Abstract
A unitary operator U, which can transform the Fock space of a four-dimensional oscillator into the space in which the Hamiltonian of four harmonically coupled identical one-dimensional harmonic oscillators is diagonalized, is found. The coordinate representation of U is presented and is used to directly derive the wave function of the energy eigenstate of the coupled oscillators. The normally ordered form of U is also deduced with the use of integration within an ordered product.