Low-temperature scanning tunneling spectroscopy and ab initio based theory reveal electron wave-vector quantization due to electron confinement in a nanostructure. A Fourier transformation of the spatial modulation pattern of the local density of states (LDOS) indicates quantization of the electron wave vector within the nanostructure, which gives rise to a discontinuous, staircaselike dispersion relation. Our results show that each step of the dispersion relation corresponds to an extremum of the LDOS and it identifes an eigenstate. The sequence of steps reveals the complete eigenstate spectrum, where states are enumerated and classified according to their even-odd parity.