Introduction to path integrals - McGill Physics: Home

1 Introduction to path integralsv. January 22, 2006 Phys 719 - M. HilkeCONTENT Classical stochastic dynamics Brownian motion (random walk) Quantum dynamics Free particle Particle in a potential Driven harmonic oscillator Semiclassical approximation Statistical description (imaginary time) Quantum dissipative systemsINTRODUCTIONPath integrals are used in a variety of fields, includingstochastic dynamics, polymer physics , protein folding,field theories, quantum mechanics, quantum field theo-ries, quantum gravity and string theory. The basic ideais to sum up all contributing paths. Here we will overviewthe technique be starting on classical dynamics, in par-ticular, the random walk problem before we discuss thequantum case by looking at a particle in a potential, andfinish with an example of a quantum dissipative system,where this technique bares all its STOCHASTIC DYNAMICSThe classical equation of motion for a particle in afluctuating environment is simply given bymdvdt=F(t) v/B(1)wheremis the mass,vthe velocity andBdescribes thefriction of the particle.

The classical equation of motion for a particle in a °uctuating environment is simply given by m dv dt = F(t)¡v=B (1) where m is the mass, v the velocity and B describes the friction of the particle. This equation is the Langevin equation if F(t) is a °uctuating force. These equa-tions are often di–cult to solve and one possibility is

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