Optical theorem

The optical theorem , also known as the Bohr-Peierls-Placzek theorem or relationship in the context of quantum mechanics (after Niels Bohr , Rudolf Peierls and George Placzek ), relates the imaginary part of the scattering amplitude to the total cross-section in scattering theory . The optical theorem is a result of wave optics or classical electrodynamics , where it is based on the conservation of the energy of scattered electromagnetic waves . Later, in quantum mechanical wave mechanics based on the conservation of probability, an analogous result for the scattering of matter waves was found, and in quantum field theory a generalization of the optical theorem for quantum fields was found. ${\displaystyle \sigma }$

In its original formulation, the optical theorem is:

${\displaystyle \sigma ={\frac {4\pi }{k}}\operatorname {Im} f_{k}(\theta =0)}$

With

• ${\displaystyle k}$: Circular wavenumber
• ${\displaystyle f_{k}(\theta =0)}$: Scatter amplitude at scatter angle . ${\displaystyle \theta =0}$

Classical electrodynamics

Light , or a general electromagnetic wave , with an electric field strength and magnetic flux density can be scattered , absorbed or transmitted by an object of finite size . The entire fields are made up of the incident fields and the scattered or transmitted fields . The power density of the field is described by the Poynting vector with the vacuum permeability . The absorbed power of the electromagnetic wave results from the area integral of the Poynting vector of the total fields over the (inwardly directed) surface of the scatterer; the scattered power as an integral of the scattered fields over the (outward-facing) surface: ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle {\vec {E}}_{i},{\vec {B}}_{i}}$ ${\displaystyle {\vec {E}}_{s},{\vec {B}}_{s}}$ ${\displaystyle {\vec {P}}={\tfrac {1}{2\mu _{0}}}\operatorname {Re} ({\vec {E}}\times {\vec {B}}^{*})}$ ${\displaystyle \mu _{0}}$

${\displaystyle P=P_{\text{abs}}+P_{\text{streu}}=-{\frac {1}{2\mu _{0}}}\operatorname {Re} \left(\int \mathrm {d} {\vec {A}}\cdot ({\vec {E}}\times {\vec {B}}^{*})\right)+{\frac {1}{2\mu _{0}}}\operatorname {Re} \left(\int \mathrm {d} {\vec {A}}\cdot ({\vec {E}}_{s}\times {\vec {B}}_{s}^{*})\right)=-{\frac {1}{2\mu _{0}}}\operatorname {Re} \left(\int \mathrm {d} {\vec {A}}\cdot ({\vec {E}}_{s}\times {\vec {B}}_{i}^{*}+{\vec {E}}_{i}^{*}\times {\vec {B}}_{s})\right)}$

With the decomposition of the electric field into plane waves

${\displaystyle {\vec {E}}_{i}=E_{0}{\vec {\varepsilon }}_{i}e^{\mathrm {i} {\vec {k}}_{i}\cdot {\vec {x}}}}$,

where the polarization vector in the direction of oscillation, the wave vector in the direction of propagation and the amplitude of the field and the relationship ${\displaystyle {\vec {\varepsilon }}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle E_{0}}$

${\displaystyle {\vec {B}}_{i}={\frac {1}{ck_{i}}}{\vec {k}}_{i}\times {\vec {E}}_{i}}$,

since the electric field, magnetic flux density and wave vector in a vacuum are perpendicular to each other in pairs, this leads to:

${\displaystyle P={\frac {1}{2\mu _{0}}}\operatorname {Re} \left(E_{0}^{*}\int \mathrm {d} A\,e^{-\mathrm {i} {\vec {k}}_{i}\cdot {\vec {x}}}\left[{\vec {\varepsilon }}_{i}^{*}\cdot ({\vec {n}}\times {\vec {B}}_{s})+{\vec {\varepsilon }}_{i}^{*}\cdot {\frac {{\vec {k}}_{i}\times ({\vec {n}}\times {\vec {E}}_{s})}{ck}}\right]\right)}$

( is the surface normal vector, ). ${\displaystyle {\vec {n}}}$ ${\displaystyle \mathrm {d} {\vec {A}}={\vec {n}}\,\mathrm {d} A}$

On the other hand, the scattering amplitude for an electromagnetic field with a polarization vector is : ${\displaystyle f}$ ${\displaystyle {\vec {\varepsilon }}}$

${\displaystyle f({\vec {\epsilon }},{\vec {k}},{\vec {k}}_{i})=\mathrm {i} {\frac {ck}{4\pi }}E^{-1}\int \mathrm {d} A\ e^{-\mathrm {i} {\vec {k}}\cdot {\vec {x}}}\left[{\vec {\varepsilon }}^{*}\cdot ({\vec {n}}\cdot {\vec {B}}_{s})+{\vec {\varepsilon }}^{*}\cdot {\frac {{\vec {k}}\times ({\vec {n}}\times {\vec {E}}_{s})}{ck}}\right]}$

By comparing these two expressions, one can see that

${\displaystyle P={\frac {2\pi }{\mu _{0}ck_{i}}}\operatorname {Im} E_{0}E_{0}^{*}F({\vec {\varepsilon }}_{i},{\vec {k}}_{i},{\vec {k}}_{i})}$

have to be. With the definition of the scattering cross section as power normalized to the incident power

${\displaystyle \sigma =P\left({\frac {EE^{*}}{2\mu _{0}c}}\right)^{-1}}$

follows the optical theorem.

Quantum field theory

In quantum field theory, the optical theorem is an exact result that is not based on perturbative approximations. In perturbation theory, the optical theorem leads to a relationship between loop diagrams and scattering cross-sections in a leading order.

Let be the matrix element of a process , then ${\displaystyle {\mathcal {M}}(i\to f)}$ ${\displaystyle i\to f}$

${\displaystyle {\mathcal {M}}(i\to f)-M^{*}(f\to i)=\mathrm {i} \sum _{X}\prod _{j\in X}\int {\frac {\mathrm {d} ^{3}{\vec {p}}_{j}}{2p_{j}^{0}}}\delta ^{(4)}(p_{i}-p_{X}){\mathcal {M}}(i\to X){\mathcal {M}}^{*}(f\to X)}$

with the sum of all possible physical (multi-particle) states and the Lorentz invariant phase space integral over all single-particle momenta in the respective multi-particle state. ${\displaystyle |X\rangle }$ ${\displaystyle {\vec {p}}_{j}}$

In particular applies to two-particle states ${\displaystyle |A\rangle }$

${\displaystyle \operatorname {Im} {\mathcal {M}}(A\to A)=2E_{\mathrm {CM} }|{\vec {p}}_{A}|\sum _{X}\sigma (A\to X)}$

in the center of gravity system with the center of gravity energy , which the optical theorem of non-relativistic quantum mechanics returns. ${\displaystyle E_{\mathrm {CM} }}$

For single-particle states , i.e. for decays, applies ${\displaystyle |B\rangle }$

${\displaystyle \operatorname {Im} {\mathcal {M}}(B\to B)=m_{B}\sum _{X}\Gamma (A\to X)=m_{B}\Gamma _{\mathrm {tot} }}$

with the mass of the decaying particle and the decay width . ${\displaystyle m_{B}}$ ${\displaystyle \Gamma }$

Derivation

The optical theorem is based on the unitarity of the S matrix of quantum field theories. Let be the nontrivial part of the S-matrix, so then from the unitarity of the S-matrix follows: ${\displaystyle {\mathcal {T}}}$ ${\displaystyle S=1+\mathrm {i} {\mathcal {T}}}$

${\displaystyle 1=S^{\dagger }S=(1-\mathrm {i} {\mathcal {T}}^{\dagger })(1+\mathrm {i} {\mathcal {T}})=1-\mathrm {i} ({\mathcal {T}}^{\dagger }-{\mathcal {T}})+{\mathcal {T}}^{\dagger }{\mathcal {T}}\quad \Leftrightarrow \quad \mathrm {i} (T^{\dagger }-{\mathcal {T}})=T^{\dagger }T}$

Multiplying and gives the left side of the equation with the definition of the matrix element as : ${\displaystyle \langle f|}$ ${\displaystyle |i\rangle }$ ${\displaystyle \langle f|{\mathcal {T}}|i\rangle =(2\pi )^{4}\delta ^{(4)}(p_{i}-p_{f}){\mathcal {M}}(i\to f)}$

${\displaystyle \langle f|\mathrm {i} ({\mathcal {T}}^{\dagger }-{\mathcal {T}})|i\rangle =\mathrm {i} (2\pi )^{4}\delta ^{(4)}(p_{i}-p_{f}){\big (}{\mathcal {M}}^{*}(f\to i)-{\mathcal {M}}(i\to f){\big )}}$

Inserting a one in the form of

${\displaystyle 1=\sum _{X}\prod _{j\in X}\int {\frac {\mathrm {d} ^{3}{\vec {p}}_{j}}{(2\pi )^{4}}}{\frac {1}{2p_{j}^{0}}}|X\rangle \langle X|}$

on the right leads to:

${\displaystyle \langle f|{\mathcal {T}}^{\dagger }{\mathcal {T}}|i\rangle =\sum _{X}\prod _{j\in X}\int {\frac {\mathrm {d} ^{3}{\vec {p}}_{j}}{2p_{j}^{0}}}(2\pi )^{4}\delta ^{(4)}(p_{i}-p_{X})\delta ^{(4)}(p_{f}-p_{X}){\mathcal {M}}(i\to X){\mathcal {M}}^{*}(f\to X)}$

The optical theorem follows by equating.

Individual evidence

1. ↑ cf. Footnote 1 in Niels Bohr, Rudolf Peierls and Georg Placzek: Nuclear Reactions in the Continuous Energy Region . In: Nature . tape 144 , 1939, pp. 200–201 , doi : 10.1038 / 144200a0 (English). The announced article in the Proceedings of the Copenhagen Academy was never published due to the outbreak of World War II.
2. ^ John David Jackson : Classical Electrodynamics . 3. Edition. John Wiley & Sons, Hoboken 1999, ISBN 978-0-471-30932-1 , pp. 500-502 (English).
3. ^ Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 454 (English).

literature

• Wolfgang Nolting: Basic Course Theoretical Physics 5/2: Quantum Mechanics - Methods and Applications , Springer, Berlin, 2006, ISBN 9783540260356 , p. 333

Web links

• Herbert Müther: Lecture notes Theoretical Physics III / IV, Quantum Mechanics I and II , 1997–1999.