Adrian Niznik-Barwicki Universität Heidelberg 09.01wolschin/qms14_11.pdf · Time-dependent...
Transcript of Adrian Niznik-Barwicki Universität Heidelberg 09.01wolschin/qms14_11.pdf · Time-dependent...
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:
l Time-dependent perturbation theory l First-order transitions l 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
Iterative Solution - Neumann series :
Interaction picture:
Hamiltonian with ''small'' perturbation H'(t):
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
First-order transitions
Solution: => Transforming to Dirac picture
=> Perturbation expansion to first order
Problem: 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>?
=> Transition rate (Fermi's Golden Rule)
=> Transition probability:
Perturbation with sinusoidal time dependence
Perturbing Hamiltionian:
Transition probability (1th order):
Result:
Absorption of light
Oscillating electric field E: Perturbed Hamiltonian as the additional energy:
Matrix element:
Transition probability:
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
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
=> exactly the same probability as in the absorption case => raises the possibility of light amplification (LASER)
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: Time-dependent perturbation theory)
Quantum Mechanics – Franz Schwabl – (Chapter 16: Interaction with the Radiation Field)
Thank you