In classical mechanics we have separate equations for wave motion and particle motion, whereas in quantum mechanics, in which the distinction between particles and waves is not clear-cut, we have a single equation— the Schrodinger equation. We have seen that the link between the Schrddinger equation and the classical wave equation is the de Broglie relation. Let us now compare Schrodinger s equation with the classical equation for particle motion. [Pg.19]

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). [Pg.11]

Recently [8-11] an alternative treatment to mix quantum mechanics with classical mechanics, based on Bohmian quantum trajectories was proposed. Briefly, the quantum subsystem is described by a time-dependent Schrodinger equation that depends parametrically on classical variables. This is similar to other approaches discussed above. The difference comes from the way the classical trajectories are calculated. In our approach, which was called mixed quantum-classical Bohmian (MQCB) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [12] which in turn are used to calculate the force acting on the classical variables. [Pg.332]

We base our quantum version of radiation on one relationship, Planck s law (Eq. 1.2) Ephoton The quantum mechanics of matter also grow out of a single equation, one that ascribes a property to matter completely alien to the classical view a wavelength. At the tiny distances over which atoms and subatomic particles interact, matter exhibits the characteristics of waves, consistent with a de Broglie wavelength. [Pg.45]

It is commonly accepted that the old quantum theory era spans from the birth of Planck s quantum hypothesis to the formulation of Schrodinger s equation. This section describes the old quantum theory in three parts the failure of classical mechanics, the birth of the quantum theory, and the completion of wave mechanics.5 8) This century obviously began with the birth of quantum theory. Many researchers appeared on the scene of quantum theory at the time, but we remember mostly the contributions of four researchers Max Planck (1901), Albert Einstein (1905), Niels Bohr (1913), and de Broglie (1923). Then Schrodinger proposed the new wave equation to conclude the age of the old quantum theory. Heisenberg established matrix mechanics and formulated the uncertainty principle. [Pg.21]

The distance that the particle can travel within the system is the particle s domain. If Ajg for a particle is much smaller than the particle s domain, then classical mechanics should be fine. But should the particle be trapped within a region no bigger than a few A g s, the classical equations will become inaccurate. This one calculation of Ajg tells us whether our system is in the quantum or classical regime use quantum mechanics when the de Broglie wavelength is comparable to the particle s domain. [Pg.47]

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