Everything about Perturbative totally explained
In
quantum mechanics,
perturbation theory is a set of approximation schemes directly related to mathematical
perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system and gradually turn on an additional "perturbing"
Hamiltonian representing a weak disturbance to the system. If the disturbance isn't too large, the various physical quantities associated with the perturbed system (for example its
energy levels and
eigenstates) will be continuously generated from those of the simple system. We can therefore study the former based on our knowledge of the latter.
Applications of perturbation theory
Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the
Schrödinger equation for
Hamiltonians of even moderate complexity. The Hamiltonians to which we know exact solutions, such as the
hydrogen atom, the
quantum harmonic oscillator and the
particle in a box, are too idealized to adequately describe most systems. Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a range of more complicated systems. For example, by adding a perturbative
electric potential to the quantum mechanical model of the hydrogen atom, we can calculate the tiny shifts in the
spectral lines of hydrogen caused by the presence of an
electric field (the
Stark effect). This is only approximate because the sum of a
Coulomb potential with a linear potential is unstable although the
tunneling time (
decay rate) is very long. This shows up as a broadening of the energy spectrum lines, something which perturbation theory fails to notice entirely.
The expressions produced by perturbation theory are not exact, but they can
lead to accurate results as long as the expansion parameter, say
, is very small. Typically, the results are expressed in terms of finite
power series in
that seem to converge to the exact values when summed to higher order. After a certain order
, however, the results become increasingly worse since the series are usually
divergent, being
asymptotic series). There exist ways to convert them into convergent series, which can be evaluated for large-expansion parameters, most efficiently by
variational perturbation theory.
In the theory of
quantum electrodynamics (QED), in which the
electron-
photon interaction is treated perturbatively, the calculation of the electron's
magnetic moment has been found to agree with experiment to eleven decimal places. In QED and other
quantum field theories, special calculation techniques known as
Feynman diagrams are used to systematically sum the power series terms.
Under some circumstances, perturbation theory is an invalid approach to take. This happens when the system we wish to describe can't be described by a small perturbation imposed on some simple system. In
quantum chromodynamics, for instance, the interaction of
quarks with the
gluon field can't be treated perturbatively at low energies because the
coupling constant (the expansion parameter) becomes too large. Perturbation theory also fails to describe states that are not generated
adiabatically from the "free model", including
bound states and various collective phenomena such as
solitons. Imagine, for example, that we've a system of free (for example non-interacting) particles, to which an attractive interaction is introduced. Depending on the form of the interaction, this may create an entirely new set of eigenstates corresponding to groups of particles bound to one another. An example of this phenomenon may be found in conventional
superconductivity, in which the
phonon-mediated attraction between
conduction electrons leads to the formation of correlated electron pairs known as
Cooper pairs. When faced with such systems, one usually turns to other approximation schemes, such as the
variational method and the
WKB approximation. This is because there's no analogue of a
bound particle in the unperturbed model and the energy of a soliton typically goes as the
inverse of the expansion parameter. However, if we "integrate" over the solitonic phenomena, the nonperturbative corrections in this case will be tiny; of the order of
Many further results may be obtained, such as
Fermi's golden rule, which relates the rate of transitions between quantum states to the density of states at particular energies, and the
Dyson series, obtained by applying the iterative method to the
time evolution operator, which is one of the starting points for the method of
Feynman diagrams.
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