This paper is the first part of a two-part study on the quantum nonlinear Schrödinger equation [the second paper follows: Lai and Haus, Phys. Rev. A 39, 854 (1989)]. The quantum nonlinear Schrödinger equation is solved analytically and is shown to have bound-state solutions. These bound-state solutions are closely related to the soliton phenomenon. This fact has not been pursued in the literature. In this paper we use the time-dependent Hartree approximation to construct approximate bound states and then superimpose these bound states to construct soliton states. This construction enables us to study the quantum effects of soliton propagation and soliton collisions.