We present the design, statistical analysis, and validation of a questionnaire to assess students' knowledge about basic aspects of quantum mechanics (QM). The QM evaluation (QME) is a true-false and multiple-choice mixed questionnaire that features 10 two-tier items spanning three relevant themes in quantum mechanics: wave behavior of matter, measurement, and atoms and electrons behavior. Its validity was assessed through a pilot administration to students and interviews with course instructors. We checked its internal consistency using both classic test theory and Rasch analysis to account for the different difficulty of each tier and for different scoring methods of the items. The questionnaire was administered to about 450 undergraduate physics students and high school physics teachers. Data show that it is a reliable instrument and all items have a good discriminatory power. Since the test does not require an advanced mathematical knowledge, it ideally lends itself to probe students' knowledge about quantum mechanics in a variety of university courses, from the introductory ones to those more formal and mathematically oriented.