Starting with a very basic statement that any physical constants cannot be written with an infinite precision, it is shown how to introduce this uncertainty into the Hamiltonian of non-relativistic atomic (NR) physics and how to estimate errors on quantum operators (energy, frequency, momenta) when an uncertainty is assigned to . The Schrödinger equation is written and the kinetic energy term is transformed into a Laplacian: . This transformation leads (as known since 1926) to the wave equation, whose solutions are wave functions. The relativity correction to the kinetic energy term is introduced and its effect is discussed. (h constant has an uncertainty value taken from CODATA.)
For a quantum mechanical particle in a classical electromagnetic field, the active and passive views of gauge invariance are compared. In the active point of view, both the wavefunction and the potentials are transformed so that the Schrödinger equation is form invariant. In the passive point of view, a change in the representation of the canonical momentum is made which just cancels the change in the gauge of the potentials. The condition that a gauge-invariant operator has the same expectation value in all gauges gives a functional equation, the solution of which shows that gauge-invariant operators depend only on the kinetic momentum or coordinates.
Two aspects of quantum Monte Carlo are discussed. First, we review a procedure for obtaining trial wavefunctions for use in quantum Monte Carlo simulations that have both smaller statistical errors and improved expectation values than commonly used functions. Second, we present a correlated sampling approach for calculating energy differences in variational Monte Carlo much more accurately than the values of the energies. This method is used to calculate the potential energy surfaces of H2 and BH. In the case of BH, at bond lengths 10% different from the reference geometry, the reduced statistical error in the relative energy results in a factor of 3000 saving in computer time.