Mueller was born in Berlin and studied engineering at the university there, gaining his PhD in 1935. He worked in Berlin at Siemens and Halske (1935–37) and at Stabilovolt (1937–46). Subsequently he was at the Altenberg Engineering School (1946–47) and the Fritz Haber Institute (1948–52), from where he moved to the Pennsylvania State University. He became a naturalized American in 1962.

He is noted for his fundamental experimental work on solid surfaces. In 1936 he invented the field-emission microscope. In this device a fine metal point is placed a distance away from a phosphorescent screen in a high vacuum with a very high negative voltage applied to the point. Electrons are emitted from the surface under the influence of the electric field (field emission) and these travel to the screen where they produce a magnified image of the surface of the tip. The instrument is used to study reactions at surfaces.

In 1951 he made a further advance using the principle of field ionization. In the field-ion microscope the tip is at a positive potential in a low pressure of inert gas. Atoms of gas adsorbing on the tip are ionized and the positive ions are repelled from the tip and produce the image. The resolution is much better than in the field-emission microscope; in 1956, by cooling the tip in liquid helium, Mueller was able to resolve individual surface atoms for the first time.

As a further refinement Mueller used a field-ion microscope with a mass spectrometer, so that individual atoms on the surface could be seen, desorbed, and identified (the atom-probe field-ion microscope).