Problems 6.2, A radial wave function with no nodes (l n-1), corresponding to a B

Question

Problems 6.2, A radial wave function with no nodes (l n-1), corresponding to a Bohr circular orbit, might be written in analytical form as a modified hydrogenic function with one parameter subject to the normalizaton R'r dr 1 By inspection, find the form of the parameter a which gives the energy in hydrogenic form, ie.. E- 171e2/(4,12ao), for an electron moving in the field of an effective central charge Z'e. Show that the peak of R'r2, the probability density per unit radial thickness, lies at ron'ao/Z', which is an effective Bohr radius for the charge distribution. Show also that with this wave function (2n +k)!

Explanation / Answer

SCHROEDINGER’S EQUATION IN SPHERICAL POLAR COORDINATES

The magnitude of a central force on an object depends on only the distance of that object

from the origin; the direction of the force is along the line joining the origin and the object. The

coulombic attraction is a central force, which implies a conservative field and which signifies that it

is expressible as the gradient of a potential energy. Schroedinger’s equation for an electron moving

in a central force field is invariably separable in spherical polar coordinates, which in

Schroedinger's paper is called simply polar coordinates [2]. We assume the electron and the

proton, or other atomic nucleus, to constitute point masses that interact according to Coulomb’s

law; a deviation from that law might imply a non-zero rest mass of a photon, for which no

evidence exists, apart from the effects of the finite volume and shape of a massive atomic nucleus,

and their isotopic variation, for which experimental evidence exists. We first relate these

coordinates, i.e. radial coordinate r, polar angular coordinate and equatorial angular coordinate

, to cartesian coordinates x, y, z as algebraic formulae, according to ISO standard 80000-2:2009,

x = r sin() sin(), y = r sin() cos(, z = r cos()

with domains 0 r < , 0 < , 0 < 2 , so that axis z in cartesian coordinates becomes the

polar axis in spherical polar coordinates. For the motion of the electron relative to the atomic

nucleus, the use of a reduced mass converts the problem of treating two interacting particles into a

treatment of effectively a single particle subject to a force field; the motion of the atom as a whole

through space is of little interest – only the internal motion produces observable properties readily

observable in atomic spectra in absorption or emission. Coordinate r signifies the distance between

reduced mass and the origin; coordinate signifies the angle of inclination between a line joining

that reduced mass to the origin and polar axis z in cartesian coordinates; coordinate signifies the

equatorial angle between a half-plane containing that line, between the reduced mass and the

origin, and half-plane x=0; a half-plane extends from the polar axis to in any direction. The

limiting cases are thus for r a point at the origin as r 0, and for a line along positive axis z as

0 and along negative axis z as . Surfaces of coordinates r, and as constant quantities

are exhibited, with definitions, in figure 1. For use within an integrating element in subsequent

integrals, the jacobian of the transformat

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