Adiabatic Approximation

The goal of this section is to solve the time dependent Schrödinger equation,

$$i \hbar \frac{\partial \psi}{\partial t}=\hat{H} \psi, \qquad$$ (17)

for a time dependent Hamiltonian, $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(x, t)$, where the potential $V(x, t)$ undergoes significant changes but in a very "large" time scale (e.g., a time scale much larger than the time associated with state transitions).R2(496)

Since V(x,t) changes very slowly, we can solve the time independent Schrödinger equation at a specific time t',

$$\hat{H}(t')\Phi_n(x,t') = E_n(t')\Phi_n(x, t').$$

Assuming that $\frac{\partial\Phi_n}{\partial t}\approx0$, since V(x,t) changes very slowly, we find that the function,

$$\psi_n(x,t)=\Phi_n(x,t)e^{-\frac{i}{\hbar}\int_0^t E_n(t')dt'},$$

is a good approximate solution to Eq. (17). In fact, it satisfies Eq. (17) exactly when $\frac{\partial\Phi_n}{\partial t} = 0$.

Expanding the general solution $\psi(x, t)$ in the basis set $\Phi_n(x,t)$ we obtain:

$$\psi(x,t)=\sum_n C_n(t)\Phi_n(x,t)e^{-\frac{i}{\hbar}\int_0^t E_n(t')dt'},$$

and substituting this expression into Eq. (17) we obtain,

$$i\hbar\sum_n(\dot{C}_n\Phi_n +C_n\dot{\Phi}_n -\frac{i}{\hbar}E_n... ...t_0^t E_n(t')dt'} = \sum_nC_nE_n\Phi_n e^{-\frac{i}{\hbar}\int_0^t E_n(t')dt'},$$

where,

$$\dot{C}_k= -\sum_nC_n <\Phi_k\vert\dot{\Phi}_n>e^{-\frac{i}{\hbar}\int_0^t dt'(E_n(t')-E_k(t'))}.$$ (18)

Note that,

$$\frac{\partial H}{\partial t}\Phi_n + H \dot{\Phi}_n =\frac{\partial E_n}{\partial t}\Phi_n + E_n \dot{\Phi}_n,$$

then,

$$<\Phi_k\vert\frac{\partial H}{\partial t}\vert\Phi_n> + <\Phi_k\v... ..._n> = \frac{\partial E_k}{\partial t}\delta_{kn} +E_n<\Phi_k\vert\dot{\Phi}_n>,$$

since$$\ <\Phi_k\vert H\vert\dot{\Phi}_n> \ = \ <\dot{\Phi}_n\vert H\vert\Phi_k>^*.$$

Furthermore, if $k \neq n$ then,

$$<\Phi_k\vert\dot{\Phi}_n> = \frac{<\Phi_k\vert\frac{\partial H}{\partial t}\vert\Phi_n>}{E_n- E_k}.$$

Substituting this expression into Eq. (18) we obtain,

$$\dot{C}_k=-C_k <\Phi_k\vert\dot{\Phi}_k> -\sum_{n\neq k}C_n \frac... ...t}\vert\Phi_n>}{(E_n- E_k)} e^{-\frac{i}{\hbar}\int_0^t dt' (E_n(t')-E_k(t'))}.$$

Let us suppose that the system starts with $C_n(0)=\delta_{nj}$, then solving by successive approximations we obtain that for $k \neq j$:

$$\dot{C}_k= \frac{<\Phi_k\vert\frac{\partial H}{\partial t}\vert\Phi_j>}{(E_k- E_j)} e^{-\frac{i}{\hbar}\int_0^t dt' (E_j(t')-E_k(t'))}.$$

Assuming that $E_j(t)$ and $E_k(t)$ are slowly varying functions in time:

$$C_k \approx \frac{<\Phi_k\vert\frac{\partial H}{\partial t}\vert\... ...j- E_k)^2} [e^{-\frac{i}{\hbar}(E_j-E_k)t} - e^{-\frac{i}{\hbar}(E_j-E_k)t_0}],$$

since$$\qquad \vert e^{-\frac{i}{\hbar}(E_j-E_k)t} - e^{-\frac{i}{\hbar}(E_j-E_k)t_0} \vert \qquad \leq \qquad 2. \qquad$$

Therefore,

$$\vert C_k\vert^2 \approx \frac{4 \hbar^2 \vert<\Phi_k\vert\frac{\partial H}{\partial t}\vert\Phi_j>\vert^2 } {(E_j- E_k)^4}.$$

The system remains in the initially populated state at all times whenever $\frac{\partial H}{\partial t}$ is sufficiently small,

$$\left \vert<\Phi_k\vert\frac{\partial H}{\partial t}\vert\Phi_j> \right \vert << \frac{(E_j -E_k)^2}{\hbar},$$ (19)

even when such state undergoes significant changes. This is the so-called adiabatic approximation. It breaks down when $E_j \approx E_k$ because the inequality introduced by Eq. (19) can not be satisfied.

Mathematically, the condition that validates the adiabatic approximation can also be expressed in terms of the frequency $\nu$ defined by the equation $E_j-E_k = h \nu=\frac{h}{\tau}$, (or the time period $\tau$ of the light emitted with frequency $\nu$) as follows,

$\frac{\tau}{2\pi}\vert<\Phi_k\vert\frac{\partial H}{\partial t}\vert\Phi_j>\vert<<(E_j -E_k).$