Closed-form solutions of the Schrödinger equation for a coupled harmonic potential in three dimensions

Using the ansatz method, we obtain the exact closed-form solutions of a time independent Schrödinger equation for a coupled harmonic potential and its variant in three dimensions. Some authors [R.S. Kaushal, Quantum mechanics of noncentral harmonic and anharmonic potentials in two dimensions, Ann. Phys. (N.Y.) 206 (1991) 90–105] have raised the difficulty to solve the Schrödinger equation for coupled potentials but we found that by imposing some restrictions on ansatz parameters and at the cost of certain constraints on the potential parameters, ground state as well as excited state solutions of the Schrödinger equation are obtained. The number of constraints increases with increase in anharmonicity of the potential.