Editing Adiabatic theorem (section)

== 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#Classical mechanics – action variables|Adiabatic invariant]] page and here.<ref name=":2">{{cite web |author=Zwiebach |first=Barton |date=Spring 2018 |title=Classical analog: oscillator with slowly varying frequency |url=https://www.youtube.com/watch?v=DYJM_P4sG-c |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211221/DYJM_P4sG-c |archive-date=2021-12-21 |publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref>

=== Quantum harmonic oscillator ===

[[Image:HO adiabatic process.gif|thumb|right|300px|'''Figure 1.''' Change in the probability density, <math>|\psi(t)|^2</math>, of a ground state quantum harmonic oscillator, due to an adiabatic increase in spring constant.]]

The [[Classical physics|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]] <math>k</math> 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 (quantum mechanics)|Hamiltonian]].

If <math>k</math> is increased adiabatically <math display="inline">\left(\frac{dk}{dt} \to 0\right)</math> then the system at time <math>t</math> will be in an instantaneous eigenstate <math>\psi(t)</math> of the ''current'' Hamiltonian <math>\hat{H}(t)</math>, corresponding to the initial eigenstate of <math>\hat{H}(0)</math>. 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, <math>n = 0</math>, 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 <math display="inline">\left(\frac{dk}{dt} \to \infty\right)</math> 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 <math>\left(|\psi(t)|^2 = |\psi(0)|^2\right)</math> for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, <math>\hat{H}(t)</math>, that resembles the initial state. The final state is composed of a [[linear superposition]] of many different eigenstates of <math>\hat{H}(t)</math> which sum to reproduce the form of the initial state.

=== Avoided curve crossing ===
{{main|Avoided crossing}}
[[File:Avoided_crossing_in_linear_field.svg|thumb|right|300px|'''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, <math>|1\rangle</math> and <math>|2\rangle</math> and the [[eigenvalues]] of the Hamiltonian, giving the energies of the eigenstates <math>|\phi_1\rangle</math> and <math>|\phi_2\rangle</math> (the adiabatic states). (Actually, <math>|\phi_1\rangle</math> and <math>|\phi_2\rangle</math> should be switched in this picture.)]]

For a more widely applicable example, consider a 2-[[Energy level|level]] atom subjected to an external [[magnetic field]].<ref name="Stenholm">{{cite journal |author=Stenholm |first=Stig |author-link=Stig Stenholm |year=1994 |title=Quantum Dynamics of Simple Systems |journal=The 44th Scottish Universities Summer School in Physics |pages=267–313}}</ref> The states, labelled <math>|1\rangle</math> and <math>|2\rangle</math> using [[bra–ket notation]], can be thought of as atomic [[Azimuthal quantum number|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:

:<math>|\Psi\rangle = c_1(t)|1\rangle + c_2(t)|2\rangle.</math>

With the field absent, the energetic separation of the diabatic states is equal to <math>\hbar\omega_0</math>; the energy of state <math>|1\rangle</math> increases with increasing magnetic field (a low-field-seeking state), while the energy of state <math>|2\rangle</math> 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

:<math>\mathbf{H} = \begin{pmatrix}
\mu B(t)-\hbar\omega_0/2 & a \\
a^* & \hbar\omega_0/2-\mu B(t)
\end{pmatrix}</math>

where <math>\mu</math> is the [[magnetic moment]] of the atom, assumed to be the same for the two diabatic states, and <math>a</math> is some time-independent [[Angular momentum coupling|coupling]] between the two states. The diagonal elements are the energies of the diabatic states (<math>E_1(t)</math> and <math>E_2(t)</math>), however, as <math>\mathbf{H}</math> is not a [[diagonal matrix]], it is clear that these states are not eigenstates of <math>\mathbf{H}</math> due to the off-diagonal coupling constant.

The eigenvectors of the matrix <math>\mathbf{H}</math> are the eigenstates of the system, which we will label <math>|\phi_1(t)\rangle</math> and <math>|\phi_2(t)\rangle</math>, with corresponding eigenvalues
<math display="block">\begin{align}
\varepsilon_1(t) &= -\frac{1}{2}\sqrt{4a^2 + (\hbar\omega_0 - 2\mu B(t))^2} \\[4pt]
\varepsilon_2(t) &= +\frac{1}{2}\sqrt{4a^2 + (\hbar\omega_0 - 2\mu B(t))^2}.
\end{align}</math>

It is important to realise that the eigenvalues <math>\varepsilon_1(t)</math> and <math>\varepsilon_2(t)</math> are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies <math>E_1(t)</math> and <math>E_2(t)</math> correspond to the [[expectation value]]s for the energy of the system in the diabatic states <math>|1\rangle</math> and <math>|2\rangle</math>.

'''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 energy level|degenerate]], and thus we have an avoided crossing. If an atom is initially in state <math>|\phi_2(t_0)\rangle</math> in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field <math display="inline">\left(\frac{dB}{dt} \to 0\right)</math> will ensure the system remains in an eigenstate of the Hamiltonian <math>|\phi_2(t)\rangle</math> throughout the process (follows the red curve). A diabatic increase in magnetic field <math display="inline">\left(\frac{dB}{dt}\to \infty\right)</math> will ensure the system follows the diabatic path (the dotted blue line), such that the system undergoes a transition to state <math>|\phi_1(t_1)\rangle</math>. For finite magnetic field slew rates <math display="inline">\left(0 < \frac{dB}{dt} < \infty\right)</math> there will be a finite probability of finding the system in either of the two eigenstates. See [[#Calculating adiabatic passage probabilities|below]] for approaches to calculating these probabilities.

These results are extremely important in [[Atomic physics|atomic]] and [[molecular physics]] for control of the energy-state distribution in a population of atoms or molecules.