| Perturbation Theory (quantum Mechanics) |
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APPLICATIONS OF PERTURBATION THEORY Perturbation theory is an extremely 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 Line s 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 &alpha 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 evalauted 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 Diagram s 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 cannot be described by a small perturbation imposed on some simple system. In Quantum Chromodynamics , for instance, the interaction of Quark s with the Gluon field cannot 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 State s and various collective phenomena such as Soliton s. Imagine, for example, that we have a system of free (i.e. 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 Electron s leads to the formation of correlated electron pairs known as Cooper Pair s. 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 is 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 or in the perturbation parameter g. Perturbation theory can only detect solutions "close" to the unperturbed solution, even if there are other solutions (which typically blow up as the expansion parameter goes to zero). The problem of non-perturbative systems has been somewhat alleviated by the advent of modern Computer s. It has become practical to obtain numerical non-perturbative solutions for certain problems, using methods such as Density Functional Theory . These advances have been of particular benefit to the field of Quantum Chemistry . Computers have also been used to carry out perturbation theory calculations to extraordinarily high levels of precision, which has proven important in Particle Physics for generating theoretical results that can be compared with experiment. There are two categories of perturbation theory: time-independent and time-dependent. In this section, we discuss time-independent perturbation theory, in which the perturbation Hamiltonian is static (i.e., possesses no time dependence.) Time-independent perturbation theory was invented by Erwin Schrödinger in 1926 , shortly after he produced his theories in wave mechanics. We begin with an unperturbed Hamiltonian ''H0'', which is also assumed to have no time dependence. It has known energy levels and eigenstates, arising from the time-independent Schrödinger equation: | ||
| :<math> \left(H 0 + \lambda V Ight) n Ang | E_n n
ang </math> |
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| :<math> n Ang | n^{(0)}
ang + \lambda n^{(1)}
ang + \lambda^2 n^{(2)}
ang + \cdots </math> |
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| \qquad\qquad | \left(E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots
ight) \left(n^{(0)}
ang + \lambda n^{(1)}
ang + \cdots
ight) |
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| :<math> H 0 n^{(1)} Ang + V n^{(0)} Ang | E_n^{(0)} n^{(1)}
ang + E_n^{(1)} n^{(0)}
ang </math> |
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| :<math> E N^{(1)} | \langle n^{(0)} V n^{(0)}
angle </math> |
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| This Is Simply The | "http://wwwinformationdelightinfo/encyclopedia/entry/expected_value" class="copylinks">Expected Value of the perturbation Hamiltonian while the system is in the unperturbed state This result can be interpreted in the following way: suppose the perturbation is applied, but we keep the system in the quantum state ''n<sup>(0)</sup>''>, which is a valid quantum state though no longer an energy eigenstate The perturbation causes the average energy of this state to increase by <''n<sup>(0)</sup>''''V''''n<sup>(0)</sup>''> However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as ''n<sup>(0)</sup>''> These further shifts are given by the second and higher order deviations |
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| :<math> Vn^{(0)} Angle | \left( \sum_{k} k^{(0)}
angle\langle k^{(0)}
ight) Vn^{(0)}
angle </math> |
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| :<math> \left(E N^{(0)} - H 0 Ight) n^{(1)} Ang | \sum_{k |
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| :<math> n^{(1)} Ang | \sum_{k |
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