arXiv subjects
Publications and source records attributed to Shaun N. Mosley.
The free Schrodinger equation has constant velocity wavepacket solutions ψ_{\bf v} of the form ψ= f({\bf r} - {\bf v}t) e^{- i m c^2 t / 2}. These solutions are eigenvectors of a momentum operator {\bf \tilde p} which is symmetric in a positive definite scalar product space. We discuss whether these ψ_{\bf v} can act as basis states rather than the usual plane wave solutions.
We find dispersion-free wavepacket solutions to the Klein-Gordon equation, with the only free parameter being the wavepacket velocity $ {\bf v} $. These wavefunctions are eigenvectors of a velocity operator with commuting components which is symmetric in a certain scalar product space. We show that this velocity operator corresponds to a classical generator which may be obtained by a canonical tranformation from $ {\bf x}, {\bf k} $.
The momentum operator $ {\bf p} = - i {\bx \nabla} $ has radial component $ {\bf \tilde p} \equiv - i {\bf \hat{r}} ({1 \over r} \partial_r r).$ We show that ${\bf \tilde p} $ is the space part of a 4-vector operator, the zero component of which is a positive operator. Their eigenfunctions are localized along an axis through the origin. The solutions of the evolution equation $ i \partial_t ψ= {\tilde p^0} ψ$ are waves along the propagation axis. Lorentz transformations of these waves yield the aberration and Doppler shift. We briefly consider spin-half and spin-one representations.
The Laplacian in spherical coordinates contains the squared radial momentum operator $ p_r^2 = - r^{-1} \partial_r^2 r $ which is Hermitian and positive. However as has been pointed out by various authors the ``radial momentum operator" $ - i r^{-1} \partial_r r $ is not Hermitian. The positive square root operator of $ p_r^2 $ is found and also its inverse. We discuss the relation of these operators with Fourier transforms, the Hilbert transform and fractional integral operators.
The electromagnetic field can be expressed in terms of two complex potentials $ α, β,$ which are related to the Debye potentials. The evolution equations for these potentials are derived, which are separable either in parabolic coordinates (leading to the radiation fields) or in radial coordinates (multipole fields). Potentials corresponding to focused wave fields as well as plane waves are discussed. A conserved radiation density can be constructed in terms of these potentials, which is positive (negative) for positive (negative) helicity radiation.