Convolution Integrals in Schrödinger’s Equation for Entangled Systems

1. Green’s Functions and Propagators

The solution to the time-dependent Schrödinger equation often involves propagators, which describe the evolution of a wavefunction over time. The propagator G(x,t;x',t') is often expressed as a convolution integral:

ψ(x,t) = ∫ G(x,t;x',t') ψ(x',t') dx'

For entangled systems, such as two-particle wavefunctions, the Green’s function method involves convolution-like integrals in both configuration and momentum space.

2. Reduced Density Matrices and Decoherence

For a bipartite quantum system, the reduced density matrix of a subsystem ρ_A(x, x') after tracing out another system B can be written as:

ρ_A(x, x') = ∫ ψ(x, y) ψ*(x', y) dy

where the convolution integral over y (the degrees of freedom of the traced-out system) leads to decoherence effects in entangled systems.

3. Convolution in the Context of Quantum Correlations

Entangled wavefunctions of two particles often involve convolution-type integrals in their evolution. For example, in momentum space:

Ψ(p₁, p₂) = ∫ K(p₁, q₁) K(p₂, q₂) Ψ(q₁, q₂) dq₁ dq₂

where K(p, q) represents a transformation kernel that could come from a propagator or measurement process.

Example: EPR State Evolution

Consider an entangled EPR state in momentum representation:

Ψ(p₁, p₂) = δ(p₁ + p₂)

The time evolution of this state under free-particle Hamiltonians involves convolution integrals with propagators:

Ψ(x₁, x₂, t) = ∫ G(x₁ - x₁', t) G(x₂ - x₂', t) Ψ(x₁', x₂', 0) dx₁' dx₂'

Summary

  • Convolution integrals appear in the propagator formalism, Green’s function methods, and reduced density matrices of entangled systems.
  • They play a crucial role in describing the time evolution, measurement effects, and decoherence of entangled states.
  • Specific cases include the evolution of EPR states and the behavior of correlated two-particle systems.