Müller was born in Berlin, where he studied at the Technische Hochschule in Charlottenburg (now Technische Universität Berlin) under Gustav Hertz. He received his degree in engineering in 1935 and his doctorate in 1936. Müller worked at the Siemens Research Laboratory, where he invented the field emission microscope in 1936 that allowed resolutions of 2 nanometers.
Müller married Klara Thüssing in 1939, and their only daughter Jutta was born in 1940. Due to the circumstances of war, he also worked at the Stabilovolt Company.
In 1947, he was appointed to the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry (renamed to Fritz-Haber-Institut on the occasion of its incorporation into the Max Planck Society in 1953) by Iwan N. Stranski. Here he developed the field ion microscope which, due to its resolution of 0.25 nm, was the first instrument used to observe atoms.
Müller, E. W. (1951). "Das Feldionenmikroskop". Zeitschrift für Physik. 131 (1): 136–142. Bibcode:1951ZPhy..131..136M. doi:10.1007/BF01329651. S2CID186215763. Müller's first FIM paper. According to Melmed, "[this paper] provided the world's first view of the atomic nature of solid matter and began an entirely new field of study."[2]
Müller, E.; Bahadur, K. (1956). "Field Ionization of Gases at a Metal Surface and the Resolution of the Field Ion Microscope". Physical Review. 102 (3): 624. Bibcode:1956PhRv..102..624M. doi:10.1103/PhysRev.102.624.