Adiabatic theorem

Adiabatic theorem This article is about the adiabatic theorem in quantum mechanics. For adiabatic processes in thermodynamics, see adiabatic process.

The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was 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.[1] 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.

Contents 1 Diabatic vs. adiabatic processes 1.1 Comparison with the adiabatic concept in thermodynamics 2 Example systems 2.1 Simple pendulum 2.2 Quantum harmonic oscillator 2.3 Avoided curve crossing 3 Mathematical statement 3.1 Proofs 4 Example applications 5 Deriving conditions for diabatic vs adiabatic passage 5.1 Diabatic passage 5.2 Adiabatic passage 6 Calculating adiabatic passage probabilities 6.1 The Landau–Zener formula 6.2 The numerical approach 7 See also 8 References Diabatic vs. adiabatic processes Comparison Diabatic Adiabatic Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the spatial 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. 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.[2] At some initial time {displaystyle t_{0}} a quantum-mechanical system has an energy given by the Hamiltonian {displaystyle {hat {H}}(t_{0})} ; the system is in an eigenstate of {displaystyle {hat {H}}(t_{0})} labelled {displaystyle psi (x,t_{0})} . Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian {displaystyle {hat {H}}(t_{1})} at some later time {displaystyle t_{1}} . The system will evolve according to the time-dependent Schrödinger equation, to reach a final state {displaystyle psi (x,t_{1})} . The adiabatic theorem states that the modification to the system depends critically on the time {displaystyle tau =t_{1}-t_{0}} during which the modification takes place.

For a truly adiabatic process we require {displaystyle tau to infty } ; in this case the final state {displaystyle psi (x,t_{1})} will be an eigenstate of the final Hamiltonian {displaystyle {hat {H}}(t_{1})} , with a modified configuration: {displaystyle |psi (x,t_{1})|^{2}neq |psi (x,t_{0})|^{2}.} The degree to which a given change approximates an adiabatic process depends on both the energy separation between {displaystyle psi (x,t_{0})} and adjacent states, and the ratio of the interval {displaystyle tau } to the characteristic time-scale of the evolution of {displaystyle psi (x,t_{0})} for a time-independent Hamiltonian, {displaystyle tau _{int}=2pi hbar /E_{0}} , where {displaystyle E_{0}} is the energy of {displaystyle psi (x,t_{0})} .

Conversely, in the limit {displaystyle tau to 0} we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged: {displaystyle |psi (x,t_{1})|^{2}=|psi (x,t_{0})|^{2}.} The so-called "gap condition" included in Born and Fock's original definition given above refers to a requirement that the spectrum of {displaystyle {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 {displaystyle {hat {H}}(t_{1})} corresponds to {displaystyle psi (t_{0})} ). In 1999 J. E. Avron and A. Elgart reformulated the adiabatic theorem to adapt it to situations without a gap.[3] Comparison with the adiabatic concept in thermodynamics The term "adiabatic" is traditionally used in thermodynamics to describe processes without the exchange of heat between system and environment (see adiabatic 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 mechanics definition[4] 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.[5] 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 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.[6] Quantum harmonic oscillator Figure 1. Change in the probability density, {displaystyle |psi (t)|^{2}} , of a ground state quantum harmonic oscillator, due to an adiabatic increase in spring constant.

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 {displaystyle 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 {displaystyle k} is increased adiabatically {textstyle left({frac {dk}{dt}}to 0right)} then the system at time {displaystyle t} will be in an instantaneous eigenstate {displaystyle psi (t)} of the current Hamiltonian {displaystyle {hat {H}}(t)} , corresponding to the initial eigenstate of {displaystyle {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, {displaystyle n=0} , 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 {textstyle left({frac {dk}{dt}}to infty right)} 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 {displaystyle left(|psi (t)|^{2}=|psi (0)|^{2}right)} for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, {displaystyle {hat {H}}(t)} , that resembles the initial state. The final state is composed of a linear superposition of many different eigenstates of {displaystyle {hat {H}}(t)} which sum to reproduce the form of the initial state.

Avoided curve crossing Main article: Avoided crossing Figure 2. An avoided energy-level crossing in a two-level system subjected to an external magnetic field. Note the energies of the diabatic states, {displaystyle |1rangle } and {displaystyle |2rangle } and the eigenvalues of the Hamiltonian, giving the energies of the eigenstates {displaystyle |phi _{1}rangle } and {displaystyle |phi _{2}rangle } (the adiabatic states). (Actually, {displaystyle |phi _{1}rangle } and {displaystyle |phi _{2}rangle } should be switched in this picture.) For a more widely applicable example, consider a 2-level atom subjected to an external magnetic field.[7] The states, labelled {displaystyle |1rangle } and {displaystyle |2rangle } using bra–ket notation, 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: {displaystyle |Psi rangle =c_{1}(t)|1rangle +c_{2}(t)|2rangle .} With the field absent, the energetic separation of the diabatic states is equal to {displaystyle hbar omega _{0}} ; the energy of state {displaystyle |1rangle } increases with increasing magnetic field (a low-field-seeking state), while the energy of state {displaystyle |2rangle } 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 {displaystyle mathbf {H} ={begin{pmatrix}mu B(t)-hbar omega _{0}/2&a\a^{*}&hbar omega _{0}/2-mu B(t)end{pmatrix}}} where {displaystyle mu } is the magnetic moment of the atom, assumed to be the same for the two diabatic states, and {displaystyle a} is some time-independent coupling between the two states. The diagonal elements are the energies of the diabatic states ( {displaystyle E_{1}(t)} and {displaystyle E_{2}(t)} ), however, as {displaystyle mathbf {H} } is not a diagonal matrix, it is clear that these states are not eigenstates of the new Hamiltonian that includes the magnetic field contribution.

The eigenvectors of the matrix {displaystyle mathbf {H} } are the eigenstates of the system, which we will label {displaystyle |phi _{1}(t)rangle } and {displaystyle |phi _{2}(t)rangle } , with corresponding eigenvalues {displaystyle {begin{aligned}varepsilon _{1}(t)&=-{frac {1}{2}}{sqrt {4a^{2}+(hbar omega _{0}-2mu B(t))^{2}}}\[4pt]varepsilon _{2}(t)&=+{frac {1}{2}}{sqrt {4a^{2}+(hbar omega _{0}-2mu B(t))^{2}}}.end{aligned}}} It is important to realise that the eigenvalues {displaystyle varepsilon _{1}(t)} and {displaystyle varepsilon _{2}(t)} are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies {displaystyle E_{1}(t)} and {displaystyle E_{2}(t)} correspond to the expectation values for the energy of the system in the diabatic states {displaystyle |1rangle } and {displaystyle |2rangle } .

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 of the Hamiltonian cannot be degenerate, and thus we have an avoided crossing. If an atom is initially in state {displaystyle |phi _{2}(t_{0})rangle } in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field {textstyle left({frac {dB}{dt}}to 0right)} will ensure the system remains in an eigenstate of the Hamiltonian {displaystyle |phi _{2}(t)rangle } throughout the process (follows the red curve). A diabatic increase in magnetic field {textstyle left({frac {dB}{dt}}to infty right)} will ensure the system follows the diabatic path (the dotted blue line), such that the system undergoes a transition to state {displaystyle |phi _{1}(t_{1})rangle } . For finite magnetic field slew rates {textstyle left(0<{frac {dB}{dt}}

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