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Quantum Mechanics for Applied Physics

- Lecture IV
- Feynman path integrals
- Feynman diagrams
- Interaction with magnetic fields

Feynman confusion

Richard Feynman 1918-1988

1965 Nobel Physics Prize!

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 Lagrangian along 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.

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

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

Derivation of the Schrödinger equation

Solving the integral over d expanding to first

order of e 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.

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

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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.

The Dyson series

- Integral equation
- Iterative solution

Time-ordering operator

- Formal solution

Freeman Dyson

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 (2k1)th

harmonic is described by a loop diagram with

(2k2) 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

Interaction with classical electromagnetic fields

Electromagnetic coupling

- Hamiltonian of spinless charge e in classical EM

fields - Electric magnetic fields (SI units)

Canonical momentum

Kinetic momentum

Gauge transformations

- Unitary generator
- States Observables

invariant

Exercise Show that

Hamiltonian

gauge anomaly

Evolution operator

Dipole interaction

- Long wavelength approximation
- EM wavelength system dimensions
- Gauge transformation

optical wavelength Bohr radius

Coulomb potential

constant

dipole operator

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

Feynman diagrams

- 1st order amplitude

absorption

emission

sum over histories

frequency ?

Richard Feynman

Feynman diagrams

- 2nd order amplitude

emission

other combinations

absorption

A useful identity

step function

frequency ?

Richard Feynman

Feynman diagrams

- 2nd order amplitude

emission

other combinations

absorption

Richard Feynman