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水素原子の2次元表示を用いる量子論 : Hの電子は平面内を運動する

正雄 岡林

Published Jan 1, 1969• 0 citations
OpenAlex

Abstract

We show that a representation in the two-dimensional coordinate is an irreducible one for the hydrogen atom. Corresponding to two dimensions of the coordinate there appear two quantum numbers, e.i., the angular momentum and the total quantum unmber; further a couple of wave functions. radial and angular. In our representation the eigenvalue for the Dirac angular momentum gives the half-integer and the same one as Schroedinger's.

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Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)

•Jan 1, 1990•15 citations
We show that a representation in the two-dimensional coordinate is an irreducible one for the hydrogen atom. Corresponding to two dimensions of the coordinate there appear two quantum numbers, e.i., the angular momentum and the total quantum unmber; further a couple of wave functions. radial and angular. In our representation the eigenvalue for the Dirac angular momentum gives the half-integer and the same one as Schroedinger's.
The wave functions for the hydrogen atom are determined in the momentum representation, using momentum variables pr{p}_{r}pr​, ${p}_{\ensuremath{\theta}}$, ${p}_{\ensuremath{\varphi}}$ which are associated with the spherical polar variables rrr, $t(={e}^{i\ensuremath{\pi}cos\ensuremath{\theta}})$ and $v({e}^{i\ensuremath{\varphi}})$. The functions may be written as ${{\ensuremath{\varphi}}_{n,l,m}({p}_{r},{p}_{\ensuremath{\theta}},{p}_{\ensuremath{\varphi}})={\ensuremath{\alpha}}_{n,l}({p}_{r}){\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})\ensuremath{\delta}({p}_{\ensuremath{\varphi}}+m),}{{\ensuremath{\alpha}}_{n,l}({p}_{r})={\left(\frac{n}{i{p}_{0}}\right)}^{3l+z}\ensuremath{\Sigma}\stackrel{n\ensuremath{-}l\ensuremath{-}1}{k=0}\left({n\ensuremath{-}l\ensuremath{-}1}{k}\right)\frac{{2}^{k\ensuremath{-}l+1}}{(2l+k+1)(2l+k)\ensuremath{\cdots}(l+k+2)}{\left(\frac{\frac{i{p}_{0}}{n}}{{p}_{r}\ensuremath{-}\frac{i{p}_{0}}{n}}\right)}^{l+k+2},}{{\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})=\ensuremath{\Sigma}\stackrel{E(\frac{l}{2})}{j=E(\frac{m}{2})}{{a}_{j}{p}_{\ensuremath{\theta}}}^{\ensuremath{-}\frac{(m+1)}{2}}{J}_{\frac{j\ensuremath{-}(m\ensuremath{-}1)}{2}}({p}_{\ensuremath{\theta}}),}$ where $\ensuremath{\delta}$ is the Dirac delta function and ${p}_{0}=Z{a}_{0}^{\ensuremath{-}1}$ (a0{a}_{0}a0​ is the Bohr radius). ${J}_{\ensuremath{\nu}}({p}_{0})$ are Bessel functions of integer or half-integer order. The radial functions are shown to have poles up to order n+1n+1n+1 at pr=ip0n{p}_{r}=i\frac{{p}_{0}}{n}pr​=inp0​​.
A partial separation of the variables is practicable for the solution of Schroedinger's temporally independent equation in cartesian coordinates x,y,z, which yields moderately simple algebraic formulae for the amplitude functions involving quantum numbers k,l,m, the same as in spherical polar coordinates. The properties of angular momentum are thus achieved with no angular variable. Several plots of surfaces of constant psi(x,y,z) are presented to illustrate the resemblance of the shapes of these surfaces to the shapes of surfaces of psi(r,theta,phi) with the corresponding quantum numbers.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
The angular momentum states may be realized by the wave functions of two dimensional harmonic oscillators in polar coordinates. Using the generating function of these wave functions, Schwinger's generating function of Clebsh-Gordan in Fock space and Gauss integral, we determine the generating function of the couplings function of two angular momentums in coordinates representations. We deduce from the last function the connection of the wave function of the hydrogen atom and the four dimensional harmonic oscillator. We also find the expression of the relation between the spherical and parabolic wave functions of hydrogen atom. In addition, we find a new bilinear generating function involving the products of Laguerre polynomials.
The hydrogen atom is a system amenable to an exact treatment within Schroedinger's formulation of quantum mechanics according to coordinates in four systems -- spherical polar, paraboloidal, ellipsoidal and spheroconical coordinates; the latter solution is reported for the first time. Applications of these solutions include angular momenta, a quantitative calculation of the absorption spectrum and accurate plots of surfaces of amplitude functions. The shape of an amplitude function, and even the quantum numbers in a particular set to specify such an individual function, depend on the coordinates in a particular chosen system, and are therefore artefacts of that particular coordinate representation within wave mechanics. All discussion of atomic or molecular properties based on such shapes or quantum numbers therefore lacks general significance
Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)