Spectra of atoms

Why is a sodium lamp yellow? How can we determine the elemental composition of the sun? How does a Helium-neon laser can work?

To some degree all of these questions require knowing the spectra of atoms, which can in theory be calculated by Quantum mechanics. However the calculations of these spectra for arbitrary systems from first principles is prohibitively difficult and computationally intensive (which is why techniques such as Density Functional Theory are used).

This post will roughly outline to calculate the spectrum of the smaller atoms by explicitly diagonalising a matrix, whose elements are simple combinatorial quantities.

The non-relativistic Hamiltonian in the Born-Oppenheimer approximation of an n-electron atom in SI units is given by H = \sum_{i=1}^{n} \left( \frac{p_i^2}{2m} - \frac{Ze^2}{4 \pi \epsilon_0} \frac{1}{r_i} + \frac{e^2}{4 \pi \epsilon_0} \sum_{j > i} r_{ij} \right) where p_i is the momentum of the ith electron, r_i is its distance from the nucleus, $r_ij$ is the distance between the ith and jth electron, m is the mass of an electron, Z is the charge of the nucleus, and e is the charge of an electron.

To simplify matters choose units such that \frac{e^2}{4 \pi \epsilon_0} = 1, m = \frac{1}{2} and \hbar = 1, these will be used in the rest of this article. Then H = \sum_{i=1}^{n} p_i^2 - \sum_{i=1}^{n} \frac{Z}{r_i} +\sum_{i=1}^{n}\sum_{j>i}^{n} \frac{1}{r_{ij}}. The terms correspond to the kinetic energy of the electrons, the electron-atom interaction, and the electron-electron interactions respectively. If we neglect the third term we recover the equation of a Hydrogenic atom which can be solved algebraically.

Our approach is to calculate the the elements of the Hamiltonian matrix in the Hydrogenic basis. We can then explicitly diagonalise the matrix in this basis; if the electron-electron term is small the matrix will be almost-diagonal. I will only cover the case of the bound states; the unbound states do need to be considered at a future point, but at least near the ground state their contribution should be negligible.

The Hydrogenic atom can be simultaneously diagonalised in a number of different basis sets (corresponding to different coordinate systems); we need a basis whose symmetry is preserved by the perturbation. Since the perturbed Hamiltonian is spherically symmetric, we choose the basis H, L, Lz where the bound states are characterised by the quantum numbers n, l, m, s (n >=1, 0 <= l < n, |m| <= l, s = \pm \frac{1}{2} with corresponding eigenvalues  \frac{Z^2}{2n^2}, l(l+1), m$). The electronic states of an n-electron atom, neglecting electron-electron interactions, are then the antisymmetric tensor products of these states.

We now proceed to calculate the matrix elements of the full Hamiltonian in this basis. The only non-trivial part of the calculation are the terms \wedge_{i=1}^{N} \langle n_i, l_i, m_i, s_i | \frac{1}{r_{st}} \wedge_{j=1}^{N} | n_j, l_j, m_j, s_j \rangle. The spins and terms i, j not equal to s, t factor through, giving Kronecker deltas. The remaining calculation is \langle n_1, l_1, m_1, n_2, l_2, m_2 | \frac{1}{r_ij} | n'_1, l'_1, m'_1, n'_2, l'_2, m'_2 \rangle. Since the term commutes with L_i and L_j, it is also proportional to \delta_{l_1}^{l'_1} \delta_{l_2}^{l'_2} \delta_{m_1}^{m'_1} \delta_{m_2}^{m'_2}.

Integrate in spherical coordinates over first r_1, then r_2 setting the z-axis of the second coordinate system along the vector r_1. Then \theta_2 is the angle between the two electrons at the nucleus, and the integrand is \frac{1}{r_{12}} = \frac{1}{\sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_2)}}, and consequently the first solid angle integral is trivial.

Thus we just need to evaluate \int d\Omega Y_{l_2}^{m_2}(\theta, \phi) {Y_{l_2}^{m_2}}^*(\theta, \phi) \int_{0}^{\infty} dr_1 r_1^2 R_{n_1}^{l_1}(r_1) {R_{n'_1}^{l_1}}^*(r_1) \int_{0}^{\infty} dr_2 r_2^2 R_{n_2}^{l_2}(r_2) {R_{n'_2}^{l_2}}^*(r_2) \frac{1}{\sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta)}} (notice that the integral must be invariant under interchange of all 1 labels with 2 labels; in practice we make the choice that makes the integral easiest).

We now separate the r_2 integral into two regions; where it is less than r_1 the \frac{1}{r_{12}} term can be expanded as \sum_{t=0}^{\infty} \frac{1}{r_1} \left( \frac{r_2}{r_1} \right)^t P_t(\cos(\theta)) where P_t is a Legendre Polynomial, and in the other region we switch r_1 with r_2.

The integral then becomes \sum_{t=0}^{\infty} \int d\Omega Y^{m_1}_{l_1} (\Omega) Y^{-m_1}_{l_1}(\Omega) \sqrt{\frac{4 \pi}{2t+1}} Y^0_t(\Omega) \int_0^{\infty} dr_1 R_{n_1}^{l_1}(r_1) {R^{l_1}_{n'_1}}^*(r_1) r_1^{2+t} \int_0^\infty dr_2 r_2^{1-t} R_{n_2}^{l_2}(r_2) {R_{n'_2}^{l_2}}^*(r_2) plus the integral switching r1 with r2 (after a fiddling change of coordinates).

The angular integral is a combinatorial quantity, which can be expressed in terms of the Clebsch-Gordan coefficients. It can be expressed using recurrance relations which can be used to compute this part of the integral. [In fact there is an explicit combinatorial representation, although in practice it would be quicker to compute it using recurrance.]

The inner radial part of the integral can be calculated by expanding the Legendre polynomials as a power series and using the relation \int_R^\infty e^{- \alpha r} r^k = \frac{e^{- \alpha R}}{\alpha^{k+1}} \sum_{j=0}^{k} (R \alpha)^j \frac{k!}{j!}, and the outer part of the integral can then be calculated using this relation again with R=0. This is simply a combinatorial factor than needs to be determined.

Thus once we have evaluated these combinatorial quantities, and combined them all to get an expression for the matrix elements of the total Hamiltonian H we can truncate it to a finite basis, and then diagonalise it computationally. It is a very interesting question as to how the truncation affects the eigenvalues.

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Regular expressions, automata and monoids

In formal language theory the task is to specify, over some given alphabet, a set of valid strings. This is useful in searching for structures textual data through files (e.g. via grep), for specifying the syntactic structure of programming languages (e.g. in Bison or pandoc), and for generating output of a specified form (e.g. automatic computer science and mathematics paper generators).

An automoton is (roughly) a set of symbols, and a set of states, along with transitions for each state that take a symbol and return another state. They can be used to model (and verify) simple processes.

Automata can be brought into correspondence with formal languages in a very natural way; given an initial state s, and a sequence of symbols (a1, a2, …, an) the automata has a naturally assigned state (… ((s a1) a2) … an) (where “(state symbol)” represents the state obtained from the transition on symbol using state). Then if we nominate an initial state, and a set of “accepting” valid states, we say a string is in the language of the automata if and only if when applied to the initial state it ends in a final state.

This gives a very useful pairing in computer science; formal languages are useful tools, and automata (often) give an efficient way to implement them on a computer.

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DVI by example

The Device Independent File Format (DVI) is the output format of Knuth’s TeX82; modern TeX engines (pdfTeX, luaTeX) output straight to Adobe’s Portable document format (PDF). However TeX82 and DVI still work as well today as they did when they were written; DVI files are easily cast to postscript or PDF.

The defining reference for DVI files is David R Fuch’s article in TUGboat Vol 3 No 2.

To find out what information is contained in a particular DVI file use Knuth’s dvitype, which outputs the operations contained in the bytecode in human readable format.

This article goes into gory detail the instructions contained in a very simple DVI file.

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Algorithms for finding the real roots of polynomials

Given an degree n polynomial over the real numbers we are guaranteed there are at most n real roots by the fundamental theorem of algebra; but how do we find them? Here we explore the Vincent-Collins-Akritas algorithm.

It uses Descartes’ rule of signs: given a polynomial p(x) = a_n x^n + \cdots + a_1 x + a_0 the number of real positive roots (counting multiplicites) is bounded above by the number of sign variations in the sequence (a_n, \ldots, a_1, a_0).

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Non-desarguesian projective planes

There are two main constructions of a projective space.

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Geometry and topology of division rings

Following from my last post (and Veblen and Young’s Projective Geometry) consider a projective plane satisfying the axioms:

  1. Given two distinct points there is a unique line that both points lie on
  2. Each line has at least three points which lie on it
  3. Given a triangle any line that intersects two sides of the triangle intersects the third.
  4. All points are spanned by d+1 points and no fewer.

Then for d>=3 is equivalent to the projective space of lines over a division ring (or skew field).

Kolmogorov asked the question what projective spaces can we do analysis on? In order to do things such as find tangent lines we are going to need some sort of topology.

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Geometry of division rings

It is fairly easy to construct a geometry from algebra: given a division ring K we form an n-dimensional vector space, the points being the elements of the field and a line being a translation of all (left) multiples of a non-zero vector, i.e. of the form \{a{\bf v} + {\bf c}| a \in K\} for some fixed vectors \bf{v} \neq 0 and c.

Interestingly it’s just as possible to go the other way, if we’re careful about what we mean by a geometry. I will loosely follow Artin’s book Geometric Algebra. In particular we have the undefined terms of point, line and the undefined relation of lies on. Then, for a fixed positive integer, the axioms are:

  1. Given two distinct points there is a unique line that both points lie on
  2. Each line has at least three points which lie on it
  3. Given a line and a point not on that line there exists a unique line lying on the plane containing them that the point lies on and no point of the first line lies on.
  4. All points are spanned by d+1 points and no fewer.

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