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Basic Theoretical Physics: A Concise Overview P24. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 236 27 The Hydrogen Atom according to Schrodinger s Wave Mechanics The characteristic values for the orbital angular momentum l 0 1 2 3 . .2 correspond in chemical language to s p d f . orbitals we also have the relation l n 1 where n 1 2 3 . is the principal quantum number see below . The magnetic quantum number m assumes the allowed values m l l 1 . l quantization of the orbital angular momentum we do not yet consider the spin momentum which will be treated below . In addition we need to express the Laplace operator in spherical coordinates V2 f rds rf 3 27V where3 4 the square of the orbital angular momentum can be written as Î2 1 d . df1 d2f -f 0 - - in 0 J 2 g 2 27.5 For the radial function R r one obtains using an auxiliary function w r r R r the following one-dimensional differential equation N.B. this depends on l but not on m h2 - w r h2l l 1 2mr2 V r w r E w r 27.6 Additionally the following boundary conditions must be satisfied w 0 0 4 and i -PR r 2 dr U dr w r 2 1 J0 0 since d3r r2 dr d and d Ylm 0 p 2 J sin 0d0 J d p Ylm 0 p 2 e-. 0 v 0 1 . From now on we concentrate on a Coulomb potential V r - e 2 4 or i.e. to an A r-potential which has already been shown to be special in classical mechanics remember that for A r-potentials the Runge-Lenz vector is conserved absence of perihelion motion of the Kepler ellipses and hyperbolae . 2 It can be shown algebraically that the commutation relations of angular momenta can also be satisfied for half-integral numbers 1 2 3 2 5 2 etc. which corresponds to the possibility of non-orbital angular momenta e.g. the spin see below. 3 The relation 27.4 has since p2 ft2V2 the following classical pendant p2 p2 r-2 r x p 2 p2 L2 r2 here L r x p is the orbital angular momentum vector. 4 Remember the footnote on the quasi one-dimensional behavior of bound s states in a 3d quantum box in the previous Chap. 25. 27.1 Product Ansatz the Radial Function 237 By analogy with the harmonic oscillator reduced lengths and energies are .
