Consider the operator
\widehat{Q}\equiv i\frac{d}{d\phi } , (3.25)
where ϕ is the usual polar coordinate in two dimensions. (This operator might arise in a physical context if we were studying the bead-on-a-ring; see Problem 2.46.) Is \widehat{Q} hermitian? Find its eigenfunctions and eigenvalues.
Here we are working with functions f(ϕ) on the finite interval 0≤ϕ≤2π , with the property that
f(\phi +2\pi )=f(\phi ), (3.26)
since ϕ and ϕ+2π describe the same physical point. Using integration by parts,
\left\langle f\mid \widehat{Q}g \right\rangle =\int_{0}^{2\pi}{f^*\left(i\frac{dg}{d\phi } \right)d\phi } = if^{*}g\mid ^{2\pi}_{0}-\int_{0}^{2\pi}{i\left(\frac{df^{*}}{d\phi } \right)}gd\phi =\left\langle \widehat{Q}f|g \right\rangle
so \widehat{Q} is hermitian (this time the boundary term disappears by virtue of Equation 3.26). The eigenvalue equation,
i\frac{d}{d\phi }=f(\phi )=qf(\phi ) (3.27)
has the general solution
f(\phi )=Ae^{-iq\phi} (3.28)
Equation 3.26 restricts the possible values of the q:
e^{-iq2\pi}=1\Rightarrow q=0,\pm 1,\pm 2,…. (3.29)
The spectrum of this operator is the set of all integers, and it is nondegenerate.