= Solution
The <wavefunction> is a <plane wave> and a simultaneous formal <eigenfunction> of the <momentum operator> and the time-translation energy operator:
$$
-i\hbar\partial_x\phi=\hbar k\phi,\qquad i\hbar\partial_t\phi=E\phi.
$$
Its <momentum> and <energy> are therefore definite, while its <probability density> is spatially uniform:
$$
\boxed{p=\hbar k,\qquad \mathcal E=E,\qquad |\phi|^2=A^2}.
$$
For a particle of mass $m$, the <probability current> is $j=(\hbar/m)\operatorname{Im}(\overline\phi\,\partial_x\phi)=\hbar kA^2/m$, directed according to the sign of $k$.
On the whole <real line>, a nonzero constant-amplitude <plane wave> is not <square-integrable> and cannot be normalised to total <probability> one. It represents an idealised <momentum eigenstate>, interpreted through <generalized eigenfunctions> or as a limit of <wave packets>. Also, arbitrary $k,E$ do not automatically solve the <Schrödinger equation> for a specified <potential energy>: for a free particle, $E=\hbar^2k^2/(2m)$, or this value plus the constant <potential energy> for a constant-potential region. If $A=0$, the zero function is not a physical quantum state.
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