The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.
Abstract Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles (θ, ϕ, ψ) that define a rotation in SO(3), and the Euclidean norm r in <?CDATA
R3?> R 3 , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form <?CDATA
H=H0⊕H1?> = 0 ⊕ 1 . Following a classical work by Schwinger, 2-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space <?CDATA
H?> is approached from the double covering group homomorphism SU(2) → SO(3) and the topology involved is taken care of by using the Hurwitz-Hopf map <?CDATA
H:R4→R3?> H : R 4 → R 3 . It is shown how to reconsider H as a 2-to-1 group map, <?CDATA
G0=R+×SU(2)→R+×SO(3)?> G 0 = R + × SU ( 2 ) → R + × SO ( 3 ) , translating it into an assignment (z 1, z 2) ↦ (r, θ, ϕ, ψ) whose domain consists of pairs (z 1, z 2) of complex variables, under the appropriate identification of <?CDATA
R4?> R 4 with <?CDATA
C2?> C 2 . It is shown how the Lie algebra of G 0 is coupled with two Heisenberg Lie algebras of 2-dimensional (Schwinger’s) harmonic oscillators generated by the operators <?CDATA
{z1,z2,zˉ1,zˉ2}?> { z 1
Chemists are aware of the solution of Schroedinger's equations for the hydrogen atom in only spherical polar coordinates, but the spatial variables are separable also in three other systems - - paraboloidal, ellipsoidal and spheroconical; we report here explicit algebraic solutions directly derived in ellipsoidal and spheroconical coordinates for the first time. Our solutions progress from those previously known in spherical polar but not entirely understood, through those little known in paraboloidal, to those in systems of ellipsoidal and spheroconical coordinates unknown before the present work. Applications of these solutions include angular momenta, a quantitative calculation of the discrete absorption spectrum and accurate plots of surfaces of amplitude functions. The shape of a surface of a particular amplitude function, and even the quantum numbers in a particular set to specify such an individual function, depend on a particular chosen system of coordinates, and are therefore artefacts of that coordinate representation within wave mechanics; a choice of a coordinate system to discuss atomic or molecular properties based on the shapes of amplitude functions or their respective quantum numbers is hence arbitrary
By invoking a standard variable transformation x → y ≡ x 2 that connects the eigenvalue problems of the hydrogen atom system (Coulomb potential) and the Kratzer oscillator system , we obtain a system of ladder operators Ω nl ± ( l = 0,1,2, ... , n − 1) that connect the degenerate energy eigenstates (belonging to different l -values) of the hydrogen atom system. Using such quantum number specific operators as the precursors, we develop a general method of constructing quantum number independent ladder operators ; and then investigate the general conditions for these operators to form a realization of the su (2) algebra. In the process, we demonstrate that, in general, a finite-dimensional linear vector space carries a realization of the su (2) algebra. Applying the method to the case at hand, we show that for the hydrogen atom system the operators and form an su (2) algebra. The Hamiltonian is identified with the Casimir operator of the algebra, thereby revealing a dynamical SU (2) group symmetry of the hydrogen atom system.
The purely algebraic technique associated with the creation and annihilation operators to resolve the radial equation of Hydrogen-like atoms (HLA) for generating the bound energy spectrum and the corresponding wave functions is suitable for many calculations in quantum physics. However, the analytical approach with series is extensively used based on wave mechanics theory in most of quantum textbooks. Indeed, much More complete than the old solution of Schrödinger's time-independent differential equation (TISE), one can simply earn all quantum information of a system by using the operational method. In addition to earlier two models, including the quantum harmonic oscillator and the total angular momentum, it can undoubtedly be a third fundamental model to solve Schrödinger's eigenvalue equation of the HLA systems in Hilbert space similar to the harmonic oscillator. We will illustrate how systematically making an appropriate groundwork to discover the coherent states can lead to providing the energy quantization and normalized radial wave functions attached to the matrix representation without additional assumptions.
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.