  In this work we report on a new bootstrap method for quantum mechanical
problems that closely mirrors the setup from conformal field theory (CFT). We
use the equations of motion to develop an analogue of the conformal block
expansion for matrix elements and impose crossing symmetry in order to place
bounds on their values. The method can be applied to any quantum mechanical
system with a local Hamiltonian, and we test it on an anharmonic oscillator
model as well as the (1+1)-dimensional transverse field Ising model (TFIM). For
the anharmonic oscillator model we show that a small number of crossing
equations provides an accurate solution to the spectrum and matrix elements.
For the TFIM we show that the Hamiltonian equations of motion, translational
invariance and global symmetry selection rules imposes a rigorous bound on the
gap and the matrix elements of TFIM in the thermodynamic limit. The bound
improves as we consider larger systems of crossing equations, ruling out more
finite-volume solutions. Our method provides a way to probe the low energy
spectrum of an infinite lattice from the Hamiltonian rigorously and without
approximation.
