@article{Jahankohan_Dizaji_Eshghi_2024, title={An Analytical Solution of the Schrödinger Equation for H_2^+ Molecular ion for n = 0, 1 and 2: The Born-Oppenheimer Approximation}, url={https://oiccpress.com/journal-of-theoretical-and-applied-physics/article/an-analytical-solution-of-the-schrodinger-equation-for-h_2-molecular-ion-for-n-0-1-and-2-the-born-oppenheimer-approximation/}, abstractNote={Hydrogen molecular ion H_2^+ is generally accepted as the simplest molecule. On the other hand, the Born-Oppenheimer approximation is almost invariably the mandatory one starting point for the quantum-mechanical treatment of molecular systems. It consists of the separation of the electronic and nuclear degrees of freedom which is possible because of the enormous difference between the masses of these two kinds of particles. In this research, we have solved the Schrödinger equation for H_2^+ molecular ion with the Born-Oppenheimer approximately. We have analytically obtained wave functions and the energies for electron and nucleus for n=0,1 and 2 by using the ansatz method. Also, we have shown that the analytical solution of the Schrödinger equation presented using the Oppenheimer approximation is a suitable solution and the total wave function of the H_2^+ ground state, first and second state is a linear combination of two linearly independent wave functions.}, journal={Journal of Theoretical and Applied Physics}, publisher={OICC Press}, author={Jahankohan, Kazem and Dizaji, Hossein Zaki and Eshghi, Mahdi}, year={2024}, month={May}, keywords={Schrödinger equation, The Born-Oppenheimer approximation, Hydrogen molecular} }