Adiabatic theorem
The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows: 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 formula_1 a quantum-mechanical system has an energy given by the Hamiltonian formula_2; the system is in an eigenstate of formula_2 labelled formula_4. Changing conditions  modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian formula_5 at some later time formula_6. The system will evolve according to the time-dependent Schrödinger equation, to reach a final state formula_7. The adiabatic theorem states that the modification to the system depends critically on the time formula_8 during which the modification takes place. For a truly adiabatic process we require formula_9; in this case the final state formula_7 will be an eigenstate of the final Hamiltonian formula_11, with a modified configuration: The degree to which a given change approximates an adiabatic process depends on both the  energy separation between formula_4 and adjacent states, and the ratio of the interval formula_14 to the characteristic time-scale of the evolution of formula_4 for a time-independent Hamiltonian, formula_16, where formula_17 is the energy of formula_4. Conversely, in the limit formula_19 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 of formula_21 is discrete and nondegenerate, such that there is no ambiguity in the ordering of the states (one can easily establish  which eigenstate of formula_5 "corresponds" to formula_23). 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 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 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 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 quantum harmonic oscillator as the spring constant  formula_24 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 formula_24 is increased adiabatically formula_26 then the system at time formula_27 will be in an instantaneous eigenstate formula_28 of the "current" Hamiltonian formula_29, corresponding to the initial eigenstate of formula_30. 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, formula_31 , 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 formula_32 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 formula_33 for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, formula_29, that resembles the initial state. The final state is composed of a linear superposition of many different  eigenstates of formula_29 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. The states, labelled formula_36 and formula_37 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: With the field absent, the energetic separation of the diabatic states is  equal to formula_39; the energy of state formula_36 increases with increasing magnetic field (a low-field-seeking state), while the energy of state formula_37 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 formula_43 is the magnetic moment of the atom, assumed to be the same for the two diabatic states, and formula_44 is some time-independent coupling between the two states. The diagonal elements are the energies of the diabatic states (formula_45 and formula_46), however, as formula_47  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 formula_47 are the eigenstates of the system, which we will label formula_49 and formula_50, with corresponding eigenvalues formula_51 It is important to realise that the eigenvalues formula_52 and formula_53 are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies formula_45 and formula_46 correspond to the expectation values for the energy of the system in the diabatic states formula_36 and formula_37. 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 formula_58 in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field formula_59 will ensure the system remains in an eigenstate of the Hamiltonian formula_50 throughout the process (follows the red curve). A diabatic increase in magnetic field formula_61 will ensure the system follows the diabatic path (the dotted blue  line), such that the system undergoes a transition to state formula_62. For finite magnetic field slew rates formula_63 there will be a finite probability of finding the system in either of the two eigenstates. See 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 formula_64 with instantaneous eigenstates formula_65 and corresponding energies formula_66, a quantum system evolves from the initial state formula_67 to the final state formula_68 where the coefficients undergo  the change of phase formula_69 with the dynamical phase formula_70 and geometric phase formula_71 In particular, formula_72, so if the system begins in an eigenstate of formula_73, it remains in an eigenstate of formula_64 during the evolution with a change of phase only. 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. This does explain many phenomena in the scope of: Deriving conditions for diabatic vs adiabatic passage. We will now pursue a more rigorous analysis. Making use of bra–ket notation, the state vector of the system at time formula_27 can be written where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the position operator It is instructive to examine the limiting cases, in which formula_14 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 formula_79, at time formula_1, to a final value formula_81, at time formula_6, where formula_8. The evolution of the system can be described in the Schrödinger picture by the time-evolution operator, defined by the integral equation which is equivalent to the Schrödinger equation. along with the initial condition formula_86. Given knowledge of the system wave function at formula_1, the evolution of the system up to a later time formula_27 can be obtained using The problem of determining the "adiabaticity" of a given process is equivalent to establishing the dependence of formula_90 on formula_14. 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 and using the definition formula_92, we have: We can expand formula_90 In the perturbative limit we can take just the first two terms and substitute them into our equation for formula_96, recognizing that is the system Hamiltonian, averaged over the interval formula_98, we have: After expanding the products and making the appropriate cancellations, we are left with: giving where formula_102 is the root mean square deviation of the system Hamiltonian  averaged over the interval of interest. The sudden approximation is valid when formula_103 (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 formula_19 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 formula_9 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 of formula_2, after a period formula_14 it will have passed into the "corresponding" eigenstate of formula_5. 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: formula_115 where formula_116 is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and formula_117 and formula_118 are the energies of the two diabatic (crossing) states. A large  formula_119 results in a large diabatic transition probability and vice versa. Using the Landau–Zener formula the probability, formula_120, of a diabatic transition is given by formula_121 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: formula_122 where formula_123  is a vector containing the adiabatic state amplitudes, formula_124 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: formula_125 for a system that began with formula_126.