Editing Adiabatic theorem (section)

== Deriving conditions for diabatic vs adiabatic passage ==
{{Disputed section|errors in the technical section|date = January 2016}}
We will now pursue a more rigorous analysis.<ref name=Messiah>{{cite book |last=Messiah |first=Albert |title=Quantum Mechanics |year=1999 |publisher=Dover Publications |isbn=0-486-40924-4 |chapter=XVII }}</ref> Making use of [[bra–ket notation]], the [[Quantum state|state vector]] of the system at time <math>t</math> can be written

:<math>|\psi(t)\rangle = \sum_n c^A_n(t)e^{-iE_nt/\hbar}|\phi_n\rangle ,</math>

where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the [[position operator]]

:<math>\psi(x,t) = \langle x|\psi(t)\rangle .</math>

It is instructive to examine the limiting cases, in which <math>\tau</math> 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 <math>\hat{H}_0</math>, at time <math>t_0</math>, to a final value <math>\hat{H}_1</math>, at time <math>t_1</math>, where <math>\tau = t_1 - t_0</math>. The evolution of the system can be described in the [[Schrödinger picture]] by the time-evolution operator, defined by the [[integral equation]]

:<math>\hat{U}(t,t_0) = 1 - \frac{i}{\hbar}\int_{t_0}^t\hat{H}(t')\hat{U}(t',t_0)dt' ,</math>

which is equivalent to the [[Schrödinger equation]].

:<math>i\hbar\frac{\partial}{\partial t}\hat{U}(t,t_0) = \hat{H}(t)\hat{U}(t,t_0),</math>

along with the initial condition <math>\hat{U}(t_0,t_0) = 1</math>. Given knowledge of the system [[wave function]] at <math>t_0</math>, the evolution of the system up to a later time <math>t</math> can be obtained using

:<math>|\psi(t)\rangle = \hat{U}(t,t_0)|\psi(t_0)\rangle.</math>

The problem of determining the ''adiabaticity'' of a given process is equivalent to establishing the dependence of <math>\hat{U}(t_1,t_0)</math> on <math>\tau</math>.

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 <math>|0\rangle \equiv |\psi(t_0)\rangle</math>, we have:

:<math>\zeta = \langle 0|\hat{U}^\dagger(t_1,t_0)\hat{U}(t_1,t_0)|0\rangle - \langle 0|\hat{U}^\dagger(t_1,t_0)|0\rangle\langle 0 | \hat{U}(t_1,t_0) | 0 \rangle.</math>

We can expand <math>\hat{U}(t_1,t_0)</math>

:<math>\hat{U}(t_1,t_0) = 1 + {1 \over i\hbar} \int_{t_0}^{t_1}\hat{H}(t)dt + {1 \over (i\hbar)^2} \int_{t_0}^{t_1}dt' \int_{t_0}^{t'}dt'' \hat{H}(t')\hat{H}(t'') + \cdots.</math>

In the [[Perturbation theory|perturbative limit]] we can take just the first two terms and substitute them into our equation for <math>\zeta</math>, recognizing that

:<math>{1 \over \tau}\int_{t_0}^{t_1}\hat{H}(t)dt \equiv \bar{H}</math>

is the system Hamiltonian, averaged over the interval <math>t_0 \to t_1</math>, we have:

:<math>\zeta = \langle 0|(1 + \tfrac{i}{\hbar}\tau\bar{H})(1 - \tfrac{i}{\hbar}\tau\bar{H})|0\rangle - \langle 0|(1 + \tfrac{i}{\hbar}\tau\bar{H})|0\rangle \langle 0|(1 - \tfrac{i}{\hbar}\tau\bar{H})|0\rangle .</math>

After expanding the products and making the appropriate cancellations, we are left with:

:<math>\zeta = \frac{\tau^2}{\hbar^2}\left(\langle 0|\bar{H}^2|0\rangle - \langle 0|\bar{H}|0\rangle\langle 0|\bar{H}|0\rangle\right) ,</math>

giving

:<math>\zeta = \frac{\tau^2\Delta\bar{H}^2}{\hbar^2} ,</math>

where <math>\Delta\bar{H}</math> is the [[root mean square]] deviation of the system Hamiltonian averaged over the interval of interest.

The sudden approximation is valid when <math>\zeta \ll 1</math> (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

:<math>\tau \ll {\hbar \over \Delta\bar{H}} ,</math>

which is a statement of the [[Heisenberg uncertainty principle#Energy-time uncertainty principle|time-energy form of the Heisenberg uncertainty principle]].

=== Diabatic passage ===

In the limit <math>\tau \to 0</math> we have infinitely rapid, or diabatic passage:

:<math>\lim_{\tau \to 0}\hat{U}(t_1,t_0) = 1 .</math>

The functional form of the system remains unchanged:

:<math>|\langle x|\psi(t_1)\rangle|^2 = \left|\langle x|\psi(t_0)\rangle\right|^2 .</math>

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:

:<math>P_D = 1 - \zeta.</math>

=== Adiabatic passage ===

In the limit <math>\tau \to \infty</math> we have infinitely slow, or adiabatic passage. The system evolves, adapting its form to the changing conditions,

:<math>|\langle x|\psi(t_1)\rangle|^2 \neq |\langle x|\psi(t_0)\rangle|^2 .</math>

If the system is initially in an [[eigenstate]] of <math>\hat{H}(t_0)</math>, after a period <math>\tau</math> it will have passed into the ''corresponding'' eigenstate of <math>\hat{H}(t_1)</math>.

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:

:<math>P_A = \zeta .</math>