We propose and demonstrate, based on the concept of a microwave bandstop filter, two quantum wave bandstop filter structures. Both structures employ nanoscale gates in a heterojunction transistor to induce a quantum cavity connected by two one-dimensional wires. As the electron wavelength is changed by the gate voltage, we observed that, at certain gate voltages, the transmission of electron waves through the cavity is partially blocked and the drain current drops as large as 50%. This phenomenon is explained in terms of the destructive quantum interference between different electron wave modes in the cavity.