Anharmonic oscillator: a solution

Alexander Turbiner

ICN-UNAM, Mexico and Stony Brook

Wed, May. 17th 2023, 14:15

Pièce 026, Bât. 774, Orme des Merisiers

Anharmonic oscillator (quantum pendulum) is one of the oldest, non-solvable exactly
problems in quantum physics. It can be shown that for the one-dimensional quartic quantum
anharmonic oscillator with V (x) = x2 + g2x4 the Perturbation Theory (PT) in powers of g2
(weak coupling regime) and the semiclassical expansion in powers of ̄h for energies coincide.
It is due to the fact that the dynamics in x-space and in (gx)-space corresponds to the
same energy spectrum with effective coupling constant ( ̄hg2). Two equations, which govern
the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB)
equations, are derived. The PT in g2 for the log of wave function leads to polynomial in x
coefficients at given order in g2 for the RB equation and to the true semiclassical expansion
in powers of ̄h for the GB equation, which corresponds to a loop expansion for the density
matrix in the path integral formalism. Matching these two expansions in a single function
leads to a extremely compact, uniform approximation of the wavefunction in x-space with
unprecedented accuracy ∼ 10−6 locally and unprecedented accuracy ∼ 10−9 −10−10 in energy
for any g2 ≥ 0.

Contact : Ioannis
TSIARES