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5.0. Wave Mechanics One cornerstone of quantum theory is the particle-wave duality [the other is the principle of uncertainty]. For example, in optical phenomena such as diffraction and interference, light behaves like waves. In collision processes such as photo-electric and Compton effects, light behaves like particles. Another example is the electron, which behaves like a particle when moving freely, but acts like standing waves when bounded in an atom. These 2 aspects are connected by the relations (5.1) E h p h k (5.2) where E and p are the energy and momentum of the particle, respectively; and are the frequency and wavelength of the wave, respectively; and k are the angular frequency and wavenumber of the wave, respectively; h h and 2 6.6256 1034 J s is called the Planck’s constant. Historically, eq(5.1) was 1st proposed by Planck as an empirical fix for the breakdown of the theory of classical statistical mechanics when applied to the problem of black body radiation. It was later applied by Einstein to the photo-electric effect, thus revealing the particle-like aspect of “waves”. Later on, de Broglie proposed the general validity of eqs(5.1-2). The implied wave-like aspects of “particles” were 1st demonstrated by Thomson and Davisson using the diffraction of electrons by a crystal lattice. The particle-wave duality is best described by the wave mechanical formulism of quantum theory invented by Schrodinger. Thus, the state of a “particle” is represented by a (complex) wave function x,t so that the probability of finding the particle in an infinitesimal volume d 3 x about x at time t is proportional to P x, t d 3 x x, t 2 d 3x (5.4) where P is called the probability density if it satisfies the normalization condition P x, t d 3 x x, t 2 d 3x 1 (5.4a) For wave functions that cannot be normalized, (5.4) indicates the relative probabilities. One example of this is a free particle described by the plane wave x, t exp i k x t (5.3) A quantity that is represented as a function A A x, p in classical mechanics is represented in wave mechanics as a differential operator Aˆ A xˆ, pˆ . In the so-called r-representation, we have xˆ x and pˆ i (5.6) The dynamics of the particle is described by the time-dependent Schrodinger equation i x, t Hˆ x, t t (5.7a) where Ĥ is the Hamiltonian operator. For a particle moving in a potential V V x , we have 1 2 p V x 2m so that (5.7a) becomes H Hˆ 2 2 i x, t V x x, t t 2m 2 2m 2 V x (5.7) Note that although some generalizations of (5.7), e.g., to the case of a system of many particles, are straightforward, some generalizations are not readily tractable, e.g., writing (5.7) in terms of generalized coordinates. Finally, eq(5.7a) can be interpreted as an additional rule supplementing those in (5.6): Ĥ i t (5.7) This will be useful when relativistic generalization of the theory is considered.