Direct Calculation of Energy Eigenvalue Spectra from Time Evolution of Nonstationary States

An arbitrary function in the eigenfunction space of some quantum-mechanical Hamiltonian may be thought to represent the initial configuration ψ(q,0) of a nonstationary state. The system develops in time according to ψ(q,t)=exp(−itH)ψ(q,0)= ∑ k=0∞(−it)kk!Hkψ(q,0).Defining F(t)≡〈ψ(q,0), ψ(q,t) 〉, and hk≡〈ψ(q,0), Hkψ(q,0) 〉, and taking the Fourier transformG(ω)= ∫ −∞∞dtexp(−iωt)F(t),we obtain G(ω)=2π ∑ k=0∞hkk!(ω).In terms of the formal expansion in the energy eingenfunctions ψ(q,0)= ∑ n=0∞cnφn(q)+ ∫ 0∞dωc(ω)φ(ω,q),the Fourier transform represents G(ω)=2π ∑ n=0∞|cn|2δ(ω+ωn)+2π|c(−ω)|2,which exhibits, in principle, the entire eigenvalue spectrum. In this paper, a direct method of calculating eigenvalue spectra, based on the foregoing principle, is proposed. Two modifications are required for computational practicability: (i) use of a finite representation for the delta function and truncation of the summation (a); (ii) replacement of the integrals hk by hk(q′)≡Hkψ(q,0)]q=q′. The modified spectral function is taken to be GN(τ,ω,q′)=2π ∑ k=0Nhk(q′)k!χτ(k)(ω),with χτ(ω)≡sinτω/πω. The sequence GN(τ,ω,q′) is shown to converge as N→∞ if in the Expansion (b) the coefficients cn and c(ω) decrease with ω as exp(—ω/λ) or faster. Assuming convergence, the spectral function represents a broadened eigenvalue spectrum.