Update tunneling section & adiabatic part

src/adiabatic_proof.md

@@ -30,7 +30,7 @@ with a time-dependent Hamiltonian $H(t)$. For notational simplicity, we will
 in the following often leave out the explicit bra-ket notation, and simple write $\Psi(t)$ instead
 of $\left|\Psi(t)\right>$.
 
-We now want to consider a Hamiltonian $H(t)$ that cheanges slowly in time. But what does *slowly* mean,
+We now want to consider a Hamiltonian $H(t)$ that changes slowly in time. But what does *slowly* mean,
 slow compared to what? In contrast to the WKB approximation this is less obvious, and
 we will find a proper criterion in the course of the proof.
 
@@ -143,57 +143,59 @@ $$
 $$
 The $n^{\text{th}}$ eigenstate stays in the $n^{\text{th}}$ eigenstate.
 
-### Adiabatic criteria: how slow is *slow*?
+### Adiabatic criterium: how slow is *slow*?
 
-In order to obtain the result above, the terms $n\neq m$ have been neglected by considering their contribution small. In principle, it should be smaller than the time associated with the energy difference of the first excitation. That is,
+In order to obtain the result above, the terms $n\neq m$ have been neglected by considering their contribution small, arguing that there should be a *large enough* energy gap between the instantaneous eigenenergies. Let us now derive a more quantitative criterion.
 
+To derive it, we will proceed in the spirit of perturbation theory: we start with the initial approximation and then insert it back to get a higher-order approximation. In our [derivation for the adiabatic theorem](adiabatic_proof.md##proof-of-the-theorem) we find as an intermediate result
 $$
-\frac{\langle \psi_m | \dot H |\psi_n\rangle}{E_m - E_n}<< \frac{E_m - E_n}{\hbar}.
+\begin{align}
+\dot{c}_{n} &= - \sum_k c_k \langle{\psi_n| \dot\psi_k} \rangle e^{i (\theta_k - \theta_n)}\\
+&= \sum_k c_k \frac{\langle \psi_n | \dot H |\psi_k \rangle}{E_n(t) - E_k(t)} e^{-\frac{i}{\hbar} \int_{0}^t E_k(t') - E_n(t') dt'}.
+\end{align}
 $$
-
-This result can be derived in the style of perturbation theory. Let us evaluate the first order solution that satiesfies $c_m(t) = 1$ and $c_{n\neq m}(t)=0$. That is,
-
+We then found the approximate solution $c_m(t) = 1$ and $c_{n\neq m}(t)=0$. Let us now put this approximate solution back into the still exact equation to find
 $$
-\dot c_n(t) = \frac{\langle \psi_m | \dot H |\psi_n \rangle}{E_m(t) - E_n(t)} e^{-\frac{i}{\hbar} \int_{0}^t E_m(t') - E_n(t') dt'}.
+\dot c_{n \neq m} = \frac{\langle \psi_n | \dot H |\psi_m \rangle}{E_n(t) - E_m(t)} e^{-\frac{i}{\hbar} \int_{0}^t E_m(t') - E_n(t') dt'}\,.
 $$
-
-The coefficient is obtained by integrating the previous equation as,
-
+Integrating this equation we then arrive at
 $$
-\rightarrow c_n(T) = \int_{0}^T \dot c_n(t) dt =  \int_{0}^T \frac{\langle \psi_m | \dot H |\psi_n\rangle}{E_m(t) - E_n(t)} e^{-\frac{i}{\hbar} \int_{0}^t E_m(t') - E_n(t') dt'} dt.
+\rightarrow c_{n \neq m}(T) = \int_{0}^T \frac{\langle \psi_n | \dot H |\psi_m\rangle}{E_n(t) - E_m(t)} e^{\frac{i}{\hbar} \int_{0}^t E_n(t') - E_m(t') dt'} dt.
 $$
+Of course we know that $c_{n \neq m}(T)$ should be zero - that was what we input into the equation to begin with! We will now be able to find our quantitative criterion for adiabaticity by finding out under whch conditions $c_{n \neq m}(T)$ will be very small.
 
-There are several time dependent quantities in the previous expression. However, we approximate some of them by using the following bounds: 
+There are several time-dependent quantities in the previous expression. To find a solution, we will  approximate some of them by using the following constant bounds: 
 
-* The largest contribution from the rate of change in the Hamiltonian will come from the largest eigenvalue.
+* The largest contribution from the rate of change in the Hamiltonian will come from the largest matrix element
 
 $$
-\langle \psi_m (t) | \dot H |\psi_n(t) \rangle \approx \overline{\langle \psi_m | \dot H |\psi_n\rangle}
+\langle \psi_n (t) | \dot H |\psi_m(t) \rangle \approx \overline{\langle \psi_n | \dot H |\psi_n\rangle}
 $$
 
 * The smallest energy difference will contribute the most.
 
 $$
-E_m(t) - E_n(t) \approx \overline{E_m - E_n}
+E_n(t) - E_m(t) \approx \overline{E_n - E_m}
 $$
 
-After such approximations, these quantities are replaced in the formula for the coefficients. Since they are not time-dependent anymore, it can be solved as,
-
+Replacing these time-dependent quantities with these extremal bounds, we arrive at a simpler problem that we can now solve analytically:
 $$
-c_n(T) = \frac{\overline{\langle \psi_m | \dot H |\psi_n\rangle}}{\overline{E_m - E_n}} e^{-\frac{i}{\hbar} \int_0^t \overline{E_m - E_n} dt'}
+\begin{align}
+c_{n\neq m}(T) &\approx \int _0^T \frac{\overline{\langle \psi_n | \dot H |\psi_m\rangle}}{\overline{E_n - E_m}} e^{\frac{i}{\hbar} \int_0^t \overline{E_n - E_m} dt'} dt\\
+&=\frac{\overline{\langle \psi_n | \dot H |\psi_m\rangle}}{\overline{E_n - E_m}} \int_0^T e^{\frac{i}{\hbar} \overline{E_n - E_m} t} dt\\
+&= \frac{\overline{\langle \psi_n | \dot H |\psi_m\rangle}}{\overline{E_n - E_m}} \frac{i\hbar }{\overline{E_n - E_m}}\left( e^{-i \overline{E_n - E_m} T /\hbar } - 1 \right)\\
+&\approx \frac{\overline{\langle \psi_n | \dot H |\psi_m\rangle}}{\overline{E_n - E_m}} \frac{i\hbar }{\overline{E_n - E_m}}
+\end{align}
 $$
-
+In the last line we made use of the fact that the last term of the previous line is oscillating. Since we are only interested in upper bounds, we can approximate it with a constant 1. To get our adiabaticity criterion, we now remember that we need to have $|c_{n\neq m}(T)| \ll 1$, so that
 $$
-c_n(T) = \frac{\overline{\langle \psi_m | \dot H |\psi_n\rangle}}{\overline{E_m - E_n}} \frac{i\hbar }{\overline{E_m - E_n}}\left( e^{-i \overline{E_m - E_n} T /\hbar } - 1 \right).
+ \frac{\hbar \overline{\langle \psi_m | \dot H |\psi_n\rangle} }{\overline{E_n - E_m}^2} \ll 1
 $$
-
-We have found an expression for the coefficients where $n \neq m$. The addibatic approximation assumes that these coefficients are small. This criteria can be explicitly written as,
-
+or in other words
 $$
-|c_n(T)| \leq \frac{\hbar \overline{\langle \psi_m | \dot H |\psi_n\rangle} }{\overline{E_m - E_n}^2} << 1
+ \frac{\overline{\langle \psi_m | \dot H |\psi_n\rangle} }{\overline{E_n - E_m}} \ll \frac{\overline{E_n - E_m}}{\hbar}\,.
 $$
-
-We recover what we first presented at this section.
+This is a quantitative estimate for the adiabaticity criterion.
 
 ## Summary
 
@@ -211,6 +213,6 @@ $$
 $$
 In order for this description to hold, we require that the system evolves slowly enough. That is, the following condition must be satisfied at all times:
 $$
-\frac{\hbar \overline{\langle \psi_m | \dot H |\psi_n\rangle} }{\overline{E_m - E_n}^2} << 1, \quad n \neq m.
+\frac{\hbar \overline{\langle \psi_n | \dot H |\psi_m\rangle} }{\overline{E_n - E_m}^2} << 1, \quad n \neq m.
 $$
 Here, the overline indicates the largest matrix element, and the smallest energy difference.
\ No newline at end of file