  We investigate an extension of the quantum Ising model in one spatial
dimension including long-range $1 / r^{\alpha}$ interactions in its statics and
dynamics with possible applications from heteronuclear polar molecules in
optical lattices to trapped ions described by two-state spin systems. We
introduce the statics of the system via both numerical techniques with finite
size and infinite size matrix product states and a theoretical approaches using
a truncated Jordan-Wigner transformation for the ferromagnetic and
antiferromagnetic case and show that finite size effects have a crucial role
shifting the quantum critical point of the external field by fifteen percent
between thirty-two and around five-hundred spins. We numerically study the
Kibble-Zurek hypothesis in the long-range quantum Ising model with Matrix
Product States. A linear quench of the external field through the quantum
critical point yields a power-law scaling of the defect density as a function
of the total quench time. For example, the increase of the defect density is
slower for longer-range models and the critical exponent changes by twenty-five
per cent. Our study emphasizes the importance of such long-range interactions
in statics and dynamics that could point to similar phenomena in a different
setup of dynamical systems or for other models.
