Update 5_operators_QM.md

src/5_operators_QM.md

@@ -252,20 +252,31 @@ $$
 \sum_{k=1}^n\left( A_{ik}B_{kj} - B_{ik}A_{kj} \right) \, .
 $$  
 
-### Matrix reprensentation of Hermitian operator
+### Matrix reprensentation of Hermitian operators
 
 As discussed above,  Hermitian operators play a central role in the physical interpretation of quantum systems.
 We can now provide the explicit expression of the Hermitian operators in the matrix representation. 
+
 Let us assume that we have an operator $\hat{A}$ in an $n$-dimensional Hilbert space
-with matrix elements $A_{ij}$ defined with respect to a set of orthonormal basis states, $\|psi_i\rangle; i=1,2,\ldots\,n}$. 
-From the matrix representation one can construct a new operator by taking the transpose and complex conjugate of the original matrix, that is:
+with matrix elements $A_{ij}$ defined with respect to a set of orthonormal basis states, $\{ |psi_i\rangle; i=1,2,\ldots\,n\}$. 
+From its matrix representation one can construct a new operator by taking the transpose and complex conjugate of the original matrix, that is:
 $$
-    \begin{pmatrix} A_{11} & A_{12} & A_{13} & \ldots \\ A_{21} & A_{22} & A_{23} & \ldots\\A_{31} & A_{32} & A_{33} & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix}\rightarrow \begin{pmatrix} A_{11}^* & A_{12}^* & A_{13}^* & \ldots \\ A_{21}^* & A_{22}^* & A_{23}^* & \ldots\\A_{31}^* & A_{32}^* & A_{33}^* & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix}
+\begin{pmatrix} A_{11} & A_{12} & A_{13} & \ldots \\ A_{21} & A_{22} & A_{23} & \ldots\\A_{31} & A_{32} & A_{33} & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix}\rightarrow \begin{pmatrix} A_{11}^* & A_{21}^* & A_{31}^* & \ldots \\ A_{12}^* & A_{22}^* & A_{32}^* & \ldots\\A_{13}^* & A_{23}^* & A_{33}^* & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix}
 $$
-This new matrix will be a new operator that will be represented by $\hat{A}^\dagger$. This new operator is called the *Hermitian adjoint*  of
+This new matrix corresponds to the matrix representation of a new operator that will be denoted by $\hat{A}^\dagger$. This new operator is called the *Hermitian adjoint*  of
 the operator $\hat{A}$.
 
-Hermitian operators are such that they are the same as their adjoint version. From the expression in its components that one gets in the matrix representation, the condition of Hermitian operator will therefore read
+Hermitian operators are such that they are *the same* as their adjoint version.
+That is, at the operator level we can write that Hermitian operators satisfy $\hat{A}=\hat{A}^\dagger$.
+From the expression in its components that one gets in the matrix representation, the condition of Hermitian operator will therefore read
+$$
+\begin{pmatrix} A_{11} & A_{12} & A_{13} & \ldots \\ A_{21} & A_{22} & A_{23} & \ldots\\A_{31} & A_{32} & A_{33} & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix} = \begin{pmatrix} A_{11}^* & A_{21}^* & A_{31}^* & \ldots \\ A_{12}^* & A_{22}^* & A_{32}^* & \ldots\\A_{13}^* & A_{23}^* & A_{33}^* & \ldots \\\vdots & \vdots & \vdots & \end{pmatrix}
+$$
+which can be expressed in a more compact way as
+\be
+A_{ij} = A_{ji}^* \, , \quad i,j=1,\ldots,n \, .
+\ee
+Any operator which satisfies this condition will be an Hermitian operator. Note that the Hermiticity of an operator is a property which is independent of the specific choice of basis.
 
 
 ##Problems