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The Quantum Mechanics of One Electron Systems

The Schroedinger equation for an electron subject to the attraction of a nucleus of positive charge Z is

∇²u + (2m/h²)(E + Ze²/r)u = 0

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:

∇²u = (∂²u/∂²r) + (2/r)(∂u/∂r)
+ (1/(r²sin(θ)))(∂(sin(θ)∂u/∂θ))/∂θ
+ (1/(r²sin²(θ)(∂²u/∂φ²)

The technique of separation of variables can be applied to the Schroedinger equation. If the wavefunction has the form

u(r, θ, φ) = R(r)Y(θ, φ)

then it must be that

(r²/R)[(d²R/dr²) + (2/r)(dR/dr) + 2(E + Z/r)R] =
−(1/Y)[(1/sin(θ))(∂(sin(θ)(∂Y/∂θ))/∂θ + (1/sin²(θ))(∂²Y/∂φ²)]

The left-hand side (LHS) of the above equation is a function of r only and the right-hand 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

(1/sin(θ))(∂(sin(θ)(∂Y/∂θ))/∂θ + (1/sin²(θ))(∂²Y/∂φ²) + λY = 0

This equation has a solution only if

λ = l(l+1)
for l = 0, 1, 2, …

If this condition is satisfied there are (2l+1) solutions, called the spherical harmonic functions Yl,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

(d²R/dr²) + (2/r)(dR/dr) + [2(E+Z/r) − l(l+1)/r²]R = 0

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 (n-1).

The solutions for the first three values of n are:

R1,0 = 2exp(-r)
R2,0 = (1/√2)exp(-r/2)(1−r/2)
R2,1 = (1/2√6)exp(-r/2)
R3,0 = (1/3√3)exp(-r/3)(1−2r/3+(2/27)r²)
R3,1 = (8/27√6)exp(-r/3)r(1−r/6)
R3,2 = (4/81√30)exp(-r/3)r²

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 Rn,l.

The spherical distribution of charge generally also depends upon Y²(θ, φ). It just happens that Y0,0 is a constant; i.e., independent of θ and φ. Thus for n=1 the charge density is spherically symmetric.

The spherical harmonic Y1,0 is equal to (3/4π)½cos(θ). The radial distribution function R2,1(r) is proportional to r*exp(-½r). Thus the probability density function P(r,θ) of an electron is given by

P(r,θ) = Kcos²(θ)r²*exp(-r)

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

cos²(θ)r²*exp(-r) = λ
which is equivalent to
r²*exp(-r) = λ/cos²(θ)

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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