Addressing #16

src/6_eigenvectors_QM.md

@@ -47,7 +47,7 @@ of vector space). We can express the previous equation in terms of its component
 assuming as usual some specific choice of basis, by using
 the rules of matrix multiplication:
 
-!!! tip "Eigenvalue equation: Eigenvalue and Eigenvector"
+!!! info "Eigenvalue equation: Eigenvalue and Eigenvector"
     $$
     \sum_{j=1}^n A_{ij} v_j = \lambda v_i \, .
     $$
@@ -57,7 +57,7 @@ the rules of matrix multiplication:
 !!! warning "Number of solutions"
     In general, there will be multiple solutions to the eigenvalue equation $A \vec{v} =\lambda \vec{v}$, each one characterised by an specific eigenvalue and eigenvectors. Note that in some cases one has  *degenerate solutions*, whereby a given matrix has two or more eigenvectors that are equal.
 
-!!! info "Characteristic equation:"
+!!! tip "Characteristic equation:"
     In order to determine the eigenvalues of the matrix $A$, we need to evaluate the solutions of the so-called *characteristic equation*
     of the matrix $A$, defined as
     $$
@@ -78,7 +78,7 @@ $$
 \sum_{j=1}^n\left( A_{ij} - \lambda \delta_{ij}\right) v_j =0 \, , \qquad i=1,2,\ldots,n\, ,
 $$
 which only admit non-trivial solutions if the determinant of the matrix $A-\lambda\mathbb{I}$ vanishes
-(the so-called Cramer condition), thus leading to the characteristic equation.
+(the so-called Cramer's condition), thus leading to the characteristic equation.
 
 Once we have solved the characteristic equation, we end up with $n$ eigenvalues $\lambda_k$, $k=1,\ldots,n$.
   
@@ -183,7 +183,7 @@ $$
 
 !!! info "Cramer's rule"
     This set of linear equations only has a non-trivial set of solutions provided that
-    the determinant of the matrix vanishes, as follows from the Cramer condition:
+    the determinant of the matrix vanishes, as follows from the Cramer's condition:
     $$
     {\rm det} \begin{pmatrix} A_{11}- \lambda_{\psi} & A_{12} & A_{13} & \ldots \\ A_{21} & A_{22}- \lambda_{\psi} & A_{23} & \ldots\\A_{31} & A_{32} & A_{33}- \lambda_{\psi} & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix}=
     \left|  \begin{array}{cccc}A_{11}- \lambda_{\psi} & A_{12} & A_{13} & \ldots \\ A_{21} & A_{22}- \lambda_{\psi} & A_{23} & \ldots\\A_{31} & A_{32} & A_{33}- \lambda_{\psi} & \ldots \\\vdots & \vdots & \vdots & \end{array} \right| = 0
@@ -198,69 +198,66 @@ $$
 |\psi_2\rangle = \begin{pmatrix} \psi_{2,1} \\  \psi_{2,2} \\  \psi_{2,3} \\ \vdots \end{pmatrix} \,, \quad \ldots \, , |\psi_n\rangle = \begin{pmatrix} \psi_{n,1} \\  \psi_{n,2} \\  \psi_{n,3} \\ \vdots \end{pmatrix} \, .
 $$
 
-An important property of eigenvalue equations is that if you have two eigenvectors
-$ |\psi_i\rangle$ and $ |\psi_j\rangle$ that have associated *different* eigenvalues,
-$\lambda_{\psi_i} \ne \lambda_{\psi_j}  $, then these two eigenvectors are orthogonal to each
-other, that is
-$$
-\langle \psi_j | \psi_i\rangle =0 \, \quad {\rm for} \quad {i \ne j} \, .
-$$
-This property is extremely important, since it suggest that we could use the eigenvectors
-of an eigenvalue equation as a *set of basis elements* for this Hilbert space.
+!!! tip "Orthogonality of eigenvectors"
+    An important property of eigenvalue equations is that if you have two eigenvectors
+    $ |\psi_i\rangle$ and $ |\psi_j\rangle$ that have associated *different* eigenvalues,
+    $\lambda_{\psi_i} \ne \lambda_{\psi_j}  $, then these two eigenvectors are orthogonal to each
+    other, that is
+    $$
+    \langle \psi_j | \psi_i\rangle =0 \, \quad {\rm for} \quad {i \ne j} \, .
+    $$
+    This property is extremely important, since it suggest that we could use the eigenvectors
+    of an eigenvalue equation as a *set of basis elements* for this Hilbert space.
 
 Recall from the discussions of eigenvalue equations in linear algebra that
 the eigenvectors $|\psi_i\rangle$ are defined *up to an overall normalisation constant*. Clearly, if $|\psi_i\rangle$ is a solution of $\hat{A}|\psi_i\rangle = \lambda_{\psi_i}|\psi_i\rangle$
-then $c|\psi_i\rangle$ will also be a solution, with $c$ some constant. In the context of quantum mechanics, we need to choose this overall rescaling constant
-to ensure that the eigenvectors are normalised, that is, that they satisfy
+then $c|\psi_i\rangle$ will also be a solution, with $c$ being a constant. In the context of quantum mechanics, we need to choose this overall rescaling constant to ensure that the eigenvectors are normalised, thus they satisfy
 $$
 \langle \psi_i | \psi_i\rangle = 1 \, \quad {\rm for~all}~i \, .
 $$
-With such a choice of normalisation, one says that the set of eigenvectors
+With such a choice of normalisation, one says that the eigenvectors in a set
 are *orthogonal* among them.
 
-The set of all the eigenvalues of an operator is called *eigenvalue spectrum* of the operator. Note that different eigenvectors can also have the same eigenvalue. If this is the case the eigenvalue is said to be *degenerate*.
+!!! tip "Eigenvalue spectrum and degeneracy"
+    The set of all eigenvalues of an operator is called the *eigenvalue spectrum* of an operator. Note that different eigenvectors can also have the same eigenvalue. If this is the case the eigenvalue is said to be *degenerate*.
 
 ***
 
-##Problems
-
-**1)** *Eigenvalues and Eigenvectors* 
-
-Find the characteristic polynomial and eigenvalues for each of the following matrices,
+##6.3. Problems
 
-$$A=\begin{pmatrix} 5&3\\2&10 \end{pmatrix}\,  \quad
-B=\begin{pmatrix} 7i&-1\\2&6i \end{pmatrix} \, \quad C=\begin{pmatrix} 2&0&-1\\0&3&1\\1&0&4 \end{pmatrix}$$
+1. *Eigenvalues and eigenvectors I* 
 
-**2)** The Hamiltonian for a two-state system is given by 
-$$H=\begin{pmatrix} \omega_1&\omega_2\\  \omega_2&\omega_1\end{pmatrix}$$
-A basis for this system is 
-$$|{0}\rangle=\begin{pmatrix}1\\0  \end{pmatrix}\, ,\quad|{1}\rangle=\begin{pmatrix}0\\1  \end{pmatrix}$$
+    Find the characteristic polynomial and eigenvalues for each of the following matrices,
+    $$A=\begin{pmatrix} 5&3\\2&10 \end{pmatrix}\,  \quad
+    B=\begin{pmatrix} 7i&-1\\2&6i \end{pmatrix} \, \quad C=\begin{pmatrix} 2&0&-1\\0&3&1\\1&0&4 \end{pmatrix}$$
 
-Find the eigenvalues and eigenvectors of the Hamiltonian $H$, and express the eigenvectors in terms of $\{|0 \rangle,|1\rangle \}$
+2. *Hamiltonian*
 
+    The Hamiltonian for a two-state system is given by 
+    $$H=\begin{pmatrix} \omega_1&\omega_2\\  \omega_2&\omega_1\end{pmatrix}$$
+    A basis for this system is 
+    $$|{0}\rangle=\begin{pmatrix}1\\0  \end{pmatrix}\, ,\quad|{1}\rangle=\begin{pmatrix}0\\1  \end{pmatrix}$$
+    Find the eigenvalues and eigenvectors of the Hamiltonian $H$, and express the eigenvectors in terms of $\{|0 \rangle,|1\rangle \}$
 
-**3)** Find the eigenvalues and eigenvectors of the matrices
+3. *Eigenvalues and eigenvectors II*
 
-$$A=\begin{pmatrix} -2&-1&-1\\6&3&2\\0&0&1 \end{pmatrix}\,  \quad B=\begin{pmatrix} 1&1&2\\2&2&2\\-1&-1&-1 \end{pmatrix} $$.
+    Find the eigenvalues and eigenvectors of the matrices
+    $$A=\begin{pmatrix} -2&-1&-1\\6&3&2\\0&0&1 \end{pmatrix}\, \quad B=\begin{pmatrix} 1&1&2\\2&2&2\\-1&-1&-1 \end{pmatrix} $$.
 
-**4)** *The Hadamard gate*
+4. *The Hadamard gate*
 
-In one of the problems of the previous section we discussed that an important operator used in quantum computation is the *Hadamard gate*, which is represented by the matrix:
-$$\hat{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix} \, .$$
-Determine the eigenvalues and eigenvectors of this operator.
+    In one of the problems of the previous section we discussed that an important operator used in quantum computation is the *Hadamard gate*, which is represented by the matrix:
+    $$\hat{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix} \, .$$
+    Determine the eigenvalues and eigenvectors of this operator.
 
-**5)** Show that the Hermitian matrix
+5.*Hermitian matrix*
 
-$$\begin{pmatrix} 0&0&i\\0&1&0\\-i&0&0 \end{pmatrix}$$
+    Show that the Hermitian matrix
+    $$\begin{pmatrix} 0&0&i\\0&1&0\\-i&0&0 \end{pmatrix}$$
+    has only two real eigenvalues and find and orthonormal set of three eigenvectors.
 
-has only two real eigenvalues and find and orthonormal set of three eigenvectors.
+6. *Orthogonality of eigenvectors*
 
-**6)** 
-Confirm, by explicit calculation, that the eigenvalues of the real, symmetric matrix
-
-$$\begin{pmatrix} 2&1&2\\1&2&2\\2&2&1 \end{pmatrix}$$
-
-are real, and its eigenvectors are orthogonal.
-
-  
-  
+    Confirm, by explicit calculation, that the eigenvalues of the real, symmetric matrix
+    $$\begin{pmatrix} 2&1&2\\1&2&2\\2&2&1 \end{pmatrix}$$
+    are real, and its eigenvectors are orthogonal.