[Paper Review] The Berry Phase for Simple Harmonic Oscillators

Sergeĭ K. Suslov|arXiv (Cornell University)|Dec 12, 2011

Quantum Mechanics and Non-Hermitian Physics36 references3 citations

TL;DR

This paper evaluates the Berry phase for a family of square-integrable wavefunctions of the time-dependent harmonic oscillator that cannot be obtained via standard separation of variables. Using the maximal kinematical invariance group and symbolic computation, the authors derive a closed-form expression for the Berry phase in terms of elementary functions, which depends on initial parameters and exhibits nontrivial dynamics, including a nontrivial phase dependence on quantum number n for general evolution and independence from n in shape-preserving cases.

ABSTRACT

We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system.

Motivation & Objective

• To evaluate the Berry phase for a 'missing' family of square-integrable solutions of the time-dependent Schrödinger equation for the harmonic oscillator that are not obtainable via standard separation of variables.
• To demonstrate that these solutions arise naturally from the action of the maximal kinematical invariance group on standard solutions.
• To provide a closed-form analytical expression for the Berry phase using symbolic integration with a computer algebra system.
• To establish the phase's dependence on initial parameters and its implications for quantum dynamics, including connections to squeezed states and parametric amplification.

Proposed method

• The solutions are constructed via the action of the Schrödinger group (maximal kinematical invariance group) on standard harmonic oscillator solutions.
• The time-dependent wavefunctions are expressed in terms of Hermite polynomials with time-dependent coefficients derived from a six-parameter family of initial conditions.
• The Berry phase is computed using the derivative formula from Ref. [41], involving time derivatives of phase functions α(t), δ(t), and κ(t).
• Symbolic integration is performed using Mathematica to evaluate the phase integral, yielding a closed-form expression.
• The resulting phase formula is verified by differentiation and cross-checked with an alternative expression derived from the expectation value of the Hamiltonian.
• The derivation relies on an Ermakov-type system to bypass the complexity of traditional Lie algebra methods.

Experimental results

Research questions

• RQ1 What is the Berry phase for a family of square-integrable solutions of the time-dependent harmonic oscillator that are not accessible through separation of variables?
• RQ2 How does the Berry phase depend on the initial parameters α₀, β₀, δ₀, ε₀, and γ₀ in the wavefunction?
• RQ3 Does the phase exhibit nontrivial dependence on the quantum number n, and under what conditions does it become independent of n?
• RQ4 Can the phase be expressed in a closed form using elementary functions, and how does this compare to alternative expressions involving the Hamiltonian expectation value?
• RQ5 What is the physical significance of the phase in relation to quantum fluctuations and parametric amplification in nonstationary media?

Key findings

• A closed-form expression for the Berry phase is derived: θₙ(t) = -(n+1/2)[arctan((2α₀ + (4α₀²+β₀⁴)tan t)/β₀²) - arctan(2α₀/β₀²) - t(4α₀²+β₀⁴+1)/(2β₀²)] + t[(2α₀ε₀ - β₀δ₀)² + ε₀²]/(2β₀²), with θₙ(0) = 0.
• For the standard solution (α₀=β₀=1, others zero), the phase reduces to θₙ(t) = 0, confirming consistency with textbook results.
• In shape-preserving evolutions (α₀=0, β₀=1), the phase becomes independent of n, a nontrivial feature of the dynamics.
• The derived phase formula is equivalent to an alternative expression based on the time integral of the Hamiltonian expectation value, confirming consistency across methods.
• The phase exhibits oscillatory behavior modulated by tan t and arctangent terms, reflecting the nontrivial time evolution of the wavefunction's global phase.
• The results are verified both analytically and computationally using Mathematica, with the notebook available from the author’s website.