  It has been found that functions can oscillate locally much faster than their
Fourier transform would suggest is possible - a phenomenon called
superoscillation. Here, we consider the case of superoscillating wave functions
in quantum mechanics. We find that they possess rather unusual properties which
raise measurement theoretic, thermodynamic and information theoretic issues. We
explicitly determine the wave functions with the most pronounced
superoscillations, together with their scaling behavior. We also address the
question how superoscillating wave functions could be produced.
