Path Integral Methods

The path integral approach was introduced by Feynman1 in a seminal paper published in 1948. It provides an alternative formulation of time-dependent quantum mechanics, equivalent to that of Schrödinger. Since its inception, the path integral has found innumerable applications in many areas of physics and chemistry. Its main attractions can be summarized as follows: the path integral formulation offers an ideal way of obtaining the classical limit of quantum mechanics; it provides a unified description of quantum dynamics and equilibrium quantum statistical mechanics; it avoids the use of wavefunctions and thus is often the only viable approach to many-body problems; and it leads to powerful influence functional methods for studying the dynamics of a low-dimensional system coupled to a harmonic bath. The path integral formulation builds on the principle of superposition, which leads to the celebrated quantum interference observed in the microscopic world. Thus, the amplitude for making a transition between two states is given by the sum of amplitudes along all possible paths connecting these states in the specified time, a concept familiar from wave optics. Classical behavior is recovered through phase cancellation among paths whose phase is not stationary. The present article presents an introduction to the path integral formulation of quantum dynamics and quantum statistical mechanics along with numerical procedures useful in these areas and in electronic structure theory. Section 2 describes the path integral formulation of the quantum mechanical propagator and its relation to the more conventional Schrödinger description. That section also derives the classical limit and discusses the connection with equilibrium properties in the canonical ensemble. Numerical techniques are described in section 3. Selective chemical applications of the path integral approach are presented in section 4 and section 5 concludes.