In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, perturbation theory is a set of approximation schemes directly related to mathematical
perturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbatio ...
for describing a complicated
quantum system
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing"
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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 level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The te ...
s and
eigenstates
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
) can be expressed as "corrections" to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as
asymptotic series In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system.
Approximate Hamiltonians
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
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
for
Hamiltonians of even moderate complexity. The Hamiltonians to which we know exact solutions, such as the
hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
, the
and the
particle in a box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
, 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.
Applying perturbation theory
Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem.
For example, by adding a perturbative
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
to the quantum mechanical model of the hydrogen atom, tiny shifts in the
spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
s of hydrogen caused by the presence of an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
(the
Stark effect
The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several compon ...
) can be calculated. This is only approximate because the sum of a
Coulomb potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
with a linear potential is unstable (has no true bound states) although the
tunneling time
Quantum tunnelling, also known as tunneling ( US) is a quantum mechanical phenomenon whereby a wavefunction can propagate through a potential barrier.
The transmission through the barrier can be finite and depends exponentially on the barrier h ...
(
decay rate
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
) is very long. This instability shows up as a broadening of the energy spectrum lines, which perturbation theory fails to reproduce 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 mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
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 In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
). There exist ways to convert them into convergent series, which can be evaluated for large-expansion parameters, most efficiently by the
variational method
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. Even convergent perturbations can converge to the wrong answer and divergent perturbations expansions can sometimes give good results at lower order.
In the theory of
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED), in which the
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
–
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
interaction is treated perturbatively, the calculation of the electron's
magnetic moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
has been found to agree with experiment to eleven decimal places. In QED and other
quantum field theories
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
, special calculation techniques known as
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s are used to systematically sum the power series terms.
Limitations
Large perturbations
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
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
, for instance, the interaction of
quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s with the
gluon
A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...
field cannot be treated perturbatively at low energies because the
coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
(the expansion parameter) becomes too large, violating the requirement that corrections must be small.
Non-adiabatic states
Perturbation theory also fails to describe states that are not generated
adiabatically from the "free model", including
bound state
Bound or bounds may refer to:
Mathematics
* Bound variable
* Upper and lower bounds, observed limits of mathematical functions
Physics
* Bound state, a particle that has a tendency to remain localized in one or more regions of space
Geography
*B ...
s and various collective phenomena such as
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
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
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
, in which the
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
-mediated attraction between
conduction electron
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in wh ...
s leads to the formation of correlated electron pairs known as
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper ...
s. When faced with such systems, one usually turns to other approximation schemes, such as the
variational method
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
and the
WKB approximation
In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...
. This is because there is no analogue of a
bound particle
Bound or bounds may refer to:
Mathematics
* Bound variable
* Upper and lower bounds, observed limits of mathematical functions
Physics
* Bound state, a particle that has a tendency to remain localized in one or more regions of space
Geography
*B ...
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 . Perturbation theory can only detect solutions "close" to the unperturbed solution, even if there are other solutions for which the perturbative expansion is not valid.
Difficult computations
The problem of non-perturbative systems has been somewhat alleviated by the advent of modern
computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
s. It has become practical to obtain numerical non-perturbative solutions for certain problems, using methods such as
density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
. These advances have been of particular benefit to the field of
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
. Computers have also been used to carry out perturbation theory calculations to extraordinarily high levels of precision, which has proven important in
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
for generating theoretical results that can be compared with experiment.
Time-independent perturbation theory
Time-independent perturbation theory is one of two categories of perturbation theory, the other being time-dependent perturbation (see next section). In time-independent perturbation theory, the perturbation Hamiltonian is static (i.e., possesses no time dependence). Time-independent perturbation theory was presented by
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theory ...
in a 1926 paper, shortly after he produced his theories in wave mechanics. In this paper Schrödinger referred to earlier work of
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
, 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.
First order corrections
The process begins with an unperturbed Hamiltonian , which is assumed to have no time dependence. It has known energy levels and
eigenstates
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
, arising from the time-independent
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
:
:
For simplicity, it is assumed that the energies are discrete. The superscripts denote that these quantities are associated with the unperturbed system. Note the use of
bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
.
A perturbation is then introduced to the Hamiltonian. Let be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. Thus, is formally a
Hermitian operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
. Let be a dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is:
:
The energy levels and eigenstates of the perturbed Hamiltonian are again given by the time-independent Schrödinger equation,
:
The objective is to express and
in terms of the energy levels and eigenstates of the old Hamiltonian. If the perturbation is sufficiently weak, they can be written as a (Maclaurin)
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
in ,
:
where
:
When , 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 the order is increased.
Substituting the power series expansion into the Schrödinger equation produces:
Expanding this equation and comparing coefficients of each power of results in an infinite series of
simultaneous equation
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
s. The zeroth-order equation is simply the Schrödinger equation for the unperturbed system,
:
The first-order equation is
:
Operating through by
, the first term on the left-hand side cancels the first term on the right-hand side. (Recall, the unperturbed Hamiltonian is
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
). This leads to the first-order energy shift,
:
This is simply the
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of the perturbation Hamiltonian while the system is in the unperturbed eigenstate.
This result can be interpreted in the following way: supposing that the perturbation is applied, but the system is kept in the quantum state
, which is a valid quantum state though no longer an energy eigenstate. The perturbation causes the average energy of this state to increase by
. However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as ''
''. These further shifts are given by the second and higher order corrections to the energy.
Before corrections to the energy eigenstate are computed, the issue of normalization must be addressed. Supposing that
:
but perturbation theory also assumes that
.
Then at first order in , the following must be true:
:
:
:
Since the overall phase is not determined in quantum mechanics,
without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate ...
, in time-independent theory it can be assumed that
is purely real. Therefore,
:
leading to
:
To obtain the first-order correction to the energy eigenstate, the expression for the first-order energy correction is inserted back into the result shown above, equating the first-order coefficients of . Then by using the
resolution of the identity
In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. ...
:
:
where the
are in the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of
, i.e., the other eigenvectors.
The first-order equation may thus be expressed as
:
Supposing that the zeroth-order energy level is not
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
, i.e. that there is no eigenstate of in the orthogonal complement of
with the energy
. After renaming the summation dummy index above as
, any
can be chosen and multiplying the first-order equation through by
gives
:
The above
also gives us the component of the first-order correction along
.
Thus, in total, the result is,
:
The first-order change in the -th energy eigenket has a contribution from each of the energy eigenstates . Each term is proportional to the matrix element
, which is a measure of how much the perturbation mixes eigenstate with eigenstate ; it is also inversely proportional to the energy difference between eigenstates and , which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies. The expression is singular if any of these states have the same energy as state , which is why it was assumed that there is no degeneracy. The above formula for the perturbed eigenstates also implies that the perturbation theory can be legitimately used only when the absolute magnitude of the matrix elements of the perturbation is small compared with the corresponding differences in the unperturbed energy levels, i.e.,
Second-order and higher-order 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
:
Up to second order, the expressions for the energies and (normalized) eigenstates are:
:
:
If an intermediate normalization is taken (it means, if we require that
), we obtain the same expression for the second-order correction to the wave function, except for the last term.
Extending the process further, the third-order energy correction can be shown to be
:
\\
E_n^ &= \frac-V_\frac \\
E_n^ &= \frac-\frac\frac-V_\frac-V_\frac+V_^2\frac \\
&=\frac-E_^\frac-2V_\frac +V_^2\frac \\
E_n^ &=\frac-\frac\frac-\frac\frac-\frac\frac \\
&\quad -V_\frac-V_\frac-V_\frac+V_\frac\frac+2V_\frac\frac \\
&\quad +V_^2\frac+V_^2\frac+V_^2\frac-V_^3\frac \\
&=\frac-2E_n^\frac-\frac\frac \\
&\quad +V_\left(-2\frac-\frac+\frac\frac+2E_n^\frac\right) \\
&\quad +V_^2\left(2\frac+\frac\right)-V_^3\frac
\end
and the states to fourth order can be written
:
All terms involved should be summed over such that the denominator does not vanish.
It is possible to relate the ''k''-th order correction to the energy ''E
n'' to the ''k''-point
connected correlation function In statistical mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It can often be obtained by summing over connected Feynman diagrams (the sum over all Feynman diagrams gives the correlation functio ...
of the perturbation ''V'' in the state
. For
, one has to consider the inverse
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
of the two-point correlator:
:
where
is the perturbing operator ''V'' in the interaction picture, evolving in Euclidean time. Then
:
Similar formulas exist to all orders in perturbation theory, allowing one to express
in terms of the inverse Laplace transform
of the connected correlation function
:
To be precise, if we write
:
then the ''k''-th order energy shift is given by
:
Effects of degeneracy
Suppose that two or more energy eigenstates of the unperturbed Hamiltonian are
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
. The first-order energy shift is not well defined, since there is no unique way to choose a basis of eigenstates for the unperturbed system. The various eigenstates for a given energy will perturb with different energies, or may well possess no continuous family of perturbations at all.
This is manifested in the calculation of the perturbed eigenstate via the fact that the operator
:
does not have a well-defined inverse.
Let denote the subspace spanned by these degenerate eigenstates. No matter how small the perturbation is, in the degenerate subspace the energy differences between the eigenstates of are non-zero, so complete mixing of at least some of these states is assured. Typically, the eigenvalues will split, and the eigenspaces will become simple (one-dimensional), or at least of smaller dimension than ''D''.
The successful perturbations will not be "small" relative to a poorly chosen basis of ''D''. Instead, we consider the perturbation "small" if the new eigenstate is close to the subspace . The new Hamiltonian must be diagonalized in , or a slight variation of ''D'', so to speak. These perturbed eigenstates in are now the basis for the perturbation expansion,
:
For the first-order perturbation, we need solve the perturbed Hamiltonian restricted to the degenerate subspace ,
:
simultaneously for all the degenerate eigenstates, where
are first-order corrections to the degenerate energy levels, and "small" is a vector of
orthogonal to ''D''. This amounts to diagonalizing the matrix
:
This procedure is approximate, since we neglected states outside the subspace ("small"). The splitting of degenerate energies
is generally observed. Although the splitting may be small,
, compared to the range of energies found in the system, it is crucial in understanding certain details, such as spectral lines in
Electron Spin Resonance
Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spi ...
experiments.
Higher-order corrections due to other eigenstates outside can be found in the same way as for the non-degenerate case,
:
The operator on the left-hand side is not singular when applied to eigenstates outside , so we can write
:
but the effect on the degenerate states is of
.
Near-degenerate states should also be treated similarly, when the original Hamiltonian splits aren't larger than the perturbation in the near-degenerate subspace. An application is found in the
nearly free electron model
In solid-state physics, the nearly free electron model (or NFE model) or quasi-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model i ...
, where near-degeneracy, treated properly, gives rise to an energy gap even for small perturbations. Other eigenstates will only shift the absolute energy of all near-degenerate states simultaneously.
Degeneracy lifted to first order
Let us consider degenerate energy eigenstates and a perturbation that completely lifts the degeneracy to first order of correction.
The perturbed Hamiltonian is denoted as
:
where
is the unperturbed Hamiltonian,
is the perturbation operator, and
is the parameter of the perturbation.
Let us focus on the degeneracy of the
-th unperturbed energy
. We will denote the unperturbed states in this degenerate subspace as
and the other unperturbed states as
, where
is the index of the unperturbed state in the degenerate subspace and
represents all other energy eigenstates with energies different from
. The eventual degeneracy among the other states with
does not change our arguments. All states
with various values of
share the same energy
when there is no perturbation, i.e., when
. The energies
of the other states
with
are all different from
, but not necessarily unique, i.e. not necessarily always different among themselves.
By
and
, we denote the matrix elements of the perturbation operator
in the basis of the unperturbed eigenstates. We assume that the basis vectors
in the degenerate subspace are chosen such that the matrix elements
are diagonal. Assuming also that the degeneracy is completely lifted to the first order, i.e. that
if
, we have the following formulae for the energy correction to the second order in
:
and for the state correction to the first order in
:
Notice that here the first order correction to the state is orthogonal to the unperturbed state,
:
Generalization to multi-parameter case
The generalization of time-independent perturbation theory to the case where there are multiple small parameters
in place of λ can be formulated more systematically using the language of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, which basically defines the derivatives of the quantum states and calculates the perturbative corrections by taking derivatives iteratively at the unperturbed point.
Hamiltonian and force operator
From the differential geometric point of view, a parameterized Hamiltonian is considered as a function defined on the parameter
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
that maps each particular set of parameters
to an Hermitian operator that acts on the Hilbert space. The parameters here can be external field, interaction strength, or driving parameters in the
quantum phase transition
In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases (phases of matter at zero temperature). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physic ...
. Let and
be the -th eigenenergy and eigenstate of respectively. In the language of differential geometry, the states
form a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
over the parameter manifold, on which derivatives of these states can be defined. The perturbation theory is to answer the following question: given
and
at an unperturbed reference point
, how to estimate the and
at close to that reference point.
Without loss of generality, the coordinate system can be shifted, such that the reference point
is set to be the origin. The following linearly parameterized Hamiltonian is frequently used
:
If the parameters are considered as generalized coordinates, then should be identified as the generalized force operators related to those coordinates. Different indices label the different forces along different directions in the parameter manifold. For example, if denotes the external magnetic field in the -direction, then should be the magnetization in the same direction.
Perturbation theory as power series expansion
The validity of perturbation theory lies on the adiabatic assumption, which assumes the eigenenergies and eigenstates of the Hamiltonian are smooth functions of parameters such that their values in the vicinity region can be calculated in power series (like
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
) of the parameters:
:
Here denotes the derivative with respect to . When applying to the state
, it should be understood as the
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...
if the vector bundle is equipped with non-vanishing
connection. All the terms on the right-hand-side of the series are evaluated at , e.g. and
. This convention will be adopted throughout this subsection, that all functions without the parameter dependence explicitly stated are assumed to be evaluated at the origin. The power series may converge slowly or even not converge when the energy levels are close to each other. The adiabatic assumption breaks down when there is energy level degeneracy, and hence the perturbation theory is not applicable in that case.
Hellmann–Feynman theorems
The above power series expansion can be readily evaluated if there is a systematic approach to calculate the derivates to any order. Using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, the derivatives can be broken down to the single derivative on either the energy or the state. The
Hellmann–Feynman theorem
In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theore ...
s are used to calculate these single derivatives. The first Hellmann–Feynman theorem gives the derivative of the energy,
:
The second Hellmann–Feynman theorem gives the derivative of the state (resolved by the complete basis with m ≠ n),
:
For the linearly parameterized Hamiltonian, simply stands for the generalized force operator .
The theorems can be simply derived by applying the differential operator to both sides of the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
which reads
:
Then overlap with the state
from left and make use of the Schrödinger equation
again,
:
Given that the eigenstates of the Hamiltonian always form an orthonormal basis
, the cases of and can be discussed separately. The first case will lead to the first theorem and the second case to the second theorem, which can be shown immediately by rearranging the terms. With the differential rules given by the Hellmann–Feynman theorems, the perturbative correction to the energies and states can be calculated systematically.
Correction of energy and state
To the second order, the energy correction reads
:
where
denotes the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
function.
The first order derivative is given by the first Hellmann–Feynman theorem directly. To obtain the second order derivative , simply applying the differential operator to the result of the first order derivative
, which reads
:
Note that for linearly parameterized Hamiltonian, there is no second derivative on the operator level. Resolve the derivative of state by inserting the complete set of basis,
:
then all parts can be calculated using the Hellmann–Feynman theorems. In terms of Lie derivatives,
according to the definition of the connection for the vector bundle. Therefore, the case can be excluded from the summation, which avoids the singularity of the energy denominator. The same procedure can be carried on for higher order derivatives, from which higher order corrections are obtained.
The same computational scheme is applicable for the correction of states. The result to the second order is as follows
:
Both energy derivatives and state derivatives will be involved in deduction. Whenever a state derivative is encountered, resolve it by inserting the complete set of basis, then the Hellmann-Feynman theorem is applicable. Because differentiation can be calculated systematically, the series expansion approach to the perturbative corrections can be coded on computers with symbolic processing software like
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
.
Effective Hamiltonian
Let be the Hamiltonian completely restricted either in the low-energy subspace
or in the high-energy subspace
, such that there is no matrix element in connecting the low- and the high-energy subspaces, i.e.
if
. Let be the coupling terms connecting the subspaces. Then when the high energy degrees of freedoms are integrated out, the effective Hamiltonian in the low energy subspace reads
:
Here
are restricted in the low energy subspace. The above result can be derived by power series expansion of
.
In a formal way it is possible to define an effective Hamiltonian that gives exactly the low-lying energy states and wavefunctions. In practice, some kind of approximation (perturbation theory) is generally required.
Time-dependent perturbation theory
Method of variation of constants
Time-dependent perturbation theory, developed by
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
, studies the effect of a time-dependent perturbation applied to a time-independent Hamiltonian
0.
Since the perturbed Hamiltonian is time-dependent, so are its energy levels and eigenstates. Thus, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory. One is interested in the following quantities:
* The time-dependent
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of some observable , for a given initial state.
* The time-dependent expansion coefficients (
w.r.t. a given time-dependent state) of those basis 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 measurement performed on a macroscopic number of copies of the perturbed system. For example, we could take to be the displacement in the -direction of the electron in a hydrogen atom, in which case the expected value, when multiplied by an appropriate coefficient, gives the time-dependent
dielectric polarization
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
of a hydrogen gas. With an appropriate choice of perturbation (i.e. an oscillating electric potential), this allows one to calculate the AC
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the gas.
The second quantity looks at the time-dependent probability of occupation for each eigenstate. This is particularly useful in
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
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 line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
s (see
line broadening
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
) and
particle decay
In particle physics, particle decay is the spontaneous process of one unstable subatomic particle transforming into multiple other particles. The particles created in this process (the ''final state'') must each be less massive than the original, ...
in
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
and
nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...
.
We will briefly examine the method behind Dirac's formulation of time-dependent perturbation theory. Choose an energy basis
for the unperturbed system. (We drop the (0) superscripts for the eigenstates, because it is not useful to speak of energy levels and eigenstates for the perturbed system.)
If the unperturbed system is an eigenstate (of the Hamiltonian)
at time = 0, its state at subsequent times varies only by a
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
(in the
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
, where state vectors evolve in time and operators are constant),
:
Now, introduce a time-dependent perturbing Hamiltonian . The Hamiltonian of the perturbed system is
:
Let
denote the quantum state of the perturbed system at time . It obeys the time-dependent Schrödinger equation,
:
The quantum state at each instant can be expressed as a linear combination of the complete eigenbasis of
:
where the s are to be determined
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
functions of which we will refer to as amplitudes (strictly speaking, they are the amplitudes in the
Dirac picture
In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) is an intermediate Dynamical pictures, representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures ...
).
We have explicitly extracted the exponential phase factors
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
and no perturbation is present, the amplitudes have the convenient property that, for all ,
= 1 and = 0 if .
The square of the absolute amplitude is the probability that the system is in state at time , since
:
Plugging into the Schrödinger equation and using the fact that ∂/∂''t'' acts by a
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
, one obtains
:
By resolving the identity in front of and multiplying through by the
bra
A bra, short for brassiere or brassière (, or ; ), is a form-fitting undergarment that is primarily used to support and cover breasts. It can serve a range of other practical and aesthetic purposes, including enhancing or reducing the appea ...
on the left, this can be reduced to a set of coupled
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s for the amplitudes,
:
where we have used equation () to evaluate the sum on in the second term, then used the fact that
.
The matrix elements of 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 , the phase winds around 0 several times. If the time-dependence of is sufficiently slow, this may cause the state amplitudes to oscillate. ( E.g., such oscillations are useful for managing radiative transitions in a
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
.)
Up to this point, we have made no approximations, so this set of differential equations is exact. By supplying appropriate initial values , we could in principle find an exact (i.e., non-perturbative) solution. This is easily done when there are only two energy levels ( = 1, 2), and this solution is useful for modelling systems like the
ammonia
Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogenous was ...
molecule.
However, exact solutions are difficult to find when there are many energy levels, and one instead looks for perturbative solutions. These may be obtained by expressing the equations in an integral form,
:
Repeatedly substituting this expression for back into right hand side, yields an iterative solution,
:
where, for example, the first-order term is
:
To the same approximation, the summation in the above expression can be removed since in the unperturbed state
so that we have
:
Several further results follow from this, such as
Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of ...
, which relates the rate of transitions between quantum states to the density of states at particular energies; or the
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
, obtained by applying the iterative method to the
time evolution operator
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
, which is one of the starting points for the method of
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s.
Method of Dyson series
Time-dependent perturbations can be reorganized through the technique of the
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
. The
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
:
has the formal solution
:
where is the time ordering operator,
:
Thus, the exponential represents the following
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
,
:
Note that in the second term, the 1/2! factor exactly cancels the double contribution due to the time-ordering operator, etc.
Consider the following perturbation problem
:
assuming that the parameter is small and that the problem
has been solved.
Perform the following unitary transformation to the
interaction picture (or Dirac picture),
:
Consequently, the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
simplifies to
:
so it is solved through the above
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
,
:
as a perturbation series with small .
Using the solution of the unperturbed problem
and
(for the sake of simplicity assume a pure discrete spectrum), yields, to first order,
:
Thus, the system, initially in the unperturbed state
, by dint of the perturbation can go into the state
. The corresponding transition probability amplitude to first order is
:
as detailed in the previous section——while the corresponding transition probability to a continuum is furnished by
Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of ...
.
As an aside, note that time-independent perturbation theory is also organized inside this time-dependent perturbation theory Dyson series. To see this, write the unitary evolution operator, obtained from the above
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
, as
:
and take the perturbation to be time-independent.
Using the identity resolution
:
with
for a pure discrete spectrum, write
:
It is evident that, at second order, one must sum on all the intermediate states. Assume
and the asymptotic limit of larger times. This means that, at each contribution of the perturbation series, one has to add a multiplicative factor
in the integrands for arbitrarily small. Thus the limit gives back the final state of the system by eliminating all oscillating terms, but keeping the secular ones. The integrals are thus computable, and, separating the diagonal terms from the others yields
:
where the time secular series yields the eigenvalues of the perturbed problem specified above, recursively; whereas the remaining time-constant part yields the corrections to the stationary eigenfunctions also given above (
.)
The unitary evolution operator is applicable to arbitrary eigenstates of the unperturbed problem and, in this case, yields a secular series that holds at small times.
Strong perturbation theory
In a similar way as for small perturbations, it is possible to develop a strong perturbation theory. Consider as usual the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
:
and we consider the question if a dual Dyson series exists that applies in the limit of a perturbation increasingly large. This question can be answered in an affirmative way
and the series is the well-known adiabatic series.
This approach is quite general and can be shown in the following way. Consider the perturbation problem
:
being . Our aim is to find a solution in the form
:
but a direct substitution into the above equation fails to produce useful results. This situation can be adjusted making a rescaling of the time variable as
producing the following meaningful equations
:
:
:
that can be solved once we know the solution of the
leading order equation. But we know that in this case we can use the
adiabatic approximation
In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
. When
does not depend on time one gets the
Wigner-Kirkwood series that is often used in
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. Indeed, in this case we introduce the unitary transformation
:
that defines a free picture as we are trying to eliminate the interaction term. Now, in dual way with respect to the small perturbations, we have to solve the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
:
and we see that the expansion parameter appears only into the exponential and so, the corresponding
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
, a dual Dyson series, is meaningful at large s and is
:
After the rescaling in time
we can see that this is indeed a series in
justifying in this way the name of dual Dyson series. The reason is that we have obtained this series simply interchanging and and we can go from one to another applying this exchange. This is called duality principle in perturbation theory. The choice
yields, as already said, a
Wigner-Kirkwood series that is a gradient expansion. The
Wigner-Kirkwood series is a semiclassical series with eigenvalues given exactly as for
WKB approximation
In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...
.
Examples
Example of first-order perturbation theory – ground-state energy of the quartic oscillator
Consider the quantum harmonic oscillator with the quartic potential perturbation and the Hamiltonian
:
The ground state of the harmonic oscillator is
:
(
), and the energy of unperturbed ground state is
:
Using the first-order correction formula, we get
:
or
:
Example of first- and second-order perturbation theory – quantum pendulum
Consider the quantum-mathematical pendulum with the Hamiltonian
:
with the potential energy
taken as the perturbation i.e.
:
The unperturbed normalized quantum wave functions are those of the rigid rotor and are given by
:
and the energies
:
The first-order energy correction to the rotor due to the potential energy is
:
Using the formula for the second-order correction, one gets
:
or
:
or
:
Potential energy as a perturbation
When the unperturbed state is a free motion of a particle with kinetic energy
, the solution of the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
:
corresponds to plane waves with wavenumber
. If there is a weak potential energy
present in the space, in the first approximation, the perturbed state is described by the equation
:
whose particular integral is
[Lifshitz, E. M., & LD and Sykes Landau (JB). (1965). Quantum Mechanics; Non-relativistic Theory. Pergamon Press.]
:
where
. In the two-dimensional case, the solution is
:
where
and
is the
Hankel function of the first kind. In the one-dimensional case, the solution is
:
where
.
Applications
*
Rabi cycle
In physics, the Rabi cycle (or Rabi flop) is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, ...
*
Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of ...
*
Muon spin spectroscopy
Muon spin spectroscopy, also known as µSR, is an experimental technique based on the implantation of spin-polarized muons in matter and on the detection of the influence of the atomic, molecular or crystalline surroundings on their spin motion. ...
*
Perturbed angular correlation
The perturbed γ-γ angular correlation, PAC for short or PAC-Spectroscopy, is a method of nuclear solid-state physics with which magnetic and electric fields in crystal structures can be measured. In doing so, electrical field gradients and the L ...
References
External links
* (lecture by
Barton Zwiebach
Barton Zwiebach (born ''Barton Zwiebach Cantor'', October 4, 1954) is a Peruvian string theorist and professor at the Massachusetts Institute of Technology.
Work
Zwiebach's undergraduate work was in Electrical Engineering at the Universidad Nac ...
)
*
{{Authority control
Mathematical physics