  Consider a free Schr\"odinger particle inside an interval with walls
characterized by the Dirichlet boundary condition. Choose a parabola as the
normalized state of the particle that satisfies this boundary condition. To
calculate the variance of the Hamiltonian in that state, one needs to calculate
the mean value of the Hamiltonian and that of its square. If one uses the
standard formula to calculate these mean values, one obtains both results
without difficulty, but the variance unexpectedly takes an imaginary value. If
one uses the same expression to calculate these mean values but first writes
the Hamiltonian and its square in terms of their respective eigenfunctions and
eigenvalues, one obtains the same result as above for the mean value of the
Hamiltonian but a different value for its square (in fact, it is not zero);
hence, the variance takes an acceptable value. From whence do these
contradictory results arise? The latter paradox has been presented in the
literature as an example of a problem that can only be properly solved by
making use of certain fundamental concepts within the general theory of linear
operators in Hilbert spaces. Here, we carefully review those concepts and apply
them in a detailed way to resolve the paradox. Our results are formulated
within the natural framework of wave mechanics, and to avoid inconveniences
that the use of Dirac's symbolic formalism could bring, we avoid the use of
that formalism throughout the article. In addition, we obtain a resolution of
the paradox in an entirely formal way without addressing the restrictions
imposed by the domains of the operators involved. We think that the content of
this paper will be useful to undergraduate and graduate students as well as to
their instructors.
