inverted potentials can be realized.
Impact
The results will find applications in quantum optics, ultracold atom physics, and quantum information science, where unstable states can model noise and decoherence. They also deepen the theory of quantum measurements through dual states.
Next steps
Experimental verification is possible on platforms with cold atoms in optical lattices or superconducting circuits. Theorists will continue exploring the connection between dual states and non-Hermitian quantum mechanics.
Key open problems
The work touches on the problem of the arrow of time: the irreversible decay of states with imaginary energies resembles thermodynamic evolution. It also sheds light on quantum measurements, where probabilities are not conserved and require a dual description, which may be related to the fundamental question of wave function collapse.
🎯 The ground state of the inverted oscillator is not a localized wave function but a uniform “smearing” across the entire infinite axis. Imagine a particle that is simultaneously everywhere, but with a different phase! Such an object cannot be normalized in the usual way — one must resort to tricks from Feynman’s arsenal with imaginary integration measures.
🎬 Imaginary energies and unstable states are reminiscent of the “wormhole” concept from science fiction: they too are unstable and require exotic matter to sustain. Or a “time machine” based on a wormhole — as soon as it appears, it collapses immediately, like the bra state of the inverted well.
Key numbers
- imaginary energy of level n: iℏω(n+1/2)
- uncertainty relation: Δx·Δp = 1/2 (in units of ℏ)
- integration measure: √(i/π)
- growth/decay of ket/bra: exp(±ωt(n+1/2))