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