Quantum Mechanics for Applied Physics

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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!
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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.
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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
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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
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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.
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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.

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The Dyson series

• Integral equation
• Iterative solution

Time-ordering operator

• Formal solution

Freeman Dyson
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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
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Interaction with classical electromagnetic fields
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Electromagnetic coupling

• Hamiltonian of spinless charge e in classical EM
  fields
• Electric magnetic fields (SI units)

Canonical momentum
Kinetic momentum
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Gauge transformations

• Unitary generator
• States Observables

invariant
Exercise Show that
Hamiltonian
gauge anomaly
Evolution operator
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Dipole interaction

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

optical wavelength Bohr radius
Coulomb potential
constant
dipole operator
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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
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Feynman diagrams

• 1st order amplitude

absorption
emission
sum over histories
frequency ?
Richard Feynman
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Feynman diagrams

• 2nd order amplitude

emission
other combinations
absorption
A useful identity
step function
frequency ?
Richard Feynman
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Feynman diagrams

• 2nd order amplitude

emission
other combinations
absorption
Richard Feynman