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

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Abstract
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 toψ(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), HHkψ(q,0) ⟩, and taking the Fourier transformG(ω)=∫−∞∞dtexp(−iωt)F(t),we obtainG(ω)=2π∑k=0∞hkk!(ω).In terms of the formal expansion in the energy eingenfunctionsψ(q,0)=∑n=0∞cnϕn(q)+∫0∞dωc(ω)ϕ(ω,q),the Fourier transform representsG(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′hk(q′)≡Hkψ(q,0)]q=q′. The modified spectral function is taken to beGN(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

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