Freely Moving Particle

A Hamiltonian for a freely moving particle and hence the corresponding Schrodinger equation ... 

Potential energy is zero for freely moving particle. So the classical Hamiltonian is... 

Although its practical applicability is not so well established as that of Eq. (11-30) for motion from rest, it represents a convenient starting point for a discussion of oscillatory motion. If the fluid oscillates in the vertical direction, and velocities are positive downwards, the equation of motion for a freely moving particle follows from Eq. (11-43) as ... 

This result is called the Stokes law. For a freely moving particle the interfacial drag formulation is normally modified and written as (5.44). 

The carbon-carbon bond distance in cyclobutadiene as 1.4 A, given and shown that it is a paramagnetic molecule by considering its p-electrons as freely moving particles in a two dimensional box ... 

Here we notice that local control theory, with the objective to stop a freely moving particle, directly leads to Stokes law describing friction by a force being proportional to the velocity. In the absence of other forces, this leads to an exponential decrease of the velocity to zero [88]. 

This is the result for a freely moving particle. Initially the mean-square displacement behaves as it would for a free particle because for very short times the solvent molecules are essentially stationary, and the particle experiences a constant force. 

Particle fnrrowing, in which the wear is caused by freely moving particles... 

The resulting Eq. (5.6) is a second-order differential equation for a freely moving particle, which treats time and spatial coordinates on the same footing. Eq. (5.6) is called the Klein-Gordon equation and was derived by W. Gordon in 1926, by O. Klein in 1927 and by others. 

The Klein-Gordon equation for a freely moving particle can also be written in a compact, explicitly covariant form. 

This corresponds then to the following energy values of the freely moving particle... 

Because of the unphysical feature of the Klein-Gordon density and the fact that spin does not emerge naturally (but would have to be included a posteriori as in the nonrelativistic framework) we are not able to deduce a fundamental relativistic quantum mechanical equation of motion for a freely moving electron. However, we may wonder which results of this section may be of importance for the derivation of such an equation of motion for the electron. Certainly, we would like to recover the plane wave solutions of Eq. (5.8) for the freely moving particle, but in order to introduce only a single integration constant (or the choice of a single initial value) for a positive definite density distribution we need to focus on first-order differential equations in time. These must also he first-order differential equations in space for the sake of Lorentz co-variance. 

The covariant form of the Dirac equation of a freely moving particle, Eq. (5.54), allows us to incorporate arbitrary external electromagnetic fields. These fields can then be used to describe the interaction of electrons with light. But it must be noted that the treatment of light is then purely classical. A fully quantized description of light and matter on an equal footing is introduced only in quantum electrodynamics and discussed in chapter 7. 

The wavefunction for a freely moving particle is sinx (exactly as for de Broglie s matter wave, sin(2jtx/A)). 

The form of the Schrodinger equation can be justified to a certain extent by showing that it implies the de Broglie relation for a freely moving particle. Free motion means motion in a region where the potential energy is zero (V=0 everywhere). Then... 

Particle cleavage (erosion) Freely moving particles, which are carried in an intermediate medium between the base and counter body cause the wear (three-body abrasion). 

Note that expression (4.110) does not contain the reduced mass //, of the colliding particle but the mass of a freely moving particle. The surface mass can be considered infinite compared to colliding molecule mass. The average collision velocity is reduced by a factor 4 compared to that for gas phase atoms. 

In recent years, several methods have been developed for simulating the transport of fully resolved freely moving particles in laminar and turbulent two-phase flows. Methods like fictions domain [17], finite difference [38], finite element [26], immersed boundary [36] as well as interface tracking [6] are only a few examples of established simulation approaches. Nevertheless, aU methods mentioned above face the same problem the limitation is that transient computations of a time-varying particle-fluid interface of moving resolved particles require a significant amount of computational time. 

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