For orthonormal functions $\psi$ and $\phi$, by definition; $$\langle \psi|\phi\rangle=0$$ However, when a wavefunction, $f$, is expressed as a linear combination of these orthonormal wavefunctions: $$f=c_1\psi+c_2\phi$$ operating on this function with an arbitrary operator, $\hat A$ results in: $$\langle f|\hat A|f\rangle=\langle c_1\psi+c_2\phi|c_1\hat A\psi+c_2\hat A\phi\rangle=c_1^2\langle\psi|\hat A|\psi\rangle+c_1c_2\langle\psi|\hat A|\phi\rangle+c_2c_1\langle\phi|\hat A|\psi\rangle+c_2^2\langle\phi|\hat A|\phi\rangle$$
My question is, do the crossed terms equal zero? (i.e does $\langle\psi|\hat A|\phi\rangle$ and $\langle\phi|\hat A|\psi\rangle$ equal zero?)
I initially thought that, yes, they do equal zero because, suppose that $\psi$ and $\phi$ are eigenfunctions of the operator $\hat A$ then it follows that: $$\langle\phi|\hat A|\psi\rangle=\langle\phi|a|\psi\rangle=a\langle\phi|\psi\rangle=a(0)=0$$ However, in a problem sheet I have just done, the question relies on these terms not being zero. Is my logic wrong or does this only equal zero in the case where the functions are eigenfunctions of the operator?