Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, 8.1 Spherical Coordinates 8.2 Schrdingers Equation in Spherical Coordinate 8.3 Separation of Variables 8.4 Three Quantum Numbers 8.5 Hydrogen Atom Wave. Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the It ensures that | ψ ( x ) | 2 | ψ ( x ) | 2 is a finite number so we can use it to calculate probabilities. This third condition follows from Born’s interpretation of quantum mechanics. The third condition requires the wave function be normalizable. (In a more advanced course on quantum mechanics, for example, potential spikes of infinite depth and height are used to model solids). The second condition requires the wave function to be smooth at all points, except in special cases. The first condition avoids sudden jumps or gaps in the wave function. ψ ( x ) ψ ( x ) must not diverge (“blow up”) at x = ± ∞.The first derivative of ψ ( x ) ψ ( x ) with respect to space, d ψ ( x ) / d x d ψ ( x ) / d x, must be continuous, unless V ( x ) = ∞ V ( x ) = ∞.ψ ( x ) ψ ( x ) must be a continuous function.The time-independent wave function ψ ( x ) ψ ( x ) solutions must satisfy three conditions: These cases provide important lessons that can be used to solve more complicated systems. In the next sections, we solve Schrӧdinger’s time-independent equation for three cases: a quantum particle in a box, a simple harmonic oscillator, and a quantum barrier. The wave-function solution to this equation must be multiplied by the time-modulation factor to obtain the time-dependent wave function. Notice that we use “big psi” ( Ψ ) ( Ψ ) for the time-dependent wave function and “little psi” ( ψ ) ( ψ ) for the time-independent wave function. This equation is called Schrӧdinger’s time-independent equation. Morrison, Atomic Many-Body Theory, 2 nd edn.Where E is the total energy of the particle (a real number). Lifshitz, Quantum Mechanics - Course of Theoretical Physics, volume 3, 3 rd edn. Cowan, The theory of atomic structure and spectra (University of California press, Berkeley, 1981) Slater, Quantum theory of atomic structure (McGraw-Hill, New York, 1960) Shortley, The theory of atomic spectra (Cambridge University Press, Cambridge, 1935) Commins, Quantum mechanics: an experimentalist’s approach (Cambridge University Press, New York, 2014)Į.U. (Prentice Hall, Harlow, England, 2003)Į.D. Joachain, Physics of Atoms and Molecules, 2 nd edn. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957)ī.H. Harris, Mathematical Methods for Physicists: A Comprehensive Guide, 7 th edn. Drake (Springer-Verlag, New York, 2006), p. Drake, in Springer Handbook of Atomic, Molecular, and Optical Physics, ed. Taylor, Journal of Physical and Chemical Reference Data 45, 043102 (2016) This chapter also establishes important nomenclature, extensively used throughout the book. This makes the one-electron atom formalism relevant as a first stepping stone for the study of multielectron atoms, and we begin this book by a brief recapitulation of the hydrogen problem. For this, we use the central-field approximation, where zeroth order states are products of individual electron wave functions. For multielectron atoms, this book principally relies on perturbation theory. For a system with a nucleus and two or more electrons, we have a multibody problem, and a direct assault on the Schrödinger equation is harder. For any quantum mechanical problem, a preferred way to approach a problem is to formulate the Schrödinger equation and to solve it analytically. It says that the solution of equation (4.25) is not simple and gives directly as equation (4.26). where P m is the associated Legendre function, defined by (4.27) P m ( x) ( 1 x 2) m / 2 ( d d x) m P ( x). As described in Potential Energy and Conservation of Energy, the force on the particle described by this equation is given by. The solution is (4.26) ( ) A P m ( cos ). Another important eigenvalue problem in physics is the Schrödinger equation. There is a fundamental difference between the theoretical approaches for atoms with at least two electrons, and systems with just a single electron. The equation describing the energy and momentum of a wave function is known as the Schrdinger equation: 2 2 m 2 ( x, t) x 2 + U ( x, t) ( x, t) i ( x, t) t.
0 Comments
Leave a Reply.AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |