Laws time-dependent Schrodinger equation

Quantum mechanics provides the law of motion for microscopic particles. Experimentally, macroscopic objects obey classical mechanics. Hence for quantum mechanics to be a valid theory, it should reduce to classical mechanics as we make the transition from microscopic to macroscopic particles. Quantum effects are associated with the de Broglie wavelength A = h/mv. Since h is very small, the de Broglie wavelength of macroscopic objects is essentially zero. Thus, in the limit A 0, we expect the time-dependent Schrodinger equation to reduce to Newton s second law. We can prove this to be so (see Prob. 7.59). 

When a molecule is excited by an ultrashort laser pulse with an appropriate center frequency, a localized wave packet can be created in the excited electronic state because of the excitation of a coherent superposition of many vibrational-rotational states. It follows from fundamental laws that the d3mamics of molecular wave packets is governed by a time-dependent Schrodinger equation (eqn 2.29), where H is the relevant Hamiltonian of the given molecule. Because molecular potential-energy surfaces are anharmonic, this molecular wave packet tends to spread both in position (coordinates) and in momentum. However, in addition to expansion or defocusing, the wave packet also suffers delocalization at a certain instant of time. Coherent quantum... 

Electrons are very light particles and cannot be described by classical mechanics. They display both wave and particle characteristics, and must be described in terms of a wave function, The quantum mechanical equation corresponding to Newtons second law is the time-dependent Schrodinger equation (h is Plancks constant divided by 27t). 

In the time concept of the pre-relativistic mechanics, the observable quantities, time t and energy E, have to be considered as another canonically conjugate pair, as in classical mechanics. The dynamic law (time-dependent energy term) of the Schrodinger equation will then completely disappear [19]. A good occasion for Weyl to introduce the relativistic view would have been his contributions to Dirac s electron theory. His other colleagues developed the method of the so-called second quantization that seemed easier for the entire community of physicists and chemists to accept. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

In general, molecular motion should be described using the laws of quantum mechanics. In quantum mechanics dynamical trajectories themselves are probabilistically defined entities. The state of the system is described by a probability amplitude function, I, which depends on coordinates and, possibly, spin states of all nuclei and electrons present in the system. I is the probability density for observing the system in a particular point in phase space. Motion of the system, or in other words its change in state with time, is described by the time-dependence of the I -function. It is determined by solving the Schrodinger equation ... 

Depending on the time and length scales, different transport laws can be used. When the objects have comparable size to the wavelength of energy carrier, wave phenomena, that is, reflection, refraction, diffraction, etc., dominate the energy transport mechanism. When the time scale of interest (t) is of the order of collision time scale (t ), time-dependent wave mechanics must be used. Schrodinger s equation must be used for electrons and phonons. Maxwell s equation must be used for photons ... 

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