The Schrödinger equation makes it possible to explain and predict a large number of phenomena of atomic physics and to calculate the experimentally observable fundamental characteristics of atomic systems, such as the energy levels of atoms and the change in atomic spectra under the influence of electric and magnetic fields. It is fully equivalent to the matrix mechanics of W. The mathematical formulation of the postulates of quantum mechanics that is based on the Schrödinger equation is called wave mechanics. This probabilistic interpretation of the wave function is one of the principal postulates of quantum mechanics. What has meaning is the square of the wave function, namely, the quantity p n( x, y, z, t) = |ψ n( x, y z, t) | 2, which is equal to the probability of finding a particle (or system) at a time t in the quantum state n at a point of space having the coordinates x, y, z. In contrast, however, to the solutions of the equation of the vibration of a string, which give the geometrical shape of the string at a given time, the solutions ψ( x, y, z, t) of the Schrödinger equation do not have direct physical meaning. The analogy between classical mechanics and geometrical optics, which is the limiting case of wave optics, played an important role in establishing the Schrödinger equation.įrom the mathematical standpoint, the Schrödinger equation is a wave equation and is similar in structure to the equation that describes the vibrations of a weighted string. The transition from the Schrödinger equation to classical trajectories is similar to the transition from wave optics to geometrical optics. The Schrödinger equation satisfies the correspondence principle and, in the limiting case where the de Broglie wavelengths are much smaller than the dimensions characteristic of the motion under examination, describes particle motion according to the laws of classical mechanics. The Schrödinger equation is a mathematical expression of a fundamental property of microparticles-the wave-particle duality-according to which all particles of matter that exist in nature are also endowed with wave properties (this hypothesis was first advanced by L. (where e is the elementary electric charge), the Schrödinger equation describes the hydrogen atom, and the E n are the energies of the stationary states of the atom. In the important particular case of the Coulomb potential A wave function ψ( x, y, z) corresponds to every value of E n, and a knowledge of the complete set of these functions makes it possible to calculate all measurable characteristics of the quantum system. the members of this series, which is virtually infinite, are numbered by the set of integral quantum numbers n. Where E is the total energy of the quantum system and ψ ( x, y, z) satisfies the stationary Schrödinger equation:įor quantum systems whose motion occurs in a bounded region of space, solutions of the Schrödinger equation exist only for certain discrete energy values: E 1, E 2.
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If the potential V is independent of time, the solutions of the Schrödinger equation may be represented in the form This equation is called the Schrödinger time-dependent equation.
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