Ch01 Newtonian mechanics

Ch01 Newtonian mechanics - Johannes Kepler(15711630 Founder...

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© E. F. Schubert 1 Johannes Kepler (1571–1630) Sir Isaac Newton (1642–1727) Founder of celestial mechanics Founder of modern mechanics 1 Classical mechanics and the advent of quantum mechanics 1.1 Newtonian mechanics The principles of classical mechanics do not provide the correct description of physical processes if very small length or energy scales are involved. Classical or newtonian mechanics allows a continuous spectrum of energies and allows continuous spatial distribution of matter. For example, coffee is distributed homogeneously within a cup. In contrast, quantum mechanical distributions are not continuous but discrete with respect to energy, angular momentum, and position. For example, the bound electrons of an atom have discrete energies and the spatial distribution of the electrons has distinct maxima and minima, that is, they are not homogeneously distributed. Quantum-mechanics does not contradict newtonian mechanics. As will be seen, quantum- mechanics merges with classical mechanics as the energies involved in a physical process increase. In the classical limit, the results obtained with quantum mechanics are identical to the results obtained with classical mechanics. This fact is known as the correspondence principle . In classical or newtonian mechanics the instantaneous state of a particle with mass m is fully described by the particle’s position [ x ( t ), y ( t ), z ( t )] and its momentum [ p x ( t ), p y ( t ), p z ( t )]. For the sake of simplicity, we consider a particle whose motion is restricted to the x -axis of a cartesian coordinate system. The position and momentum of the particle are then described by x ( t ) and p ( t ) = p x ( t ) . The momentum p ( t ) is related to the particle’s velocity v ( t ) by p ( t ) = m v ( t ) = m [d x ( t ) / d t ] . It is desirable to know not only the instantaneous state-variables x ( t ) and p ( t ), but also their functional evolution with time. Newton’s first and second law enable us to determine this functional dependence. Newton’s first law states that the momentum is a constant, if there are no forces acting on the particle, i. e.
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