Approximating the Universe: Perturbation Theory

The theory of quantum mechanics is humanity’s most honourable attempt at describing the nature of the most fundamental building blocks of the Universe. Providing a quantitative understanding of subatomic particles and their interactions, the theory has stood up to one hundred years of experiments testing its validity with only minor deviation from observations. The agreement between theory and experiment is all the more impressive given the huge amount of approximation present in the mathematics of quantum physics.

Approximation in Physics

Quantum systems are parametrised completely by their wavefunctions. All of the measurable quantities of the system, such as the different energy levels the system can occupy, can be deduced from the wavefunction. The nature of the wavefunction, and the subsequent properties of the system, can be determined by solving the Schrödinger equation. Sound easy enough?

Unfortunately, solving the Schrodinger equation is no walk in the park. It turns out that even the simplest of quantum systems, such as a hydrogen atom, require significant mathematical effort to solve. Finding solutions for larger systems, such as atoms with a higher atomic number or, if you like, around 1030 electrons moving through a piece of metal, is not possible without the use of multiple approximations.

Perturbation Theory

Perturbation theory is a cunning approach to finding approximations to the wavefunction and energies of quantum systems that differ only slightly from a system that we can solve exactly using the Schrodinger equation. The new system is modelled identically to the solved system bar the presence of a small perturbation or modification. The altered properties of the new system, or corrections, can subsequently be deduced from the form of the perturbation. This method has provided physicists with monumental insight into the nature of previously unsolvable systems and its results have been tested to display impressive precision given its simplicity. The rest of the article is dedicated to presenting a couple of the most fruitful applications of perturbation theory

The Helium Atom

Helium is the second lightest element with an atomic number of 2. A single atom of helium contains two protons and two electrons (uncharged neutrons ignored). Each electron orbits the nucleus at a specific radius in a particular energy level; the energy values for each level is of interest to physicists. Regrettably, determining the energy level values turns out to be impractical via directly solving the Schrodinger equation.

As previously mentioned, it is possible to find exact solutions to the Schrodinger equation for the hydrogen atom, which contains just a single electron and proton. It seems reasonable to expect the energy levels for helium to resemble those for hydrogen to some level, modified by the addition of an extra proton and the interaction of the pair of electrons. Perturbation theory can therefore be applied.

The known energy levels of hydrogen are first scaled up to account for the presence of the extra proton, which contributes to a larger electric attraction between the nucleus and orbiting electrons. The perturbation is then introduced to take the mutual electron repulsion into consideration. The resultant energy corrections are calculated and combined with the scaled up hydrogen energy levels to predict an approximate energy level spectrum for the helium atom. After a single approximation, the calculated energies are found to agree with experiment to within 7%, a remarkable result considering the modesty of the calculation.

Electrons in Metals

Combine a vast, ordered array of atoms with a sea of delocalised electrons and you have yourself a metal. The electric properties of metals are determined by the ease at which electrons can flow through the material, largely a product of the energy levels electrons occupy within the material due to the attractive forces supplied by atomic nuclei. It should come as no surprise that obtaining the energy level spectrum by directly solving the Schrodinger equation for a network of over 1030 atoms and electrons is wildly impractical.

Noting that the motion of the electrons within metals suggests the forces of attraction are very weak, we can model the system as a collection of free electrons in the presence of a weak periodic perturbation arising from interactions with nearby nuclei. This idea is the primary ingredient in nearly-free electron theory. The energy level spectrum for free electrons is very simple to obtain from the Schrodinger equation and forms the basis of free electron theory.

Upon addition of the corrections that stem from the perturbation due to nearby atoms, the energies of slow-moving electrons resemble that of free electrons . For faster moving electrons, the energy level spectrum splits into two bands. This result leads to the emergence of band structure within materials, forming the basis of semiconductor physics and its countless applications to modern technology.

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