Investigation the order of approximants affects in the ITP method for calculating the wave function of the ground state

In order to show the effects of different degrees of expansion of the imaginary time propagation operator on convergence, the imaginary time propagation method is used, for example, for a simple harmonic oscillator. Considering the potential energy, the wave function of the ground state of the system as well as the wave function that is close to the wave function of the ground state of the system can be seen that each time the imaginary time operator is applied, the wave of initial guess function approaches the wave function of the ground state of the system. Examining the different levels of expansion of the operator, it can be seen that the higher the level used, the less repetition of the operations of this operator to reach the wave function of the base state of the system. The Monte Carlo computational method was used to calculate the integrals. Finally, this method has been tested on the hydrogen atom to investigate the physical problem.