  While wave-packet solutions for relativistic wave equations are oftentimes
thought to be approximate (paraxial), we demonstrate that there is a family of
such solutions, which are exact, by employing a null-plane (light-cone)
variables formalism. A scalar Gaussian wave-packet in transverse plane is
generalized so that it acquires a well-defined z-component of the orbital
angular momentum (OAM), while may not acquire a typical "doughnut" spatial
profile. Such quantum states and beams, in contrast to the Bessel ones, may
have an azimuthal-angle-dependent probability density and finite quantum
uncertainty of the OAM, which is determined by the packet's width. We construct
a well-normalized Airy wave-packet, which can be interpreted as a one-particle
state for relativistic massive boson, show that its center moves along the same
quasi-classical straight path and, what is more important, spreads with time
and distance exactly as a Gaussian wave-packet does, in accordance with the
uncertainty principle. It is explained that this fact does not contradict the
well-known "non-spreading" feature of the Airy beams. While the effective OAM
for such states is zero, its quantum uncertainty (or the beam's OAM bandwidth)
is found to be finite, and it depends on the packet's parameters. A link
between exact solutions for the Klein-Gordon equation in the
null-plane-variables formalism and the approximate ones in the usual approach
is indicated, generalizations of these states for a boson in external field of
a plane electromagnetic wave are also presented.
