The existence of one-dimensional (1D) electronic states between self-organized Pt nanowires spaced 1.6 or 2.4 nm apart on a Ge(001) surface is revealed by low-temperature scanning tunneling microscopy. These perfectly straight Pt nanowires act as barriers for a surface state (located just below the Fermi level) of the underlying terrace. The energy positions of the 1D electronic states are in good agreement with the energy levels of a quantum particle in a well. Spatial maps of the differential conductivity of the 1D electronic states conclusively reveal that these states are exclusively present in the troughs between the Pt nanowires.