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Formalism
Published in Shabnam Siddiqui, Quantum Mechanics, 2018
The previous inner product is a function of a continuous variable and is also called as wave function. To further understand the meaning of the wave function, consider H as the energy operator of a system, also called as the Hamiltonian. Such that, H^|En⟩=En|En⟩ where |En⟩ are energy eigenvectors of the system that form a discrete complete orthonormal set. Such that, 1=∑n|En⟩⟨En|
Basis functions, operators, and quantum dynamics
Published in David K Ferry, Quantum Mechanics, 2001
We note that in the previous paragraph the Heisenberg uncertainty relation holds only for non-commuting operators, such as position and momentum. What about the often cited uncertainty in energy and time? In the non-relativistic quantum mechanics with which we are dealing, time is not an operator, and therefore it commutes with the total-energy operator—the Hamiltonian itself. Thus, there is no Heisenberg uncertainty principle for energy and time! We will, however, see in a later chapter that an eigenstate that possesses a lifetime can give rise to an uncertainty relation that greatly resembles the Heisenberg uncertainty relation—but there is a basic difference. The Heisenberg uncertainty relation describes basic fundamental uncertainties, while this latter relationship is related to a measurement of the energy alone, and is not therefore a fundamental uncertainty relation (Landau and Lifshitz 1958).
Introduction to the Kinetic Theory of Gases
Published in Caroline Desgranges, Jerome Delhommelle, A Mole of Chemistry, 2020
Caroline Desgranges, Jerome Delhommelle
Around 1830, Hamilton (1805–1865) proposes a new theory that unifies mechanics and optics. Like Lagrange, his theory is based on a variational principle (integral minimization in mathematics which corresponds to finding the shortest path in mechanics). With Jacobi (1804–1851), they develop a new formulation now known as Hamiltonian mechanics. This new mechanics revolves around the concept of energy. Later on, the increasing importance of mathematics will lead to the concept of operators to study position, momentum and energy. One of the most significant operators is undoubtedly the energy operator in quantum mechanics, that we all know through the famous equation Ĥψ = Eψ, where Ĥ (energy operator) is called the Hamiltonian!
The infrared spectrum of PF3 and analysis of rotational energy clustering effect
Published in Molecular Physics, 2020
Barry P. Mant, Katy L. Chubb, Andrey Yachmenev, Jonathan Tennyson, Sergei N. Yurchenko
The ro-vibrational Hamiltonian was constructed numerically as implemented in TROVE [72]. The Hamiltonian was expanded using a power series around the equilibrium geometry of the molecule. The coordinates used were linearised versions of the stretching coordinates; – and bending coordinates; –. The kinetic energy operator was expanded to 6th order and the potential energy operator to 8th order. Morse coordinates of form (i=1–3 and a=1.0 Å−1 is the Morse parameter) were used in the potential expansion for the stretching coordinates with the bending coordinates expanded as (i=4–6) themselves. Atomic masses were used throughout.
Propagation of nonstationary electronic and nuclear states: attosecond dynamics in LiF
Published in Molecular Physics, 2018
Ksenia G. Komarova, F. Remacle, R. D. Levine
Next we need to propagate the nuclear wave functions on the grid. Can we confine the propagation to near neighbours? The action of the nuclear kinetic energy operator is usually evaluated by transforming the wave function of the grid to the momentum space [12]. There the action of the kinetic energy operator is a local multiplication by . But the transformation to momentum space requires knowing the nuclear wave function throughout the grid. A possible alternative is to evaluate the derivatives in the momentum and the kinetic energy operators using a finite difference expression [9]. In the supplementary information, we briefly outline these expressions using only near neighbours (which is correct to order of a2) and also the results using next near neighbours. The nuclear kinetic energy operator has the matrix elements in the electronic basis when we use only near neighbours.
Dissociation of HF molecule in position and momentum representation by an optimally controlled polychromatic field: study in the dual space using simulated annealing
Published in Molecular Physics, 2022
Dipayan Seal, Pulak Naskar, Pinaki Chaudhury, Subhasree Ghosh
The time-independent Hamiltonian has two parts, kinetic energy part and potential energy part. The kinetic energy part is diagonal in momentum basis and the potential energy part is diagonal in the position grid basis as is the kinetic energy operator and is the value of the kinetic energy at . Similarly, is the potential energy operator.