Early-time properties of quantum evolution

Abstract

Approximate formulas are given for the effective Hamiltonian Hll(t} governing the time evolution in a subspace JVil of the state space %. It is proved that this approximation is correct for any Hamiltonian H

The true quantum face of the “exponential” decay: Unstable systems in rest and in motion

“…Results of numerical calculations performed for ω(µ) = ω BW (µ) and presented in Fig (4) confirm the conclusion (31). In these Figures the function Summing up the oscillating decay curves of one component unstable system can not be considered as something extraordinary or as anomaly: It seems to be a universal feature of the decay process.…”

“…and h(t) is the effective Hamiltonian governing the time evolution in the subspace of unstable states H ∥ = PH, where P = |ϕ⟩⟨ϕ| (see [31] and also [32,33] and references therein). H ⊖ H ∥ = H ⊥ ≡ QH is the subspace of decay products.…”

“…H ⊖ H ∥ = H ⊥ ≡ QH is the subspace of decay products. Here Q = I − P. There is the following equivalent formula for h(t) [31][32][33]:…”

“…One meets the effective Hamiltonian h(t) when one starts with the Schrödinger equation for the total state space H and looks for the rigorous evolution equation for a distinguished subspace of states H || ⊂ H [27,31]. In general h(t) is a complex function of time and in the case of H ∥ of dimension two or more the effective Hamiltonian governing the time evolution in such a subspace it is a nonhermitian matrix H ∥ or non-hermitian operator.…”

“…are the instantaneous mass (energy) µ ϕ (t) and the instantaneous decay rate, γ ϕ (t) [31][32][33]. Here ℜ (z) and ℑ (z) denote the real and imaginary parts of z respectively.…”