Path Integrals in Quantum Mechanics (16-27 Aug.)

Module content

• Learn about the extension of the Lagrangian and Hamiltonian picture of classical mechanics to the quantum level
• Discuss how the transition amplitude between two quantum-mechanical states can be written in terms of a path integral
• Establish a general relation between path integrals and the Green functions of the underlying Hamilton operator
• Study quantum-mechanical harmonic and anharmonic oscillators, Brownian motion, geometrical optics, instantons, and scattering theory

In quantum mechanics, path integrals provide a physically very intuitive alternative to the common operator-based method of quantization. This appealing feature has lead to a proliferation of the use of path integrals in various fields of physics beyond their origins in quantum mechanics. It is therefore of great use to become acquainted with the method of path integration.

In this module, we briefly review the Lagrangian and Hamiltonian picture of classical mechanics and recall the basic ideas of the underlying variational principles. Based on these concepts, we extend our considerations to the quantum level and discuss how the transition amplitude between two quantum-mechanical states can be written in terms of a path integral. We establish a general relation between path integrals and the Green functions of the underlying Hamilton operator, which, in turn, can be used as a starting point for solving Schrödinger’s equation. Path integrals can be exactly solved for free particles, when their Hamiltonian is quadratic in both coordinates and momenta. In other cases, one has to apply approximation schemes such as the semi-classical WKB approximation or perturbation theory. The formalism will be illustrated by means of numerous interesting examples, such as the quantum-mechanical harmonic and anharmonic oscillators, Brownian motion, geometrical optics, instantons, and scattering theory. In the last part, an outlook to further applications of path integrals in quantum field theory and statistical field theory will be presented.