Quantum Mechanics
Quantum mechanics has the distinction of being considered both the most empirically successful and the most poorly understood theory in the history of physics.
To take an oft-cited example of the first point: The theoretically calculated value of the anomalous magnetic moment of the electron using quantum electrodynamics matches the observed value to twelve decimal places, arguably the best confirmed empirical prediction ever made. To illustrate the second point, we have the equally oft-cited remarks of Niels Bohr, "Anyone who says that they can contemplate quantum mechanics without becoming dizzy has not understood the concept in the least," and of Richard Feynman, "[We] have always had (secret, secret, close the doors!) we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents." How could both of these circumstances obtain?
For the purposes of making predictions, quantum theory consists in a mathematical apparatus and has clear enough rules of thumb about how to apply the mathematical apparatus in various experimental situations. If one is doing an experiment or observing something, one must first associate a mathematical quantum state or wave function with the system under observation. For example, if one prepares in the laboratory an electron beam with a fixed momentum, then the quantum state of each electron in the beam will be something like a sine wave.
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