Adiabatic theorem


Adiabatic theorem

The adiabatic theorem is an important concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928),cite journal |author=M. Born and V. A. Fock |title=Beweis des Adiabatensatzes |journal=Zeitschrift für Physik A Hadrons and Nuclei |volume=51 |issue=3-4 |pages=165–180 |year=1928 |url=http://www.springerlink.com/content/m4x427124n456704/fulltext.pdf|doi=|format=PDF] can be stated as follows:

:"A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum."

It may not be immediately clear from this formulation but the adiabatic theorem is, in fact, an extremely intuitive concept. Simply stated, a quantum mechanical system subjected to gradually changing external conditions can adapt its functional form, while in the case of rapidly varying conditions, there is no time for the functional form of the state to adapt, so the probability density remains unchanged.

The consequences of this apparently simple result are many, varied and extremely subtle. In order to make this clear we will begin with a fairly qualitative description, followed by a series of example systems, before undertaking a more rigorous analysis. Finally we will look at techniques used for adiabaticity calculations.

Diabatic vs. adiabatic processes

Diabatic process: Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the probability density remains unchanged. Typically there is no eigenstate of the final Hamiltonian with the same functional form as the initial state. The system ends in a linear combination of states that sum to reproduce the initial probability density.

Adiabatic process: Gradually changing conditions allow the system to adapt its configuration, hence the probability density is modified by the process. If the system starts in an eigenstate of the initial Hamiltonian, it will end in the "corresponding" eigenstate of the final Hamiltonian.cite journal |author=T. Kato |title=On the Adiabatic Theorem of Quantum Mechanics |journal=Journal of the Physical Society of Japan |volume=5 |issue=6 |pages=435–439 |year=1950 |url=http://jpsj.ipap.jp/link?JPSJ/5/435/pdf|doi=]

At some initial time scriptstyle{t_0} a quantum mechanical system has an energy given by the Hamiltonian scriptstyle{hat{H}(t_0)}; the system is in an eigenstate of scriptstyle{hat{H}(t_0)} labelled scriptstyle{psi(x,t_0)}. Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian scriptstyle{hat{H}(t_1)} at some later time scriptstyle{t_1}. The system will evolve according to the Schrödinger equation, to reach a final state scriptstyle{psi(x,t_1)}. The adiabatic theorem states that the modification to the system depends critically on the time scriptstyle{ au = t_1 - t_0} during which the modification takes place.

For a truly adiabatic process we require scriptstyle{ au ightarrow infty}; in this case the final state scriptstyle{psi(x,t_1)} will be an eigenstate of the final Hamiltonian scriptstyle{hat{H}(t_1)}, with a modified configuration:

:|psi(x,t_1)|^2 eq |psi(x,t_0)|^2.

The degree to which a given change approximates an adiabatic process depends on both the energy separation between scriptstyle{psi(x,t_0)} and adjacent states, and the ratio of the interval scriptstyle{ au} to the characteristic time-scale of the evolution of scriptstyle{psi(x,t_0)} for a time-independent Hamiltonian, scriptstyle{ au_{int} = 2pihbar/E_0}, where scriptstyle{E_0} is the energy of scriptstyle{psi(x,t_0)}.

Conversely, in the limit scriptstyle{ au ightarrow 0} we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged:

:|psi(x,t_1)|^2 = |psi(x,t_0)|^2quad.

The so called "gap condition" included in Born and Fock's original definition given above refers to a requirement that the spectrum of scriptstyle{hat{H is discrete and nondegenerate, such that there is no ambiguity in the ordering of the states (one can easily establish which eigenstate of scriptstyle{hat{H}(t_1)} "corresponds" to scriptstyle{psi(t_0)}). In 1990 J. E. Avron and A. Elgart reformulated the adiabatic theorem, eliminating the gap condition.cite journal |author=J. E. Avron and A. Elgart |title=Adiabatic Theorem without a Gap Condition |journal=Communications in Mathematical Physics |volume=203 |issue=2 |pages=445–463 |year=1999 |url=http://www.springerlink.com/content/ad0jyug24jg97nt6/fulltext.pdf|doi=10.1007/s002200050620|format=PDF]

Note that the term adiabatic is traditionally used in thermodynamics to describe processes without the exchange of heat between system and environment (see adiabatic process). The quantum mechanical definition is closer to the thermodynamical concept of a quasistatic process, and has no direct relation with heat exchange. These two different definitions can be the source of much confusion, especially when the two concepts (heat exchange and sufficiently slow processes) are present in a given problem.

Example systems

Simple pendulum

As an example, consider a pendulum 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. This is referred to as an adiabatic process.cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |year=2005 |publisher=Pearson Prentice Hall |location= |isbn=0-13-111892-7 |chapter=10 ]

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 quantum harmonic oscillator as the spring constant scriptstyle{k} is increased. Classically this is equivalent to increasing the stiffness of a spring; quantum mechanically the effect is a narrowing of the potential energy curve in the system Hamiltonian.

If scriptstyle{k} is increased adiabatically scriptstyle{left(frac{dk}{dt} ightarrow 0 ight)} then the system at time scriptstyle{t} will be in an instantaneous eigenstate scriptstyle{psi(t)} of the "current" Hamiltonian scriptstyle{hat{H}(t)}, corresponding to the initial eigenstate of scriptstyle{hat{H}(0)}. For the special case of a system, like the quantum harmonic oscillator, described by a single quantum number, this means the quantum number will remain unchanged. Figure 1 shows how a harmonic oscillator, initially in its ground state, scriptstyle{n = 1}, 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 scriptstyle{left(frac{dk}{dt} ightarrow infty ight)} 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 scriptstyle{left(|psi(t)|^2 = |psi(0)|^2 ight)} for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, scriptstyle{hat{H}(t)}, that resembles the initial state. In fact the final state is composed of a linear superposition of many different eigenstates of scriptstyle{hat{H}(t)} which sum to reproduce the form of the initial state.

Avoided curve crossing

For a more widely applicable example, consider a 2-level atom subjected to an external magnetic field.cite journal |author=S. Stenholm |title=Quantum Dynamics of Simple Systems |journal=The 44th Scottish Universities Summer School in Physics |volume= |issue= |pages=267–313 |year=1994 |url= |pmid= |doi=] The states, labelled scriptstyle = {a^2/hbar over left|frac{dq}{dt}frac{partial}{partial q}(E_2 - E_1) ight\ &= {a^2 over hbar|alpha\end{align}

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:

:ihbardot{underline{c^A(t) = mathbf{H}_A(t)underline{c}^A(t),

where scriptstyle{underline{c}^A(t)} is a vector containing the adiabatic state amplitudes, scriptstyle{mathbf{H}_A(t)} 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:

:P_D = |c^A_2(t_1)|^2quad

for a system that began with scriptstyle{|c^A_1(t_0)|^2 = 1}.

See also

* Landau–Zener formula
* Berry phase
* Quantum stirring, ratchets, and pumping
* Born-Oppenheimer approximation

References


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