Addressing #16

src/6_eigenvectors_QM.md

@@ -14,10 +14,16 @@ and at the end of the lecture notes, there is a set of corresponding exercises:
 
 - [6.3. Problems](#63-problems)
 
+***
+
 The contents of this lecture are summarised in the following **video**:
 
 - [Eigenvalues and eigenvectors](https://www.dropbox.com/s/n6hb5cu2iy8i8x4/linear_algebra_09.mov?dl=0)
 
+*The total length of the videos: ~3 minutes 30 seconds*
+
+***
+
 In the previous lecture, we discussed a number of *operator equations*, which have the form
 $$
 \hat{A}|\psi\rangle=|\varphi\rangle \, ,
@@ -65,7 +71,7 @@ the rules of matrix multiplication:
     $$
     where $\mathbb{I}$ is the identity matrix of dimensions $n\times n$, and ${\rm det}$ is the determinant.
 
-This relations follows from the eigenvalue equation in terms of components
+This relation follows from the eigenvalue equation in terms of components
 $$
 \begin{align}
 \sum_{j=1}^n A_{ij} v_j &= \lambda v_i \, , \\
@@ -73,7 +79,7 @@ $$
 \to \quad \sum_{j=1}^n\left( A_{ij} - \lambda \delta_{ij}\right) v_j &=0 \, .
 \end{align}
 $$
-Therefore the eigenvalue condition can be written as a set of coupled linear equations
+Therefore, the eigenvalue condition can be written as a set of coupled linear equations
 $$
 \sum_{j=1}^n\left( A_{ij} - \lambda \delta_{ij}\right) v_j =0 \, , \qquad i=1,2,\ldots,n\, ,
 $$
@@ -98,7 +104,7 @@ $$
 = A_{11}A_{22} - A_{12}A_{21} \, ,
 $$
 while the corresponding expression for a matrix belonging to a vector
-space in $n=3$ dimensions will be given in terms of the previous expression
+space in $n=3$ dimensions in terms of the previous expression will be given as
 $$
 {\rm det}\left( A \right) = \left|  \begin{array}{ccc} A_{11}  & A_{12}  & A_{13} \\ A_{21}  &  A_{22}
 &  A_{23} \\ A_{31}  &  A_{32}
@@ -171,14 +177,14 @@ corresponding eigenvalue problem for this operator, what we called above as the
 This is most efficiently done in the matrix representation of this operation, where we have
 that the above operator equation can be expressed in terms of its components as
 $$
-\begin{pmatrix} A_{11} & A_{12} & A_{13} & \ldots \\ A_{21} & A_{22} & A_{23} & \ldots\\A_{31} & A_{32} & A_{33} & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix} \begin{pmatrix} \psi_{k,1}\\\psi_{k,2}\\\psi_{k,3} \\\vdots\end{pmatrix}=  \lambda_{\psi_k}\begin{pmatrix} \psi_{k,1}\\\psi_{k,2}\\\psi_{k,3} \\\vdots\end{pmatrix} \, , \quad k=1,\ldots,n \, .
+\begin{pmatrix} A_{11} & A_{12} & A_{13} & \ldots \\ A_{21} & A_{22} & A_{23} & \ldots\\A_{31} & A_{32} & A_{33} & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix} \begin{pmatrix} \psi_{k,1}\\\psi_{k,2}\\\psi_{k,3} \\\vdots\end{pmatrix}=  \lambda_{\psi_k}\begin{pmatrix} \psi_{k,1}\\\psi_{k,2}\\\psi_{k,3} \\\vdots\end{pmatrix} \, .
 $$
 
 As discussed above, this condition is identical to solving a set of linear equations
 for the form
 $$
 \begin{pmatrix} A_{11}- \lambda_{\psi_k} & A_{12} & A_{13} & \ldots \\ A_{21} & A_{22}- \lambda_{\psi_k} & A_{23} & \ldots\\A_{31} & A_{32} & A_{33}- \lambda_{\psi_k} & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix}
-\begin{pmatrix} \psi_{k,1}\\\psi_{k,2}\\\psi_{k,3} \\\vdots\end{pmatrix}=0 \, , \quad k=1,\ldots,n \, .
+\begin{pmatrix} \psi_{k,1}\\\psi_{k,2}\\\psi_{k,3} \\\vdots\end{pmatrix}=0 \, .
 $$
 
 !!! info "Cramer's rule"
@@ -250,7 +256,7 @@ are *orthogonal* among them.
     $$\hat{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix} \, .$$
     Determine the eigenvalues and eigenvectors of this operator.
 
-5.*Hermitian matrix*
+5. *Hermitian matrix*
 
     Show that the Hermitian matrix
     $$\begin{pmatrix} 0&0&i\\0&1&0\\-i&0&0 \end{pmatrix}$$