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Quantum Chemistry Methods for Molecular Disordered Materials
Published in Alexander Bagaturyants, Vener Mikhail, Multiscale Modeling in Nanophotonics, 2017
Alexander Bagaturyants, Vener Mikhail
Because ap†aq†|0⟩=ϕpϕq=-ϕqϕp=-aq†ap†|0⟩, $ a_{p}^{\dag } a_{q}^{\dag } |0\rangle = \left| {\phi_{p} \phi_{q} } \right\rangle = - \left| {\phi_{q} \phi_{p} } \right\rangle = - a_{q}^{\dag } a_{p}^{\dag } |0\rangle , $ creation and annihilation operators obey anticommutation relations ap†aq†+aq†ap†=0,apaq+aqap=0ap†aq+aqap†=δpq. $ a_{p}^{\dag } a_{q}^{\dag } + a_{q}^{\dag } a_{p}^{\dag } = 0,a_{p} a_{q} + a_{q} a_{p} = 0a_{p}^{\dag } a_{q} + a_{q} a_{p}^{\dag } = \delta_{pq} . $
Electromagnetism and Quantum Field Theory
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
Describe a quantum electromagnetic field in terms of creation and annihilation operators in wave-vector space or equivalently momentum space. Do the following steps: Write down the Hamiltonian H of a single quantum harmonic oscillator: H=p2/2m+mω2x2/2,p=−ihd/dxwhere h is Planck’s constant divided by 2π. Solve the eigenvalue equation Hψ(x)=eψ(x)∫ℝ|ψ(x)|2dx<∞as follows. Assume ψ(x)=u(x).exp(−ax2/2)where a is determined so that the coefficient of x2 cancels out in the differential equation for u(x). Then, write down an infinite series expansion for u(x) and show that if the infinite series does not terminate with some term being zero, then ψ(x) = exp(−ax2/2).u(x) explodes as |x| → ∞ and hence cannot be square integrable. Show that the infinite series terminates to a polynomial iff E = (n + 1/2)hω for some non-negative integer n.
Variational approaches to quantum impurities: from the Fröhlich polaron to the angulon
Published in Molecular Physics, 2019
Xiang Li, Giacomo Bighin, Enderalp Yakaboylu, Mikhail Lemeshko
The Fröhlich Hamiltonian, describing an impurity immersed in a bosonic bath, is given by: Here the first term represents the kinetic energy of an impurity with mass m. The second term, with , corresponds to the kinetic energy of the bosons, as parametrised by the dispersion relation . The bosonic creation and annihilation operators, and , obey the commutation relation . Finally, the last term is the impurity-bath interaction, where determines the coupling strength, and is the position operator of the impurity with respect to the laboratory frame.
Corrections to the Casimir–Polder potential arising from electric octupole coupling
Published in Molecular Physics, 2019
The starting point is to recognize that in the molecular QED formalism [8-12], the interaction of an m-th order electric multipole moment of particle , positioned at , with the transverse electric displacement field, , is where the m-th order electric multipole moment is defined as where is the generalised coordinate vector of the electron, and the Latin subscripts denote Cartesian tensor components in the space-fixed frame of reference. A sum is implied for indices that repeat. The second quantized transverse electric displacement field operator is given by the mode expansion where for electromagnetic radiation of mode , is the wave vector or direction of propagation, and is the polarisation index, is the complex unit electric polarisation vector, and V is the quantisation volume. The creation and annihilation operators and , respectively increase and decrease by one the number of photons of a particular mode in the radiation field. The overbar designates the complex conjugate quantity.
A quantum description of linear, and non-linear optical interactions in arrays of plasmonic nanoparticles
Published in Journal of Modern Optics, 2018
Ehsan Arabahmadi, Zabihollah Ahmadi, Bizhan Rashidian
This is exactly the commutation relation for standard bosonic creation and annihilation operators. So, we call , the annihilation operator of a plasmon with wave vector k and polarization r, and as its corresponding creation operator. Using (19), the Hamiltonian takes the following familiar form: