The tunneling current is measured as a function of voltage, lateral position, and vertical separation between a tungsten probe tip and a Si(111)2 × 1 surface. A rich spectrum is obtained in the ratio of differential to total conductivity, revealing the structure of the surface-state bands. The magnitude of the parallel wave vector for certain surface states is determined from the decay length of the tunneling current. Real-space images of the surface states reveal a phase reversal between those states on either side of the surface-state band gap. © 1986 The American Physical Society.