Part I The Wave Function and Schrodinger Equation
　　第一部分波函數(shù)及薛定諤方程
　　Chapter 1 The wave function and Schrodinger equation
　　波函數(shù)和薛定諤方程
　　The wave-particle duality and matter wave
　　波粒二象性與物質(zhì)波
　　Particles and waves are two distinct entities in classical physics. As we have learned from both text books and daily experiences,a particle is a localized bundle of energy and momentum. At any tnstant, it can be described by state parameters,such as position q and momentum p (or velocity v if the mass is a constant). The parameters q and p evolve \n time according to some equations of motion,such as Newton’s law F = dp /dt. Given the tnitial values q (tz) and p (t ) at time tt, the position q (t) and momentum pit) at any time t can be deduced from the equations of motion. In contrast to the particle, a wave is considered as a periodic disturbance spread over space. It usually appears as some kind of periodic movement transferring energy from one point to another. A wave satisfies the superposition rules and presents jnterference and diffraction phenomena.
　　1.1.1 The wave-particle duality of light 光的波粒二象性
　　In the 17lh century, two competing theories of light were proposed during the debate about the nature of light: one was offered by Christian Huygens and the other by Isaac Newton. According to the theories,light was thought either to consist of waves (Huygens) or of corpuscles/particles (Newton). Huygens proposed that each point on a light wave front acted as a spherical point source
　　for the progressing wave. Newton’s argument satisfactorily and more simply
　　explained geometric optics and did not presume a medium for particle travel.At the beginning of 1 9lh century (801?1805),Thomas Young conducted the famous double-slit experiment showing that light from two slits interfere to produce a fringe pattern on a screen,a phenomenon that cannot be described by classical particles. In 1909?Geoffrey Ingram Taylor conducted an experiment that showed the interference phenomena by individual photons. The double-slit experiment has been repeated in many ways over the years and has become a standard demonstration of wave-like motion. But neither Newton nor Young were quite convincing about the nature of light. Light could not be described purely as a wave or as consisting of particles.
　　In 1900, Max Planck postulated that the energy of oscillators in a black body is an integer multiple of hv,where h is the Planck constant and v is the frequency of the oscillator. The problem was that the wave nature of light was widely accepted at that time,the concept have been completely formulated in Maxwell’s theory and confirmed by interference and diffraction experiments. After Planck’s quanta hypothesis, Albert Einstein postulated the concept of the photon to explain the photoelectric effect in 1 905, which also was used later to explain Compton scattering. According to Einstein’s assumption, electromagnetic radiation of frequency v consists of discrete units (quanta) of energy hv. That is, electromagnetic energy itself is quantized,and a single quantum is called photon.
　　Thus, mater can absorb energy from a monochromatic beam of light of
　　frequency v only in units of hv because the light arrives in the form of discrete quanta, each with energy hv. Einstein had re-introduced the problem of wave- particle duality for light! The relation between the light wave (v,A ) and the light photon (e,_p)is presented by the equations
　　where h is the Planck constant. Eqs (1.1.1) and (1.1.2) are called the Planck- Einstein relations,which reflect the wave-particle duality of light.
　　1.1.2 Matter wave and the wave -particle duality of matter 物質(zhì)波及物質(zhì)的波粒二象性
　　If the light could sometimes behave like particles, then should matter
　　particles also show wave-like behavior? Upon examining Einstein’s idea of the parallelism between the light and matter,Louis de Broglie in 1 924 proposed that
　　electrons,which generally are believed to be particles, should exhibit wave-like behavior. The wavelength and frequency of a quantum mechanical particle are associated with its momentum and energy by de Broglie’s hypothesis
　　Where e and p are the energy and momentum of the particle,and v represents the frequency. The wavelength of the matter wave associated with the particle, A, is also called de Broglie wavelength. In the case of non-relativistic theory,the de Broglie wavelength for a free particle with mass m and energy e is
　　The wave nature of an electron and de Broglie’s hypothesis have been experimentally confirmed by electron diffraction experiments by G. P. Thompson, and C. Davisson & L. Germer. In short, any matter must be considered as having both the particle-like and wave -like properties.
　　You may want to argue that common sense tells us that billiard balls and ping-pong balls travel along definite trajectories and do not show any wave-like properties! The point is that the wave nature of matter is not apparent for macroscopic phenomena since Planck’s constant h is so small. For example, s

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