Truncated anharmonic oscillator

This represents a truncated anharmonic oscillator with anharmonicity controlled... 

The free energy of a Ginzburg-Landau field describing a system of weakly coupled chains in a plane is identified with the ground-state energy of a linear array of quantum anharmonic oscillators. The equivalent Hamiltonian is simplified for both the real and complex fields using a truncated basis of states. Results for both the real and complex fields will be discussed. In addition, the behavior of the specific heat and inverse correlation length for finite numbers of weakly coupled chains will be discussed. 

Tbus the anharmonicity in the vibration is captured by the coefficient x, which adopts values typically less than 1 % of ft>e for bond stretches, but may be up to 5% for those involving hydrogen. It can be shown that any function capable of representing the variation of potential energy with displacement will lead to energy levels given by a power series in (v + 1/2), so the simple harmonic and Morse oscillators are particular cases of this general anharmonic oscillator, with the power series truncated after the first and second terms, respectively. 

The key observation is that the higher-order corrections to the energy, in powers of 1/D, arise from anharmonic corrections to the normal mode harmonic oscillator motion. Now a given anharmonic correction to the energy, as we all learned long ago when we studied quantum mechanics, can be computed exactly from a finite number of excited harmonic oscillator functions. This means that a truncated basis which contains properly scaled harmonic oscillator functions can be used to compute exactly a finite number of anharmonic corrections. One simply pre-determines to which order one wants to compute the anharmonic corrections, calculates how many excited... 

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