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The Schroedinger equation for an electron subject to the attraction of a nucleus of positive charge Z is
where u is the wavefunction for the electron, m is the electron's mass, h is Planck's constant
divided by 2π, E is the total energy of the system, e is the charge of the electron and r is the separation distance
of the electron from the nucleus. It is presumed that the mass of the nucleus is so large that the center of mass
of the system is the center of the nucleus. The symbol ∇² stands for the Laplacian operator which in
spherical coordinates of (r, θ, φ) is:
The technique of separation of variables can be applied to the Schroedinger equation. If the wavefunction has the form
then it must be that
The lefthand side (LHS) of the above equation is a function of r only and the righthand side (RHS) is a function of only θ and φ. Therefore the common value must be a constant independent of r, θ and φ, say λ. The RHS of the above equation then reduces to
This equation has a solution only if
If this condition is satisfied there are (2l+1) solutions, called the spherical harmonic functions Y_{l,m}(θ, φ). The variable m is an integer that can take on any value between −l and +l.
The fact that λ = l(l+1) means that the radially dependent component of the wavefunction must satisfy the equation
The term l(l+1)/r² corresponds to the centrifugal force of classical physics and l corresponds to angular momentum.
The solutions for R are dependent upon a positive integer n and l as well as r. The parameter l can take on any integer value from 0 to (n1).
The solutions for the first three values of n are:
The wavefunction u(r,θ,φ) is generally of complex value. The square of its magnitude is the probability density. Thus the charge density for an electron is proportional to r²R²(r). This quantity is plotted for the first few solutions for R_{n,l}.
The spherical distribution of charge generally also depends upon Y²(θ, φ). It just happens that Y_{0,0} is a constant; i.e., independent of θ and φ. Thus for n=1 the charge density is spherically symmetric.
The spherical harmonic Y_{1,0} is equal to (3/4π)^{½}cos(θ). The radial distribution function R_{2,1}(r) is proportional to r*exp(½r). Thus the probability density function P(r,θ) of an electron is given by
where K is a constant.
A good way of visually depicting the probability density function is by displaying the curves of constant probability density. This would involve solving the equation
for r as a function of θ. The function r²*exp(r) has a maximum value. Therefore cos²(θ) must have a minimum value. Likewise cos²(θ) has a maximum value of 1 at θ equal to 0 and π. This determines the minimum value of r.
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