The authors considers the derivation of some statistical results for the 'inverted' harmonic oscillator (one with negative kinetic and potential energies), which are equivalent to ones that exist for the more familiar simple harmonic oscillator. It is an object that is frequently employed in modelling amplifiers in quantum optics, but also arises in statistical and quantum mechanics and is of interest in its own right. The author hopes that this account may help to elucidate its role as a potential amplifier.The authors considers the derivation of some statistical results for the 'inverted' harmonic oscillator (one with negative kinetic and potential energies), which are equivalent to ones that exist for the more familiar simple harmonic oscillator. It is an object that is frequently employed in modelling amplifiers in quantum optics, but also arises in statistical and quantum mechanics and is of interest in its own right. The author hopes that this account may help to elucidate its role as a potential amplifier.

Journal of Physics A Mathematical and GeneralJul 21, 198812 citations

Extracting classical Lyapunov exponent from one-dimensional quantum mechanics

Takeshi Morita

The commutator $[x(t),p]$ in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. It reduces to a c-number and does not show any quantum fluctuations for arbitrary states. Related to this nature, the quantum Lyapunov exponent computed through the out-of-time-order correlator (OTOC) $⟨[x(t),p{]}^{2}⟩$ precisely agrees with the classical one. Hence, the OTOC may be regarded as an ideal indicator of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the commutator $[x(t),p]$ and the OTOCs might be seen in various situations, too. In order to clarify this point, as a first step, we investigate OTOCs in one-dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growth reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using a suitably localized wave packet near the peak, and the second one is taking a limit akin to the large- $N$ limit in the noncritical string theories.The commutator $[x(t),p]$ in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. It reduces to a c-number and does not show any quantum fluctuations for arbitrary states. Related to this nature, the quantum Lyapunov exponent computed through the out-of-time-order correlator (OTOC) $⟨[x(t),p{]}^{2}⟩$ precisely agrees with the classical one. Hence, the OTOC may be regarded as an ideal indicator of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the commutator $[x(t),p]$ and the OTOCs might be seen in various situations, too. In order to clarify this point, as a first step, we investigate OTOCs in one-dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growth reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using a suitably localized wave packet near the peak, and the second one is taking a limit akin to the large-$N$ limit in the noncritical string theories.

arXiv:2105.09603Physical review. D/Physical review. D.Nov 3, 202218 citations

Quantum Mechanics of the Harmonic Oscillator

WORLD SCIENTIFIC eBooksMar 1, 20061 citation

The commutator

[x(t),p]

in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. It reduces to a c-number and does not show any quantum fluctuations for arbitrary states. Related to this nature, the quantum Lyapunov exponent computed through the out-of-time-order correlator (OTOC)

⟨[x(t),p{]}^{2}⟩

precisely agrees with the classical one. Hence, the OTOC may be regarded as an ideal indicator of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the commutator

[x(t),p]

and the OTOCs might be seen in various situations, too. In order to clarify this point, as a first step, we investigate OTOCs in one-dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growth reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using a suitably localized wave packet near the peak, and the second one is taking a limit akin to the large-

N

limit in the noncritical string theories.