Abstract
The present chapter concentrates on electric dipole transitions (E1), while Sect. 5.4 will also treat electric quadrupole (E2) and magnetic dipole (M1) transitions. After some basics and terminology on electromagnetic radiation, polarization, and photon spin (Sect. 4.1), the essentials of spectroscopy are introduced in Sect. 4.2, the Einstein A and B coefficients are defined, and the classical model of a radiating oscillator is reviewed. The advanced reader may jump over this section and ignore also Sect. 4.3.1–4.3.5, where the elements of time dependent perturbation theory are summarized. However, Sect. 4.3.6 with terminology and some key results as well Sect. 4.4 with essentials on selection rules for dipole (E1) transitions are needed in the following sections and should be read carefully. The same holds for Sect. 4.5 where the angular characteristics of dipole radiation are presented. Section 4.6 may be used by the expert reader just as a source of reference with details on the evaluation of matrix elements and Einstein coefficients. In Sect. 4.7 photoinduced linear combinations of states are discussed – a theme of broad relevance. In this context we also introduce quantum beats and indicate some spectroscopic perspectives. Finally, we ask the very fundamental, almost philosophical question whether electrons may really ‘jump’ from one stationary state into another – and present experiments illuminating this profoundly.
A quantum system, such as an atom, may only be observed by changing it. Electromagnetic waves can induce transitions between stationary states and are the basis of spectroscopy – one of the most important methods for studying quantum systems in general. In this chapter we want to recapitulate the quantum mechanical tools needed and then treat in detail the rules and phenomena which underly light induced, electric dipole (E1) transitions.
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Notes
- 1.
We find this approach (leading to correct results for E1 transitions) conceptually more accessible than the general, rigorous treatment of all transition types based on the vector potential. The latter is outlined in Appendix H, while in Chaps. 1 and 2, Vol. 2, we shall generalize (4.1) and learn how to treat spatial distributions and quasi-monochromasy of real light beams adequately.
- 2.
- 3.
Methods for generating and detecting polarized light will be discussed in Chap. 1, Vol. 2.
- 4.
Unfortunately the terms “alignment” and “orientation” are mixed up time and again in the literature: Alignment refers to the direction of a polar vector (e.g. the E vector in the case of linearly polarized light). In contrast, orientation specifies the sense of rotation of an axial vector (e.g. the angular momentum of left and right circularly polarized light).
- 5.
For clarity, throughout this chapter we shall use \(\bar{R}_{ba}\), \(\bar{B}_{ba}\) etc. for rates and coefficients averaged over initial and summed over final substates, since in general we have to account for degenerate energy levels. In contrast, R ab (e) refers to transitions between specific substates b and a with polarization e.
- 6.
The bar on top of quantities indicates temporal averaging.
- 7.
In the literature often \(\tilde{u} ( \nu) =2\pi\tilde{u} ( \omega) \) per unit frequency is used. Then one has to replace \(\bar{B}_{ba}\rightarrow\bar{B} _{ba}^{(\nu)}=\bar {B}_{ba}/2\pi\) in all relevant equations.
- 8.
A somewhat heuristic derivation of eE 0 and \(\widehat{\mathsf{D}}(\widehat{{\boldsymbol {p}}},{\boldsymbol {r}})\) for electric dipole transitions will be presented below in Sect. 4.3.4, while a more rigorous, general derivation and specialization is found in Appendix H.
- 9.
- 10.
More precisely, e r ba is called dipole length matrix element. According to (H.25) the corresponding dipole velocity matrix element is
$$e \langle b\vert\widehat{{\boldsymbol {p}}}\vert a \rangle=\mathrm {i}\omega_{ba}m_{\mathrm{e}}e \langle b\vert {\boldsymbol {r}}\vert a \rangle. $$Both formulations lead to identical results if exact wave functions are used. For approximate solutions (i.e. quite generally) significant differences may occur.
- 11.
This is possible as long as no other preferential direction is enforced by a particular experiment. More generally, linearly polarized light propagating in z (at) direction may be written with (4.15) as a linear combination of σ + and σ − light (\(\cos\beta=\sin\beta=1/\sqrt{2}\)). Absorption or emission of this light involves superpositions of Δm=±1 transitions (sometimes called σ) and leads to the generation of linear combinations of states – as we shall explain in more detail in Sect. 4.5.2 and Sect. 4.7.1.
- 12.
More details for evaluating these matrix elements are made available in Appendix C.
- 13.
The overall phase factor −iexp(±iφ k ) is here of no significance for measurable quantities, since these are proportional to the absolute squares of the matrix elements.
- 14.
A detailed discussion of coherence will be given in Chap. 2, Vol. 2.
References
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Acronyms and Terminology
- CW:
-
‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc.
- E1:
-
‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation.
- E2:
-
‘Electric quadrupole’, transitions induced by the interaction of a quadrupolar charge distribution with the electromagnetic radiation field.
- IP:
-
‘Ionization potential’, of free atoms or molecules (in solid state physics the equivalent is called “workfunction”).
- IR:
-
‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between \(760\operatorname{nm}\) and \(1\operatorname{mm}\) according to ISO 21348 (2007).
- LHC:
-
‘Left hand cicularly’, polarized light, also σ + light.
- LIF:
-
‘Laser induced fluorescence’, radiation emitted from a quantum system after excitation by laser radiation (see Sect. 5.5.1, Vol. 2).
- M1:
-
‘Magnetic dipole’, transitions induced by the interaction of a magnetic dipole with the magnetic field component of electromagnetic radiation.
- NIST:
-
‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html.
- OMA:
-
‘Optical multichannel analyzer’, spectrometer which allows simultaneous registration of a whole spectrum.
- QED:
-
‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction.
- RHC:
-
‘Right hand cicularly’, polarized light, also σ − light.
- UV:
-
‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between \(100\operatorname{nm}\) and \(400\operatorname{nm}\) according to ISO 21348 (2007).
- VIS:
-
‘Visible’, spectral range of electromagnetic radiation. Wavelengths between \(380\operatorname{nm}\) and \(760\operatorname{nm}\) according to ISO 21348 (2007).
- VUV:
-
‘Vacuum ultraviolet’, spectral range of electromagentic radiation. part of the UV spectral range. Wavelengths between \(10\operatorname{nm}\) and \(200\operatorname{nm}\) according to ISO 21348 (2007).
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Hertel, I.V., Schulz, CP. (2015). Non-stationary Problems: Dipole Excitation with One Photon. In: Atoms, Molecules and Optical Physics 1. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54322-7_4
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