We provide a detailed description of a general procedure by which a nano/micro-mechanical resonator can be calibrated using its thermal motion. A brief introduction to the equations of motion for such a resonator is presented, followed by a detailed
derivation of the corresponding power spectral density (PSD) function. The effective masses for a number of different resonator geometries are determined using both finite element method (FEM) modeling and analytical calculations.
We discuss a projector Monte Carlo method for quantum spin models formulated in the valence bond basis, using the S=1/2 Heisenberg antiferromagnet as an example. Its singlet ground state can be projected out of an arbitrary basis state as the trial s
tate, but a more rapid convergence can be obtained using a good variational state. As an alternative to first carrying out a time consuming variational Monte Carlo calculation, we show that a very good trial state can be generated in an iterative fashion in the course of the simulation itself. We also show how the properties of the valence bond basis enable calculations of quantities that are difficult to obtain with the standard basis of Sz eigenstates. In particular, we discuss quantities involving finite-momentum states in the triplet sector, such as the dispersion relation and the spectral weight of the lowest triplet.
