deletions | additions       diff --git a/section_Density_of_states_In__.tex b/section_Density_of_states_In__.tex  index 4805662..e523d3a 100644  --- a/section_Density_of_states_In__.tex  +++ b/section_Density_of_states_In__.tex 
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  We first compute the solution to the time-independent Schrödinger equation for a semi-infinte chain. Our results should coincide with the previous in this report. Let us assume a quite general solution for $\mathcal{H}_W$:  \begin{equation}  |\Psi\rangle = \sum_{i=1}^{i=\infty} c_{i} |i\rangle \sum_{j=1}^{j=\infty} c_{j} |j\rangle  \end{equation}  Then   \begin{equation} 
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\end{equation}  or  \begin{equation}  H_{W} \left(\sum_{i=1}^{i=\infty} c_{i} |i>\right) \left(\sum_{j=1}^{j=\infty} c_{j} |j\rangle\right)  =E \left(\sum_{i=1}^\{i=\infty} c_{i} |i>\right) \left(\sum_{j=1}^{j=\infty} c_{j} |j\rangle\right)  \end{equation}  We project onto a state $|m>$ and use the fact that $\langle m | j\rangle=\delta_{m,j}$, the states are an orthornormal basis. Then, we obtain the following   \begin{equation}  c_{i+1}+c_{i-1}=\frac{E-\epsilon_0}{t} c_{j+1}+c_{j-1}=\frac{E-\epsilon_0}{t}  \end{equation}  Let us name $\epsilon=(E-\epsilon_0)/t$  The solution of this equation are left and right plane waves   \begin{equation}  c_j=A e^{ik j} +B e^{-ikj}  \end{equation}  We employ this solution to obtain the eigenenergies  \begin{equation}  A e^{ik (j+1)} +B e^{-ik(j+1)} + A e^{ik (j+1)} +B e^{-ik(j+1)} = \epsilon A e^{ikj)} +B e^{-ikj}  \end{equation}  Then the solution for the energy is $(E-\epsilon_0)/t=2\cos k$, with $E=\epsilon_0+2\cos k$  We determine now the A and B coefficients in $c_j$. For such purpose we just employ the Schrödinger equation for the site $1$:  \begin{equation}  c_2=[(E-\epsilon_0)/t] c_1 = A e^{2ik}+Be^{-2ik}= A e^{ik} + t B e^{-i k} (A e^{ik}+Be^{-ik})  \end{equation}  Then we infer that $A=-B$, and therefore $c_j= 2Ai\sin k j$ with $E=\epsilon+2t\cos k$