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 is not too large, the various physical quantities associated with the perturbed system (e.g. 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. For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ...

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. This box:      For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with... This box:      At a point in space, the electric potential is the potential energy per unit of charge that is associated with a static (time-invariant) electric field. ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... The Stark effect is the splitting of a spectral line into several components in the presence of an electric field. ... Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... Radioactive decay is the set of various processes by which unstable atomic nuclei emit subatomic particles (radiation). ...

The expressions produced by perturbation theory are not exact, but they can lead to accurate results as long as the expansion parameter, say α, is very small. Typically, the results are expressed in terms of finite power series in α 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 method. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... It has been suggested that this article or section be merged with 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. Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... For other uses, see Electron (disambiguation). ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... A bar magnet. ... Quantum field theory (QFT) is the quantum theory of fields. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ...

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 quarks 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 states and various collective phenomena such as solitons. 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 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 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 e ^{− 1 / g} 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). Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... For other uses, see Quark (disambiguation). ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ... In quantum mechanics, an adiabatic process is an infinitely slow change in the Hamiltonian of a system. ... In physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. ... In mathematics and physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed; solitons are caused by a delicate balance between nonlinear and dispersive effects in the medium. ... A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. ... Normal modes of vibration progression through a crystal. ... Electrical conduction is the current (movement of charged particles) through a material in response to an electric field. ... A Cooper pair is the name given to electrons that are bound together in a certain manner first described by Leon Cooper. ... It has been suggested that this article or section be merged with Variational_perturbation_theory. ... In physics, the WKB (Wentzel-Kramers-Brillouin) approximation, also known as WKBJ (Wentzel-Kramers-Brillouin-Jeffreys) approximation, is the most familiar example of a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is... In physics, a bound state is a composite of two or more building blocks (particles) that behaves as a single object. ...

The problem of non-perturbative systems has been somewhat alleviated by the advent of modern computers. 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. This article is about the machine. ... Density functional theory (DFT) is a quantum mechanical theory used in physics and chemistry to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules and the condensed phases. ... Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...

Time-independent perturbation theory

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 presented by Erwin Schrödinger in a 1926 paper,^[1] shortly after he produced his theories in wave mechanics. In this paper Schrödinger referred to earlier work of Lord Rayleigh,^[2] who investigated harmonic vibrations of a string perturbed by small inhomogeneities. This is why this perturbation theory is often referred to as Rayleigh-Schrödinger perturbation theory. Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... Year 1926 (MCMXXVI) was a common year starting on Friday (link will display the full calendar) of the Gregorian calendar. ... See also Rayleigh fading Rayleigh scattering Rayleigh number Rayleigh waves Rayleigh-Jeans law External links Nobel website bio of Rayleigh About John William Strutt MacTutor biography of Lord Rayleigh Categories: People stubs | 1842 births | 1919 deaths | Nobel Prize in Physics winners | Peers | British physicists | Discoverer of a chemical element ...

First order corrections

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: This box:      For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

For simplicity, we have assumed that the energies are discrete. The (0) superscripts denote that these quantities are associated with the unperturbed system.

We now introduce a perturbation to the Hamiltonian. Let V be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. (Thus, V is formally a Hermitian operator.) Let λ be a dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...

H = H₀ + λV.

The energy levels and eigenstates of the perturbed Hamiltonian are again given by the Schrödinger equation:

Our goal is to express E_n and |n> in terms of the energy levels and eigenstates of the old Hamiltonian. If the perturbation is sufficiently weak, we can write them as power series in λ: In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

where

and

When λ = 0, these reduce to the unperturbed values, which are the first term in each series. Since the perturbation is weak, the energy levels and eigenstates should not deviate too much from their unperturbed values, and the terms should rapidly become smaller as we go to higher order.

Plugging the power series into the Schrödinger equation, we obtain

Expanding this equation and comparing coefficients of each power of λ results in an infinite series of simultaneous equations. The zeroth-order equation is simply the Schrödinger equation for the unperturbed system. The first-order equation is In mathematics, simultaneous equations are a set of equations where variables are shared. ...

Multiply through by <n⁽⁰⁾|. The first term on the left-hand side cancels with the first term on the right-hand side. (Recall, the unperturbed Hamiltonian is hermitian). This leads to the first-order energy shift: A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

This is simply the expectation 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⁽⁰⁾>, 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⁽⁰⁾|V|n⁽⁰⁾>. However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as |n⁽⁰⁾>. These further shifts are given by the second and higher order corrections to the energy. In quantum mechanics, the expectation value is the predicted mean value of the result of an experiment. ...

Before we compute the corrections to the energy eigenstate, we need to address the issue of normalization. We may suppose <n⁽⁰⁾|n⁽⁰⁾>=1, but perturbation theory assumes we also have <n|n>=1. It follows that at first order in λ, we must have <n⁽⁰⁾|n⁽¹⁾>+<n⁽¹⁾|n⁽⁰⁾>=0. Since the overall phase is not determined in quantum mechanics, without loss of generality, we may assume <n⁽⁰⁾|n> is purely real. Therefore, <n⁽⁰⁾|n⁽¹⁾>=<n⁽¹⁾|n⁽⁰⁾>, and we deduce

To obtain the first-order correction to the energy eigenstate, we insert our expression for the first-order energy correction back into the result shown above of equating the first-order coefficients of λ. We then make use of the resolution of the identity, It has been suggested that this article or section be merged with Borel functional calculus. ...

where is in the orthogonal complement of . The result is In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ...

For the moment, suppose that the zeroth-order energy level is not degenerate, i.e. there is no eigenstate of H₀ in the orthogonal complement of with the energy . We multiply through by <k⁽⁰⁾|, which gives The energy levels of two or more physical states are said to be degenerate when they have the same value. ...

and hence the component of the first-order correction along |k⁽⁰⁾> since by assumption . In total we get

The first-order change in the n-th energy eigenket has a contribution from each of the energy eigenstates k ≠ n. Each term is proportional to the matrix element <k⁽⁰⁾|V|n⁽⁰⁾>, which is a measure of how much the perturbation mixes eigenstate n with eigenstate k; it is also inversely proportional to the energy difference between eigenstates k and n, which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies. We see also that the expression is singular if any of these states have the same energy as state n, which is why we assumed that there is no degeneracy.

Second order and higher corrections

We can find the higher-order deviations by a similar procedure, though the calculations become quite tedious with our current formulation. Our normalization prescription gives that 2<n⁽⁰⁾|n⁽²⁾>+<n⁽¹⁾|n⁽¹⁾>=0. Up to second order, the expressions for the energies and (normalized) eigenstates are:

Extending the process further, the third order energy correction can be shown to be ^[3]

Effects of degeneracy

Suppose that two or more energy eigenstates are degenerate. Our above calculation for the first-order energy shift is unaffected, but the calculation of the change in the eigenstate is problematic because the operator The energy levels of two or more physical states are said to be degenerate when they have the same value. ...

does not have a well-defined inverse.

This is actually a conceptual, rather than mathematical, problem. Imagine that we have two or more perturbed eigenstates with different energies, which are continuously generated from an equal number of unperturbed eigenstates that are degenerate. Let D denote the subspace spanned by these degenerate eigenstates. The problem lies in the fact that there is no unique way to choose a basis of energy eigenstates for the unperturbed system. In particular, we could construct a different basis for D by choosing different linear combinations of the spanning eigenstates. In such a basis, the unperturbed eigenstates would not continuously generate the perturbed eigenstates.

We thus see that, in the presence of degeneracy, perturbation theory does not work with an arbitrary choice of energy basis. We must instead choose a basis so that the perturbation Hamiltonian is diagonal in the degenerate subspace D. In other words,

In that case, our equation for the first-order deviation in the energy eigenstate reduces to

The operator on the left hand side is not singular when applied to eigenstates outside D, so we can write

Time-dependent perturbation theory

Time-dependent perturbation theory, developed by Paul Dirac, studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian H₀. Since the perturbed Hamiltonian is time-dependent, so are its energy levels and eigenstates. Therefore, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory. We are interested in the following quantities: Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...

• The time-dependent expected value of some observable A, for a given initial state.
• The time-dependent amplitudes of those quantum states that are energy eigenkets (eigenvectors) in the unperturbed system.

The first quantity is important because it gives rise to the classical result of an A measurement performed on a macroscopic number of copies of the perturbed system. For example, we could take A to be the displacement in the x-direction of the electron in a hydrogen atom, in which case the expected value, when multiplied by an appropriate coefficient, gives the time-dependent electrical polarization of a hydrogen gas. With an appropriate choice of perturbation (i.e. an oscillating electric potential), this allows us to calculate the AC permittivity of the gas. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

The second quantity looks at the time-dependent probability of occupation for each eigenstate. This is particularly useful in laser physics, where one is interested in the populations of different atomic states in a gas when a time-dependent electric field is applied. These probabilities are also useful for calculating the "quantum broadening" of spectral lines (see line broadening). For other uses, see Laser (disambiguation). ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ...

We will briefly examine the ideas behind Dirac's formulation of time-dependent perturbation theory. Choose an energy basis {|n>} for the unperturbed system. (We will drop the (0) superscripts for the eigenstates, because it is not meaningful to speak of energy levels and eigenstates for the perturbed system.)

If the unperturbed system is in eigenstate |j> at time t = 0, its state at subsequent times varies only by a phase (we are following the Schrödinger picture, where state vectors evolve in time and operators are constant): This article is about a portion of a periodic process. ... Heisenbergs form for the equations of motion We have seen that in Schrödingers scheme the dynamical variables of the system remain fixed during a period of undisturbed motion. ...

We now introduce a time-dependent perturbing Hamiltonian V(t). The Hamiltonian of the perturbed system is

H = H₀ + V(t)

Let |ψ(t)> denote the quantum state of the perturbed system at time t. It obeys the time-dependent Schrödinger equation,

The quantum state at each instant can be expressed as a linear combination of the eigenbasis {|n>}. We can write the linear combination as

where the c_n(t)s are undetermined complex functions of t which we will refer to as amplitudes (strictly speaking, they are the amplitudes in the Dirac picture). We have explicitly extracted the exponential phase factors exp(-iE_nt/h) on the right hand side. This is only a matter of convention, and may be done without loss of generality. The reason we go to this trouble is that when the system starts in the state |j> and no perturbation is present, the amplitudes have the convenient property that, for all t, c_j(t) = 1 and c_n(t) = 0 if n≠j. A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1] Every complex number can be... In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. ...

The absolute square of the amplitude c_n(t) is the probability that the system is in state n at time t, since

Plugging into the Schrödinger equation and using the fact that ∂/∂t acts by a chain rule, we obtain In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

By resolving the identity in front of V, this can be reduced to a set of partial differential equations for the amplitudes: In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

The matrix elements of V play a similar role as in time-independent perturbation theory, being proportional to the rate at which amplitudes are shifted between states. Note, however, that the direction of the shift is modified by the exponential phase factor. Over times much longer than the energy difference E_k-E_n, the phase winds many times. If the time-dependence of V is sufficiently slow, this may cause the state amplitudes to oscillate. Such oscillations are useful for managing radiative transitions in a laser. For other uses, see Laser (disambiguation). ...

Up to this point, we have made no approximations, so this set of differential equations is exact. By supplying appropriate initial values c_n(0), we could in principle find an exact (i.e. non-perturbative) solution. This is easily done when there are only two energy levels (n = 1, 2), and the solution is useful for modelling systems like the ammonia molecule. However, exact solutions are difficult to find when there are many energy levels, and one instead looks for perturbative solutions, which may be obtained by putting the equations in an integral form: For other uses, see Ammonia (disambiguation). ...

By repeatedly substituting this expression for c_n back into right hand side, we get an iterative solution

where, for example, the first-order term is

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. In quantum physics, Fermis golden rule is a way to calculate the transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system into a continuum of energy eigenstates, due to a perturbation. ... It has been suggested that this article or section be merged with Dyson operator. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ...

See also

• Fermi's golden rule

In quantum physics, Fermis golden rule is a way to calculate the transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system into a continuum of energy eigenstates, due to a perturbation. ...

References

1. ^ E. Schrödinger, Annalen der Physik, Vierte Folge, Band 80, p. 437 (1926)
2. ^ J. W. S. Rayleigh, Theory of Sound, 2nd edition Vol. I, pp 115-118, Macmillan, London (1894)
3. ^ L. D. Landau, E. M. Lifschitz, ``Quantum Mechanics: Non-relativistic Theory", 3rd ed.