Semiclassical expansion
for arbitrary anharmonic oscillator in one dimension

Enter expression for the potential (see description of allowed functions)
V(x) = 
Enter oscillator quantum number n = n = 0 for the ground or 1, 2, 3, ... for excited states
Enter order of perturbation theory N = 1, 2, 3, ...
Optionally, enter xmin ~ - expected location of minimum of the potential
Default response
Use multiple precision arithmetic (many times slower)

H = -h2/2 d2/dx2 + V(x) Hamiltonian of the problem
E = E0 + E1 h + E2 h2 + ... + EN hN Semiclassical expansion of energy
V(x) Potential. It should have a minimum
h Plank's constant
E0 = Vmin Classical limit of energy
E1 h = (n + 1/2) h omega Quantum energy of harmonic vibration
omega = [V''(xmin)]1/2 Frequency of small vibrations
n Harmonic oscillator quantum number
N Order of perturbation theory

Mathematica programs for this calculation

600 coefficients for n=0, n=1, and n=2 for the quartic anharmonic oscillator V(x) = x2/2 + x4 calculated earlier.
1000 coefficients for n=0, n=1, and n=2 for the quartic anharmonic oscillator, in gzip-format.

435 coefficients of the ground-state energy expansion for the sextic anharmonic oscillator V(x) = x2/2 + x6 calculated earlier. Only non-zero coefficients E1, E3, E5, ... are listed there.

350 coefficients of the ground-state energy expansion for the octic anharmonic oscillator V(x) = x2/2 + x8 calculated earlier. Only non-zero coefficients E1, E4, E7, ... are listed there.

380 coefficients of the ground-state energy expansion for the cubic anharmonic oscillator V(x) = x2/2 + x3 calculated earlier.

251 coefficients of the ground-state energy expansion for the Mathieu potential V(x) = cos x calculated earlier.

RSPTexpansionOn-line calculation of Rayleigh - Schrödinger perturbation series

Online calculations

Designed by A. Sergeev