Quantum Mechanics for Applied Physics

1. Quantum Mechanics for Applied Physics Lecture IV Feynman path integrals Feynman diagrams Interaction with magnetic fields Feynman confusion 1965 Nobel Physics Prize! Richard Feynman 1918-1988

2. Feynman path integrals • Feynman proposed the following postulates: • The probability for any fundamental event is given by the square modulus of a complex amplitude. • The amplitude for some event is given by adding together the contributions of all the histories which include that event. • The amplitude a certain history contributes is proportional to • Where S is the action of that history, given by the time integral of the Lagrangianalong the corresponding path in the phase space of the system. Feynman showed that his formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action. Three out of the many paths included in the path integral used to calculate the quantum amplitude for a particle moving from point A to point B.

3. Classical Action for WKB and path Integrals The action is a particular quantity in a physical system that can be used to describe its operation. Action is an alternative to differential equations. The values of the physical variable at all intermediate points may then be determined by "minimizing" the action. In classical mechanics, the input function is the evolution of the system between two times t1 and t2, where represent the generalize coordinates. The action is defined as the Integral of the Lagrangian L for an input evolution between the two times, where the endpoints of the evolution are fixed and defined. When the total energy E is conserved, the HJ equation can be solved with the folowing variable separation

4. Definition The probability to go from point (xa,ta) to (xb,tb) is P(a,b) All Paths contribute equally in magnitude but the phase is changing The phase is the classical action in quantum units

5. Derivation of the Schrödinger equation Solving the integral over δ expanding to first order of ε we get the Schrödinger equation Photonic information processing needs quantum design rules Neil Gunther, Edoardo Charbon, Dmitri Boiko, and Giordano Beretta The quantum nature of light requires engineers to have a special set of design rules for fabricating photonic information processors that operate correctly. This device includes a 32 × 32 array of CMOS single-photon detectors.

6. Feynman Feynman diagrams are graphical ways to represent exchange forces. Each point at which lines come together is called a vertex, and at each vertex one may examine the conservation laws which govern particle interactions. • The intermediate stages in any diagram cannot be observed = virtual particles • The Initial and final particles can be observed = real particles

8. Feynman diagrams for electron-electron scattering • The illustration shows Feynman diagrams for electron-electron scattering. • In each diagram, the straight lines represent space-time trajectories of noninteracting electrons, and the wavy lines represent photons, particles that transmit the electromagnetic interaction. • External lines at the bottom of each diagram represent incoming particles (before the interactions), and lines at the top, outgoing particles (after the interactions). • Interactions between photons and electrons occur at the vertices where photon lines meet electron lines.

9. The Dyson series • Integral equation: • Iterative solution: Time-ordering operator • Formal solution: Freeman Dyson

10. Generation of harmonics by a focused laser beam in the vacuum A.M. Fedotova, and N.B. Narozhny, a, A Moscow Engineering Physics Institute, 115409 Moscow, Russia Received 18 September 2006;  accepted 22 September 2006.  Available online 5 October 2006. Abstract We consider generation of odd harmonics by a super strong focused laser beam in the vacuum. The process occurs due to the plural light-by-light scattering effect. In the leading order of perturbation theory, generation of (2k+1)th harmonic is described by a loop diagram with (2k+2) external incoming, and two outgoing legs. A frequency of the beam is assumed to be much smaller than the Compton frequency, so that the approximation of a constant uniform electromagnetic field is valid locally. Analytical expressions for angular distribution of generated photons, as well as for their total emission rate are obtained in the leading order of perturbation theory. Influence of higher-order diagrams is studied numerically using the formalism of Intense Field QED. It is shown that the process may become observable for the beam intensity of the order of 1027 W/cm2. Keywords: Super strong laser field

11. Interaction with classical electromagnetic fields

12. Electromagnetic coupling • Hamiltonian of spinless charge e in classical EM fields • Electric & magnetic fields (SI units): Canonical momentum Kinetic momentum

13. Gauge transformations invariant • Unitary generator: • States: Observables: Exercise: Show that Hamiltonian: gauge `anomaly’ Evolution operator:

14. Dipole interaction • Long wavelength approximation: EM wavelength system dimensions Gauge transformation: optical wavelength Bohr radius Coulomb potential constant dipole operator

15. Absorption and emission • 2-level system in resonant monochromatic EM field: • Radiation-induced transition amplitude • Absorptn/emission rate: • validity: Target state is in continuous spectrum • or: rate much slower than natural width frequency  polarization

16. Feynman diagrams • 1st order amplitude absorption emission sum over histories frequency  Richard Feynman

17. Feynman diagrams emission • 2nd order amplitude + other combinations absorption A useful identity: step function frequency  Richard Feynman

18. Feynman diagrams emission • 2nd order amplitude + other combinations absorption Richard Feynman