Download Interaction with the radiation field

Interaction with the radiation field
Adrian Niznik-Barwicki
Universität Heidelberg
09.01.2015
What happens when light meets
matter?
Review: Light as an
Electromagnetical Wave (1865)
Maxwell's equation in free space:
Derivation for a E-Field:
EM-waves :
traveling at speed:
Classical picture of light-atom
interaction
=> electron as an damped harmonic oscillator driven at frequency ω
=> the so-called The Lorentz Oscillator
=> large portion of the observed effects in atom-field interactions not
supported (e.g quantized transitions) – QM model needed!
Interaction with the radiaton field
Hamiltonian of an electron in an electromagnetic field:
It can be splitted in the unperturbed and the perturbed term:
time-independent (unperturbed) term
time-dependent interaction term
Mathematical reminder:
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Time-dependent perturbation theory
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First-order transitions
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Perturbation with sinusoidal time dependence
Motivation:
We want to calculate the transition probability
Pm->n
(the probability that a particle which started out in the state |m> will
be found, at time t, in state |m>)
Time-dependent perturbation theory
Hamiltonian with ''small'' perturbation H'(t):
Interaction picture:
Iterative Solution
- Neumann series :
Interaction picture of QM
=> Suited when a Hamiltonian consists of a simple "free" Hamiltonian
and a perturbation.
=> Suited to quantum field theory and many-body physics.
=> The interaction picture does not always exist (Haag's theorem)
=> Introduced by Dirac in 1926
Differences among the three pictures
Problem:
First-order transitions
The system |Ψ> starts out in the state |m>.
''Small'' perturbation H'(t) takes place.
What is the probability of landing in the state |n>?
Solution:
=> Transforming to Dirac picture
=> Perturbation expansion to first order
=> Transition probability:
=> Transition rate (Fermi's Golden Rule)
Perturbation with sinusoidal time dependence
Perturbing Hamiltionian:
Transition probability (1th order):
Result:
Absorption of light
Oscillating electric field E:
Matrix element:
Transition probability:
Perturbed Hamiltonian as the additional energy:
Transition probability Pm->n as a function of the light frequency
Transition probability Pa->b as a function of the frequency
Transition probability Pa->b as a function of the frequency of time
Stimulated Emission
=> exactly the same probability as in the absorption case
=> raises the possibility of light amplification (LASER)
Spontaneous Emission
=> quantization of radiation field required (QFT)
=> fields are nonzero in the ground state
=> there is not such thing as a really spontanous emission
LASER
History
1865 - Light is an EM wave (Maxwell)
1900 - Classical model of atom-field interaction (Lorentz)
1905 - Einstein assumes that that an electromagnetic field
consists of quanta of energy (keyword: photoelectric effect )
1916 - Stimulated Emission (Einstein)
1926 - Interaction picture (Dirac)
1927 - Quantization of radiation field – we have photons! (Dirac)
1927 - Fermi Golden Rule (Dirac)
1950 – MASER is invented by Charles Townes and Arthur
Schawlow (Nobel Prize in Physics later)
1960 - Theodore H. Maiman operates the first functioning LASER
Paul A. M. Dirac
Bibliography
Introduction to Electromagnetism – David Griffiths (Chapter 9: Electromagnetic
Waves)
Introduction to Quantum Mechanics – David Griffiths (Chapter 9: Timedependent perturbation theory)
Quantum Mechanics – Franz Schwabl – (Chapter 16: Interaction with the
Radiation Field)
Thank you