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Examples and applications
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Determine the commutator [a+, a−]. These operators are called ladder operators. The operator a+ raises the energy level and a− lowers it. Determine the relationship of these operators to the stationary states of the harmonic oscillator, expressed in terms of the Hermite polynomials from Exercise 4.15 of Chapter 5.
The electronic system
Published in Molecular Physics, 2022
Julia Gerschmann, Erik Schwanke, Silke Ospelkaus, Eberhard Tiemann
The parameters for the non-diagonal matrix elements between Σ and Π states contain the spin–orbit (A) and rotational (B) interaction, where p stands for the expectation value of the ladder operator of the orbital angular momentum . Assuming a weak variation of the spin–orbit interaction with internuclear separation we will model the v-dependence of this interaction by the overlap integral V of the vibrational states v and v and simplify this also for the rotational part B. By introducing the fit parameters we reduce the number of free parameters to two, where the latter one is now dimensionless and the effective overlap integral has the dimension of energy. Moreover, is assumed to be the same for both states and and is expected to be close to , since these states are related to an electronic atom state with L = 1.
Unravelling open-system quantum dynamics of non-interacting Fermions
Published in Molecular Physics, 2018
To demonstrate the validity of the method, we report calculations on systems of N non-interacting spin-up Fermions of mass m = 1me (atomic units are used in all reported numerical results) trapped in a 1D potential V(x) (see Equation (8)) and using only one Lindblad operator to be used in Equation (7). Note that is the lowering ladder operator for a harmonic oscillator of frequency ωℓ (although it is still a Fermionic operator). In the results shown below, we use ωℓ = 1Eh/ℏ and γ = 0.2Eh/ℏ and N = 8 Fermions. The calculation was carried out using a high-order numerical implementation of the algorithm depicted in the previous section, where the single particle wave functions and operators were represented on a Fourier grid and the non-unitary time propagation was performed using a high-degree interpolating polynomial in the Newton form [37,38].
A biaxial nematic liquid crystal composed of matchbox-symmetric molecules
Published in Molecular Physics, 2020
Robert A. Skutnik, Immanuel S. Geier, Martin Schoen
Based upon symmetry considerations [21,34] and utilising a ladder-operator formalism [29] borrowed from angular-momentum theory in quantum mechanics [35], we can derive an expression for the anisotropic interaction potential in terms of a full contraction of Cartesian second-rank tensors. Thus, our potential is equivalent to the one used in the mean-field calculations of Straley [24] and of Sonnet et al. [5] (see also Ref. [36]). We focus on uniaxial and biaxial order parameters [22,23], the orientation distribution function (odf), and the orientation correlation function.