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Method Applied to a Helium Atom |
The Hamiltonian function for the two electrons of a helium atom is easy to specify. Let p_{1} and p_{2} be the momentum of the electrons. The kinetic energy of the atom is then
where m is the electron mass. Let r_{1} and r_{2} be the position vectors of the two electrons with respect to an origin at the center of the nucleus. The magnitudes of r_{1} and r_{2} are denoted as r_{1} and r_{2}. The potential energy of the atom is then
where q is the product of the constant for the electrostatic force and the square of the unit charge.
The Schrödinger equation for the system is also easily derived, but obtaining a solution is nearly impossible.
The Hartree procedure consists of considering a single electron with the effect on it of the other electron being replaced by its effect on the potential energy function.
where R is the average distance between the electron and the average position of the other electron. The average position of the other electron may be the center of the atom, in which case R would be equal to r. However if the other electron is considered as a spherical distribution of charge that part which is closer to the origin than r would have an effect but that which is farther away than r would have no effect. The value of R would then be ½r.
This Hamiltonian function is then converted to its Hamiltonian operator by replacing p with −ih∂/∂r
where h is Planck's constant divided by 2π and i is the imaginary unit √−1. The exponent of 2 for p
results in the second derivative with respect to r. The time independent Schrödinger equation for the system is then
where ψ is the wave function of the electron and ε is real-valued onstant, the energy of the system. This is converted into matrix form by letting Ψ represent ψ(r) as an infinite dimensional vector. Likewise V is an infinite dimensional diagonal matrix with 1/r on the principal diagonal. This means that points for the arguments of the function must straddle the origin to avoid having a term involving division by zero. This can be done by taking the points nearest the origin to be +δ/2 and −δ/2. Thus the points corresponding to the vector components are …, 2&fract12;δ, 1&fract12;δ, &fract12;δ, −&fract12;δ, −1&fract12;δ, −2&fract12;δ, ….
The second derivative operation can be represented as
The matrix version of the system is then
where J is a matrix of zeroes except for (… 1, −2; 1, …) centered on the principal diagonal.
As it happens the model with an electron in the same shell shielding a half unit of charge is equivalent to the Bohr model for hydrogen with the central charge being 3/2 rather than 1. The ionization energies can be computed and ccompared with the experimental values. For the details see helium model.
Comparison of Measured Helium Spectrum Lines with Values Computed from a Modified Version of the Bohr Model |
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Measured wavelength | Computed wavelength | Error |
438.793 nm | 433.937 nm | -1.1% |
471.314 nm | 486.009 nm | +3.1% |
492.193 nm | 486.009 nm | -1.3% |
501.5675 nm | 486.009 nm | -3.2% |
667.815 nm | 656.112 nm | -1.8% |
(To be continued.)
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