Anharmonic Oscillator Spectrum via Diagonalization of Amplitudes
Anharmonic Oscillator Spectrum via Diagonalization of Amplitudes
The energy spectrum of a quantum system can be accurately calculated by the numerical diagonalization of the space-discretized matrix of its evolution operator, that is, the matrix of its transition amplitudes. Here we calculate the spectrum of a one-dimensional anharmonic oscillator with the potential , using level effective action. For a general quantum system described by the Hamiltonian , the probability for a transition from an initial state to a final state in time is calculated as , with the transition amplitude . In a recently developed effective action approach, the amplitude is expressed in terms of the effective potential. Then a set of recursive relations allows systematic analytic derivation of terms in the expansion of the effective potential in the time . The effective action thus obtained is characterized by a chosen level corresponding to the maximal order occurring in its expansion.
V(x)=+
1
2
2
x
g
24
4
x
p=8
H
a〉
b〉
t
A(ab;t)
2
|
A(ab;t)=〈ba〉
-tH/ℏ
e
t
p
p
t