The adiabatic theorem is a concept 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, ...
. Its original form, due to
Max Born
Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...
and
Vladimir Fock (1928), was stated as follows:
:''A physical system remains in its instantaneous
eigenstate
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 in ...
if a given
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
* Perturbat ...
is acting on it slowly enough and if there is a gap between the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
and the rest of the
Hamiltonian's
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
.''
In simpler terms, a quantum mechanical system subjected to gradually changing external conditions adapts its functional form, but when subjected to rapidly varying conditions there is insufficient time for the functional form to adapt, so the spatial probability density remains unchanged.
Diabatic vs. adiabatic processes
At some initial time
a quantum-mechanical system has an energy given by the Hamiltonian
; the system is in an eigenstate of
labelled
. Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian
at some later time
. The system will evolve according to the time-dependent
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 th ...
, to reach a final state
. The adiabatic theorem states that the modification to the system depends critically on the time
during which the modification takes place.
For a truly adiabatic process we require
; in this case the final state
will be an eigenstate of the final Hamiltonian
, with a modified configuration:
:
The degree to which a given change approximates an adiabatic process depends on both the energy separation between
and adjacent states, and the ratio of the interval
to the characteristic time-scale of the evolution of
for a time-independent Hamiltonian,
, where
is the energy of
.
Conversely, in the limit
we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged:
:
The so-called "gap condition" included in Born and Fock's original definition given above refers to a requirement that the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of
is
discrete and
nondegenerate, such that there is no ambiguity in the ordering of the states (one can easily establish which eigenstate of
''corresponds'' to
). In 1999 J. E. Avron and A. Elgart reformulated the adiabatic theorem to adapt it to situations without a gap.
Comparison with the adiabatic concept in thermodynamics
The term "adiabatic" is traditionally used in
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
to describe processes without the exchange of heat between system and environment (see
adiabatic process
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, ...
), more precisely these processes are usually faster than the timescale of heat exchange. (For example, a pressure wave is adiabatic with respect to a heat wave, which is not adiabatic.) Adiabatic in the context of thermodynamics is often used as a synonym for fast process.
The
classical and
quantum
In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizat ...
mechanics definition
is closer instead to the thermodynamical concept of a
quasistatic process, which are processes that are almost always at equilibrium (i.e. that are slower than the internal energy exchange interactions time scales, namely a "normal" atmospheric heat wave is quasi-static and a pressure wave is not). Adiabatic in the context of Mechanics is often used as a synonym for slow process.
In the quantum world adiabatic means for example that the time scale of electrons and photon interactions is much faster or almost instantaneous with respect to the average time scale of electrons and photon propagation. Therefore, we can model the interactions as a piece of continuous propagation of electrons and photons (i.e. states at equilibrium) plus a quantum jump between states (i.e. instantaneous).
The adiabatic theorem in this heuristic context tells essentially that quantum jumps are preferably avoided and the system tries to conserve the state and the quantum numbers.
The quantum mechanical concept of adiabatic is related to
adiabatic invariant, it is often used in the
old quantum theory and has no direct relation with heat exchange.
Example systems
Simple pendulum
As an example, consider a
pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
oscillating in a vertical plane. If the support is moved, the mode of oscillation of the pendulum will change. If the support is moved ''sufficiently slowly'', the motion of the pendulum relative to the support will remain unchanged. A gradual change in external conditions allows the system to adapt, such that it retains its initial character. The detailed classical example is available in the
Adiabatic invariant page and here.
Quantum harmonic oscillator
The
classical nature of a pendulum precludes a full description of the effects of the adiabatic theorem. As a further example consider a
as the
spring constant is increased. Classically this is equivalent to increasing the stiffness of a spring; quantum-mechanically the effect is a narrowing of the
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potenti ...
curve in the system
Hamiltonian.
If
is increased adiabatically
then the system at time
will be in an instantaneous eigenstate
of the ''current'' Hamiltonian
, corresponding to the initial eigenstate of
. For the special case of a system like the quantum harmonic oscillator described by a single
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can ...
, this means the quantum number will remain unchanged. Figure 1 shows how a harmonic oscillator, initially in its ground state,
, remains in the ground state as the potential energy curve is compressed; the functional form of the state adapting to the slowly varying conditions.
For a rapidly increased spring constant, the system undergoes a diabatic process
in which the system has no time to adapt its functional form to the changing conditions. While the final state must look identical to the initial state
for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian,
, that resembles the initial state. The final state is composed of a
linear superposition
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
of many different eigenstates of
which sum to reproduce the form of the initial state.
Avoided curve crossing
For a more widely applicable example, consider a 2-
level
Level or levels may refer to:
Engineering
*Level (instrument), a device used to measure true horizontal or relative heights
*Spirit level, an instrument designed to indicate whether a surface is horizontal or vertical
* Canal pound or level
*Reg ...
atom subjected to an external
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
.
The states, labelled
and
using
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. Mathem ...
, can be thought of as atomic
angular-momentum states, each with a particular geometry. For reasons that will become clear these states will henceforth be referred to as the diabatic states. The system wavefunction can be represented as a linear combination of the diabatic states:
:
With the field absent, the energetic separation of the diabatic states is equal to
; the energy of state
increases with increasing magnetic field (a low-field-seeking state), while the energy of state
decreases with increasing magnetic field (a high-field-seeking state). Assuming the magnetic-field dependence is linear, the
Hamiltonian matrix for the system with the field applied can be written
:
where
is the
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 electroma ...
of the atom, assumed to be the same for the two diabatic states, and
is some time-independent
coupling between the two states. The diagonal elements are the energies of the diabatic states (
and
), however, as
is not a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
, it is clear that these states are not eigenstates of the new Hamiltonian that includes the magnetic field contribution.
The eigenvectors of the matrix
are the eigenstates of the system, which we will label
and
, with corresponding eigenvalues
It is important to realise that the eigenvalues
and
are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies
and
correspond to the
expectation values for the energy of the system in the diabatic states
and
.
Figure 2 shows the dependence of the diabatic and adiabatic energies on the value of the magnetic field; note that for non-zero coupling the
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
of the Hamiltonian cannot be
degenerate, and thus we have an avoided crossing. If an atom is initially in state
in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field
will ensure the system remains in an eigenstate of the Hamiltonian
throughout the process (follows the red curve). A diabatic increase in magnetic field
will ensure the system follows the diabatic path (the dotted blue line), such that the system undergoes a transition to state
. For finite magnetic field slew rates
there will be a finite probability of finding the system in either of the two eigenstates. See
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
for approaches to calculating these probabilities.
These results are extremely important in
atomic and
molecular physics for control of the energy-state distribution in a population of atoms or molecules.
Mathematical statement
Under a slowly changing Hamiltonian
with instantaneous eigenstates
and corresponding energies
, a quantum system evolves from the initial state
to the final state
where the coefficients undergo the change of phase
with the dynamical phase
and
geometric phase
In particular,
, so if the system begins in an eigenstate of
, it remains in an eigenstate of
during the evolution with a change of phase only.
Proofs
:
:
:
Example applications
Often a solid crystal is modeled as a set of independent valence electrons moving in a mean perfectly periodic potential generated by a rigid lattice of ions. With the Adiabatic theorem we can also include instead the motion of the valence electrons across the crystal and the thermal motion of the ions as in the
Born–Oppenheimer approximation
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and elect ...
.
This does explain many phenomena in the scope of:
* thermodynamics: Temperature dependence of
specific heat
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
,
thermal expansion
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kin ...
,
melting
Melting, or fusion, is a physical process that results in the phase transition of a substance from a solid to a liquid. This occurs when the internal energy of the solid increases, typically by the application of heat or pressure, which in ...
* transport phenomena: the temperature dependence of
electric resistivity of
conductors, the temperature dependence of
electric conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
in
insulators, Some properties of low temperature
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 ...
* optics: optic
absorption
Absorption may refer to:
Chemistry and biology
*Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which s ...
in the
infrared
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of Light, visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from ...
for
ionic crystals,
Brillouin scattering Brillouin scattering (also known as Brillouin light scattering or BLS), named after Léon Brillouin, refers to the interaction of light with the material waves in a medium (e.g. electrostriction and magnetostriction). It is mediated by the refra ...
,
Raman scattering
Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by ...
Deriving conditions for diabatic vs adiabatic passage
We will now pursue a more rigorous analysis.
Making 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. Mathem ...
, the
state vector of the system at time
can be written
:
where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the
position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.
When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues ...
:
It is instructive to examine the limiting cases, in which
is very large (adiabatic, or gradual change) and very small (diabatic, or sudden change).
Consider a system Hamiltonian undergoing continuous change from an initial value
, at time
, to a final value
, at time
, where
. The evolution of the system can be described 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 ma ...
by the time-evolution operator, defined by the
integral equation
:
which is equivalent 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 th ...
.
:
along with the initial condition
. Given knowledge of the system
wave function at
, the evolution of the system up to a later time
can be obtained using
:
The problem of determining the ''adiabaticity'' of a given process is equivalent to establishing the dependence of
on
.
To determine the validity of the adiabatic approximation for a given process, one can calculate the probability of finding the system in a state other than that in which it started. Using
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. Mathem ...
and using the definition
, we have:
:
We can expand
:
In the
perturbative limit we can take just the first two terms and substitute them into our equation for
, recognizing that
:
is the system Hamiltonian, averaged over the interval
, we have:
:
After expanding the products and making the appropriate cancellations, we are left with:
:
giving
:
where
is the
root mean square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
deviation of the system Hamiltonian averaged over the interval of interest.
The sudden approximation is valid when
(the probability of finding the system in a state other than that in which is started approaches zero), thus the validity condition is given by
:
which is a statement of the
time-energy form of the Heisenberg uncertainty principle.
Diabatic passage
In the limit
we have infinitely rapid, or diabatic passage:
:
The functional form of the system remains unchanged:
:
This is sometimes referred to as the sudden approximation. The validity of the approximation for a given process can be characterized by the probability that the state of the system remains unchanged:
:
Adiabatic passage
In the limit
we have infinitely slow, or adiabatic passage. The system evolves, adapting its form to the changing conditions,
:
If the system is initially in an
eigenstate
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 in ...
of
, after a period
it will have passed into the ''corresponding'' eigenstate of
.
This is referred to as the adiabatic approximation. The validity of the approximation for a given process can be determined from the probability that the final state of the system is different from the initial state:
:
Calculating adiabatic passage probabilities
The Landau–Zener formula
In 1932 an analytic solution to the problem of calculating adiabatic transition probabilities was published separately by
Lev Landau and
Clarence Zener,
for the special case of a linearly changing perturbation in which the time-varying component does not couple the relevant states (hence the coupling in the diabatic Hamiltonian matrix is independent of time).
The key figure of merit in this approach is the Landau–Zener velocity:
where
is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and
and
are the energies of the two diabatic (crossing) states. A large
results in a large diabatic transition probability and vice versa.
Using the Landau–Zener formula the probability,
, of a diabatic transition is given by
The numerical approach
For a transition involving a nonlinear change in perturbation variable or time-dependent coupling between the diabatic states, the equations of motion for the system dynamics cannot be solved analytically. The diabatic transition probability can still be obtained using one of the wide variety of
numerical solution algorithms for ordinary differential equations.
The equations to be solved can be obtained from the time-dependent Schrödinger equation:
where
is a
vector containing the adiabatic state amplitudes,
is the time-dependent adiabatic Hamiltonian,
and the overdot represents a time derivative.
Comparison of the initial conditions used with the values of the state amplitudes following the transition can yield the diabatic transition probability. In particular, for a two-state system:
for a system that began with
.
See also
*
Landau–Zener formula
The Landau–Zener formula is an analytic solution to the equations of motion governing the transition dynamics of a two-state quantum system, with a time-dependent Hamiltonian varying such that the energy separation of the two states is a linea ...
*
Berry phase
*
Quantum stirring, ratchets, and pumping
*
Adiabatic quantum motor
*
Born–Oppenheimer approximation
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and elect ...
*
Diabatic
*
Eigenstate thermalization hypothesis
The eigenstate thermalization hypothesis (or ETH) is a set of ideas which purports to explain when and why an isolated quantum mechanical system can be accurately described using equilibrium statistical mechanics. In particular, it is devoted to ...
References
{{reflist, 2
Theorems in quantum mechanics