Selection rule

In quantum mechanics, a selection rule is a rule that provides information on whether a transition between two states of a given system (for example, atomic shell , atomic nucleus or vibration state) is possible through the emission or absorption of electromagnetic radiation . When “forbidden” transitions are mentioned, these prohibitions are often “softened” by various effects and the respective transitions can nevertheless be observed; however, the transition probability is usually very small. With a given multipole order , the rules can be theoretically justified by calculating the transition matrix elements according to Fermi's golden rule .

Selection rules for electric dipole radiation

Electronic transitions in the orbitals occur primarily through electrical dipole radiation . For one-electron transitions, if the electron spin is neglected , the following selection rules apply :

${\displaystyle \,\Delta l=\pm 1}$

${\displaystyle \,\Delta m_{l}=0,\pm 1}$

In this case, referred to the angular momentum quantum number , the magnetic orbital angular momentum quantum number of the system. The first selection rule can be understood from the fact that an angular momentum must always be transmitted through the emission or absorption of a photon, for example from an atomic shell, since the photon as a boson itself has a spin and angular momentum conservation must apply. However, it must be noted that a direct transfer of an orbital angular momentum from the photon to the electron is rather improbable due to the different orders of magnitude of wavelengths in the optical range compared to atomic or molecular radii . In the case of electrical dipole transitions, the absorption or emission of a photon takes place without orbital angular momentum transmission. ${\displaystyle l}$ ${\displaystyle m_{l}}$ ${\displaystyle \,\Delta l=\pm 1}$ ${\displaystyle s=1}$ ${\displaystyle \lambda _{optisch}\approx 10^{-7}-10^{-6}m}$ ${\displaystyle r_{atomar}\approx 10^{-10}m}$

Selection rules for any multipole radiation

The following selection rules apply to any multipole transitions (in the following E k or M k for electric or magnetic radiation, e.g. E1 for electric dipole radiation, E2 for electric quadrupole radiation, M3 for magnetic octupole radiation, etc.): ${\displaystyle 2^{k}}$

${\displaystyle |I_{i}-I_{f}|\leq k\leq I_{i}+I_{f}}$

${\displaystyle P_{i}\cdot P_{f}=(-1)^{k}}$for E k ,

${\displaystyle P_{i}\cdot P_{f}=(-1)^{k+1}}$for M k .

${\displaystyle I_{i}}$and it is the total angular momentum of the states of the system involved, or the parity of the initial or final state. k denotes the (integer) angular momentum of the radiation field. ${\displaystyle I_{f}}$ ${\displaystyle P_{i}}$ ${\displaystyle P_{f}}$

basis

The selection rules according to which a transition is characterized as allowed or prohibited are derived from the transition matrix elements ${\displaystyle |\Psi _{i}\rangle \to |\Psi _{f}\rangle }$

${\displaystyle M_{if}=\langle \Psi _{f}|\mu |\Psi _{i}\rangle }$

derived. It is the transition moment operator, the initial state and the final state. ${\displaystyle \mu }$ ${\displaystyle |\Psi _{i}\rangle }$ ${\displaystyle |\Psi _{f}\rangle }$

A transition is prohibited if the transition matrix element disappears, otherwise it is allowed. The exact value is often of no interest, since the selection rules are weakened by considering higher orders of the transition operator.

The transition matrix element can be solved for idealized models such as the harmonic oscillator , the rigid rotator and the hydrogen atom through simple symmetry considerations.

For a one-electron system e.g. B. the transition matrix element is given by the integral over the position wave functions of the electron after the transition , the transition moment operator and the starting position wave function of the electron ${\displaystyle \Psi _{f}^{*}({\vec {r}})}$ ${\displaystyle \mu }$ ${\displaystyle \Psi _{i}({\vec {r}})}$

${\displaystyle M_{if}=\int _{\mathbb {R} ^{3}}\Psi _{f}^{*}({\vec {r}})\mu \Psi _{i}({\vec {r}})\,d^{3}r}$

The product must have even symmetry , because with odd symmetry the integral vanishes and the transition is not allowed. The symmetry of is the direct product of the symmetries of the three components (see also: character table ). ${\displaystyle m_{if}({\vec {r}})=\Psi _{f}^{*}({\vec {r}})\mu \Psi _{i}({\vec {r}})}$ ${\displaystyle m_{if}(-{\vec {r}})=m_{if}({\vec {r}})}$ ${\displaystyle m_{if}(-{\vec {r}})=-m_{if}({\vec {r}})}$ ${\displaystyle m_{if}}$

Symmetry of the transition moment operator

crossing	µ transforms like	comment
electric dipole	x, y, z	optical spectra
electric quadrupole	x 2 , y 2 , z 2 , xy, xz, yz	Constraint x 2 + y 2 + z 2 = 0
electrical polarizability	x 2 , y 2 , z 2 , xy, xz, yz	Raman spectra
magnetic dipole	R x , R y , R z	optical spectra (weak)

R _x , R _y and R _z mean rotations about the x, y and z directions.

Overview

In the following, the selection rules for the lowest orders of multipole radiation are given for hydrogen-like atoms. It is

• ${\displaystyle J}$ the total angular momentum quantum number,
• ${\displaystyle L}$ the total orbital angular momentum quantum number,
• ${\displaystyle S}$ the total spin quantum number and
• ${\displaystyle M_{J}}$ the total magnetic quantum number,
• ${\displaystyle l}$ the orbital angular momentum quantum number.

	Electric dipole (E1)	Magnetic dipole (M1)	Electric quadrupole (E2)	Magnetic Quadrupole (M2)	Electric octupole (E3)	Magnetic octupole (M3)
(1)	${\displaystyle {\begin{matrix}\Delta J=0,\pm 1\\(J=0\not \leftrightarrow 0)\end{matrix}}}$		${\displaystyle {\begin{matrix}\Delta J=0,\pm 1,\pm 2\\(J=0\not \leftrightarrow 0,0\not \leftrightarrow 1;\ {\begin{matrix}{1 \over 2}\end{matrix}}\not \leftrightarrow {\begin{matrix}{1 \over 2}\end{matrix}})\end{matrix}}}$		${\displaystyle {\begin{matrix}\Delta J=0,\pm 1,\pm 2,\pm 3\\(J=0\not \leftrightarrow 0,0\not \leftrightarrow 1,0\not \leftrightarrow 2;\\{\begin{matrix}{1 \over 2}\end{matrix}}\not \leftrightarrow {\begin{matrix}{1 \over 2}\end{matrix}},{\begin{matrix}{1 \over 2}\end{matrix}}\not \leftrightarrow {\begin{matrix}{3 \over 2}\end{matrix}};\ 1\not \leftrightarrow 1)\end{matrix}}}$
(2)	${\displaystyle {\begin{matrix}\Delta M_{J}=0,\pm 1\\(M=0\not \leftrightarrow 0,{\text{wenn }}\Delta J=0)\end{matrix}}}$		${\displaystyle \Delta M_{J}=0,\pm 1,\pm 2}$		${\displaystyle \Delta M_{J}=0,\pm 1,\pm 2,\pm 3}$
(3)	${\displaystyle \pi _{\mathrm {f} }=-\pi _{\mathrm {i} }\,}$	${\displaystyle \pi _{\mathrm {f} }=\pi _{\mathrm {i} }\,}$		${\displaystyle \pi _{\mathrm {f} }=-\pi _{\mathrm {i} }\,}$		${\displaystyle \pi _{\mathrm {f} }=\pi _{\mathrm {i} }\,}$
(4)	${\displaystyle \Delta l=\pm 1}$, any ${\displaystyle \Delta n}$	${\displaystyle \Delta l=0,\,\Delta n=0}$	${\displaystyle \Delta l=0,\pm 2}$, any ${\displaystyle \Delta n}$	${\displaystyle \Delta l=\pm 1}$, any ${\displaystyle \Delta n}$	${\displaystyle \Delta l=\pm 1,\pm 3}$, any ${\displaystyle \Delta n}$	${\displaystyle \Delta l=0,\pm 2}$, any ${\displaystyle \Delta n}$
(5)	If ${\displaystyle \Delta S=0}$ ${\displaystyle {\begin{matrix}\Delta L=0,\pm 1\\(L=0\not \leftrightarrow 0)\end{matrix}}}$	If ${\displaystyle \Delta S=0}$ ${\displaystyle \Delta L=0\,}$	If ${\displaystyle \Delta S=0}$ ${\displaystyle {\begin{matrix}\Delta L=0,\pm 1,\pm 2\\(L=0\not \leftrightarrow 0,1)\end{matrix}}}$		If ${\displaystyle \Delta S=0}$ ${\displaystyle {\begin{matrix}\Delta L=0,\pm 1,\pm 2,\pm 3\\(L=0\not \leftrightarrow 0,1,2;\ 1\not \leftrightarrow 1)\end{matrix}}}$
(6)	If ${\displaystyle \Delta S=\pm 1}$ ${\displaystyle \Delta L=0,\pm 1,\pm 2\,}$		If ${\displaystyle \Delta S=\pm 1}$ ${\displaystyle {\begin{matrix}\Delta L=0,\pm 1,\\\pm 2,\pm 3\\(L=0\not \leftrightarrow 0)\end{matrix}}}$	If ${\displaystyle \Delta S=\pm 1}$ ${\displaystyle {\begin{matrix}\Delta L=0,\pm 1\\(L=0\not \leftrightarrow 0)\end{matrix}}}$	If ${\displaystyle \Delta S=\pm 1}$ ${\displaystyle {\begin{matrix}\Delta L=0,\pm 1,\\\pm 2,\pm 3,\pm 4\\(L=0\not \leftrightarrow 0,1)\end{matrix}}}$	If ${\displaystyle \Delta S=\pm 1}$ ${\displaystyle {\begin{matrix}\Delta L=0,\pm 1,\\\pm 2\\(L=0\not \leftrightarrow 0)\end{matrix}}}$

To (2): The size provides information about the polarization of the EM radiation. means linearly polarized light, means circularly polarized light. ${\displaystyle \Delta M}$ ${\displaystyle \Delta M=0}$ ${\displaystyle \Delta M=\pm 1}$

In (3) the parity is considered, i.e. the behavior of the wave function with spatial reflections . ${\displaystyle P\psi ({\vec {r}})\equiv \psi (-{\vec {r}})=\pm \psi ({\vec {r}})}$

For one-electron systems, (4) applies without exception. For multi-electron systems consider (5) and (6).

For only light atoms, (5) holds strictly; means that transitions from the singlet to the triplet system are not allowed because the spin-orbit coupling is small (only then can the wave function be written as the product of the position and spin function). ${\displaystyle \Delta S=0}$

For heavy atoms with large spin-orbit coupling there is intercombination (6), i. H. Transitions between different multiplet systems. The transition probability is, however, much lower than with (5).

Quantum mechanical consideration

Analysis of the Hamilton operator

For a particle with the charge in the electromagnetic field, the Hamilton operator ( SI units ) is given by: ${\displaystyle q}$

${\displaystyle H={\frac {1}{2m}}\left({\vec {p}}-q{\vec {A}}\right)^{2}+q\Phi ({\vec {r}})=\left({\frac {{\vec {p}}^{\,2}}{2m}}-{\frac {q}{2m}}{\vec {p}}\cdot {\vec {A}}-{\frac {q}{2m}}{\vec {A}}\cdot {\vec {p}}+{\frac {q^{2}}{2m}}{\vec {A}}^{2}\right)+q\Phi ({\vec {r}})}$,

where the mass of the particle, the momentum operator, the vector potential operator , are the electrostatic potential. ${\displaystyle m}$ ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {A}}}$ ${\displaystyle \Phi ({\vec {r}})}$

With the commutation relation of and  : ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {A}}}$

${\displaystyle \sum _{i=1}^{3}[p_{i},A_{i}]={\frac {\hbar }{i}}\sum _{i}{\frac {\partial A_{i}}{\partial q_{i}}}={\frac {\hbar }{i}}\nabla \cdot {\vec {A}}}$,

and the Coulomb calibration :

${\displaystyle \nabla \cdot {\vec {A}}=0}$ ,

applies:

${\displaystyle {\vec {p}}\cdot {\vec {A}}={\vec {A}}\cdot {\vec {p}}}$.

In addition, the field should not be extremely strong, so that applies and the quadratic term in can be neglected. ${\displaystyle {\vec {A}}\cdot {\vec {p}}\gg {\frac {e}{c}}{\vec {A}}^{2}}$ ${\displaystyle {\vec {A}}}$

Thus the approximated Hamilton operator is the same

${\displaystyle H={\frac {{\vec {p}}^{\,2}}{2m}}-{\frac {q}{m}}{\vec {A}}\cdot {\vec {p}}+q\Phi ({\vec {r}})}$,

wherein a time-dependent periodic disturbance corresponds to the transitions of the electronic states of an atom or molecule capable of inducing. ${\displaystyle {\frac {q}{m}}{\vec {A}}\cdot {\vec {p}}}$

Vector potential of the electromagnetic field

Classic

The radiated field is now a plane wave , e.g. B. classic

${\displaystyle {\vec {A}}({\vec {r}},t)=2A_{0}{\hat {\epsilon }}\cos({\vec {k}}\cdot {\vec {r}}-\omega t)=A_{0}{\hat {\epsilon }}\left(e^{i({\vec {k}}\cdot {\vec {r}}-\omega t)}+e^{-i({\vec {k}}\cdot {\vec {r}}-\omega t)}\right)}$

The unit vector indicates the direction of the vector potential, thus the polarization . is the angular frequency and the wave vector of the electromagnetic radiation . This consideration would be sufficient for stimulated emission and absorption . ${\displaystyle {\hat {\epsilon }}}$ ${\displaystyle \omega }$ ${\displaystyle {\vec {k}}}$

Quantum mechanics

In order to be able to explain the effect of the spontaneous emission one has to consider the EM field quantized. The above disturbance leads to the emission or absorption of photons of energy ; d. H. Energy quanta of magnitude are added or subtracted from the EM field . ${\displaystyle \hbar \omega }$ ${\displaystyle \hbar \omega }$

Now we postulate that the vacuum contains an infinite number of harmonic oscillators , namely one for any wavenumber (or frequency), since precisely the harmonic oscillator has equidistant energy jumps ( and between two neighboring energy levels). The number of photons in a volume now corresponds to the quantum number of the harmonic oscillator. ${\displaystyle E_{n}=\hbar \omega (n+1/2)}$ ${\displaystyle \Delta E=\hbar \omega }$ ${\displaystyle N}$ ${\displaystyle V}$ ${\displaystyle n}$

In the quantized form is an operator that has parts of the bosonic creation and annihilation operators . ${\displaystyle {\vec {A}}}$

${\displaystyle {\vec {A}}({\vec {r}},t)=A_{0}{\hat {\epsilon }}\left({\sqrt {N}}e^{i({\vec {k}}\cdot {\vec {r}}-\omega t)}+{\sqrt {N+1}}e^{-i({\vec {k}}\cdot {\vec {r}}-\omega t)}\right)\quad {\text{mit}}\quad A_{0}={\sqrt {\frac {2\pi c^{2}\hbar }{\omega V}}}}$

The first term describes the absorption of a photon by the atom (a photon and the energy is withdrawn from the EM field - annihilation) and the second term describes the emission of a photon by the atom (the EM field becomes a photon and the energy added - generation). ${\displaystyle \hbar \omega }$ ${\displaystyle \hbar \omega }$

In the quantized case, the energy of the oscillators is never zero (minimum for ) and thus the interference field is never zero either - spontaneous emission can take place - because it applies to : ${\displaystyle E_{0}=\hbar \omega /2}$ ${\displaystyle N=0}$ ${\displaystyle N=0}$

${\displaystyle {\vec {A}}({\vec {r}},t)=A_{0}{\hat {\epsilon }}e^{-i({\vec {k}}\cdot {\vec {r}}-\omega t)}}$

Transition rates

The above perturbation operators are periodic in time because of the factors . According to Fermi's golden rule , the transition rate (= transition probability per time) from state to state is the same: ${\displaystyle e^{\pm i\omega t}}$ ${\displaystyle i}$ ${\displaystyle f}$

${\displaystyle \Gamma _{i\rightarrow f}={\frac {2\pi }{\hbar }}\,|\langle \Psi _{f}|{\frac {e}{mc}}{\vec {A}}\cdot {\vec {p}}e^{\mp i\omega t}|\Psi _{i}\rangle |^{2}\,\delta (E_{f}-E_{i}+\hbar \omega )}$

Especially for the spontaneous emission you get:

${\displaystyle \Gamma _{i\rightarrow f}=A_{0}^{2}{\frac {2\pi }{\hbar }}{\frac {e^{2}}{m^{2}c^{2}}}\,|\langle \Psi _{f}|e^{-i{\vec {k}}\cdot {\vec {r}}}{\hat {\epsilon }}\cdot {\vec {p}}|\Psi _{i}\rangle |^{2}\,\delta (E_{f}-E_{i}+\hbar \omega )}$

The matrix elements are therefore the decisive factor in how likely a transition will take place. ${\displaystyle \langle \Psi _{f}|e^{-i{\vec {k}}\cdot {\vec {r}}}{\hat {\epsilon }}\cdot {\vec {p}}|\Psi _{i}\rangle }$

Dipole approximation

The dipole approximation is an approximation method from quantum optics . The exponential function can be expanded into a series :

${\displaystyle e^{-i{\vec {k}}\cdot {\vec {r}}}=1-i{\vec {k}}\cdot {\vec {r}}-{\frac {1}{2}}({\vec {k}}\cdot {\vec {r}})^{2}\mp \ldots }$

For hydrogen-like atoms, the wave number and radius can be estimated as follows - for the ground state energy, for the Bohr radius; is the fine structure constant : ${\displaystyle E}$ ${\displaystyle r}$ ${\displaystyle \alpha }$

${\displaystyle |{\vec {k}}|={\frac {E}{\hbar c}}\approx {\frac {mc\alpha Z^{2}}{2\hbar }}\ ,\quad |{\vec {r}}|\lessapprox {\frac {\hbar }{mc\alpha Z}}\quad \Rightarrow {\vec {k}}\cdot {\vec {r}}\leq |{\vec {k}}||{\vec {r}}|\lessapprox {\frac {\alpha Z}{2}}={\frac {Z}{274}}}$

For you can break off the series after the first term: ${\displaystyle Z\ll 274}$

${\displaystyle e^{-i{\vec {k}}\cdot {\vec {r}}}\approx 1}$

The same potential acts on the atomic nucleus and the electrons. This is the electrical dipole approximation. It is justified if the variation of the potential to the order of the atom can be neglected. This clearly means that the wavelength of the radiation must be significantly larger than the dimensions of the atom .

The undisturbed Hamilton operator (without spin-orbit coupling) has the form ; the commutators : and apply . The momentum operator can thus be expressed by a commutator: ${\displaystyle H={\frac {{\vec {p}}^{2}}{2m}}+V({\vec {r}})}$ ${\displaystyle [V,{\vec {r}}]=0}$ ${\displaystyle [H,{\vec {r}}]={\frac {\hbar }{i}}{\frac {\vec {p}}{m}}}$

${\displaystyle {\begin{aligned}\langle \Psi _{f}|{\hat {\epsilon }}\cdot {\vec {p}}|\Psi _{i}\rangle &={\hat {\epsilon }}\cdot \langle \Psi _{f}|{\vec {p}}|\Psi _{i}\rangle ={\frac {im}{\hbar }}{\hat {\epsilon }}\cdot \langle \Psi _{f}|[H,{\vec {r}}]|\Psi _{i}\rangle \\&={\frac {im(E_{f}-E_{i})}{\hbar }}{\hat {\epsilon }}\cdot \langle \Psi _{f}|{\vec {r}}|\Psi _{i}\rangle ={\frac {im(E_{f}-E_{i})}{\hbar }}\langle \Psi _{f}|{\hat {\epsilon }}\cdot {\vec {r}}|\Psi _{i}\rangle \end{aligned}}}$

The vector in the matrix element explains the designation of the electrical dipole transition. The electric dipole moment contains precisely the first power of the position vector. ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {p}}=\int _{V}{\vec {r}}\rho ({\vec {r}})\,d^{3}r}$

Now the matrix elements have to be analyzed. Their size is a measure of the probability of the transition . If the matrix element disappears (at least in the dipole approximation) the transition by means of the one-photon process is not possible. ${\displaystyle \langle \Psi _{f}|{\hat {\epsilon }}\cdot {\vec {r}}|\Psi _{i}\rangle }$ ${\displaystyle |\Psi _{i}\rangle \to |\Psi _{f}\rangle }$

If one takes into account the next term of the development, one obtains electrical quadrupole and magnetic dipole transitions.

literature

• Hook, Wolf: Atomic and Quantum Physics , Springer.
• Hook, Wolf: Molecular Physics and Quantum Chemistry , Springer.
• Cohen-Tannoudji: Quantum Mechanics 2 , de Gruyter.
• Schwabl: Quantum Mechanics , Springer.
• Spectral Lines: Selection Rules, Intensities, Transition Probabilities, Values, and Line Strengths in WC Martin and WL Wiese: Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data and Formulas. National Institute of Standards and Technology . Last accessed on December 10, 2010.

Individual evidence

1. ^ Peter Zimmermann: Introduction to Atomic and Molecular Physics Akademische Verlagsgesellschaft, Wiesbaden 1978, ISBN 3-400-00400-6 , pp. 55–56
2. ↑ Chapter 4 of the script for the lecture Introduction to Nuclear and Elementary Particle Physics in the winter semester 2007/08 by Prof. Dr. Hermann Kolanosk, p.78 German Electron Synchrotron, Research Center of the Helmholtz Association. Retrieved December 3, 2018.
3. ↑ JA Salthouse, Ware, MJ: Point group character tables and related data . Cambridge University Press, 1972, ISBN 0521081394 .
4. ^ Pierre Meystre, Murray Sargent: Element of Quantum Optics . 4th edition. Springer, Berlin / Heidelberg / New York 2007, ISBN 978-3-540-74209-8 , pp. 74 .
5. ↑ Christopher C. Gerry, Peter L. Knight: Introductory Quantum Optics . 3. Edition. Cambridge University Press, Cambridge 2008, ISBN 978-0-521-52735-4 , pp. 76 .