1st major update to lecture note 4

src/4_vector_spaces_QM.md

@@ -116,43 +116,31 @@ The inner product (probability amplitude) $\langle \psi | \phi \rangle$ exhibits
           You can see from the properties of complex algebra that this length must be a real number. A physically valid state $|\psi \rangle$ must be normalized to unity, that is $\langle \psi | \psi \rangle=1$. Note that a state that cannot be normalized to unity does not represent a physically acceptable state.
      6. **Orthonormality:** A set of orthonormal basis vectors $\{|\psi_i\rangle\text{;}\; i=1,2,3,...,n\}$ will have the property $\langle \psi_i |\psi_j \rangle=\delta_{ij}$ where $\delta_{ij}$ is a mathematical symbol known as the *Kronecker delta*, which equals unity if $i=j$ and zero if $i\neq j$.
    
-From all the above conditions we see that a Hilbert space is a so-called *complex inner product space*, which is nothing but a complex vector space equipped with a inner product. All the vectors belonging to a Hilbert space $\mathcal{H}$ have a finite norm, that is they can be normalized to unity. This normalisation condition is essential is we are to apply the probabilistic interpretation of the state vectors described above.
+From all the above conditions, we see that a Hilbert space is a so-called *complex inner product space*, which is nothing else but a complex vector space equipped with a inner product. All the vectors belonging to a Hilbert space $\mathcal{H}$ have a finite norm, which means that they can be normalized to unity. This normalisation condition is essential is we are to apply the probabilistic interpretation of the state vectors described above.
 
-## Matrix representation of ket and bra vectors
+## 4.3. Matrix representation of ket and bra vectors
  
-As we have discussed, in quantum mechanics a general state vector $|\psi\rangle$ can be represented in terms of the basis vectors, $\{|\phi_i\rangle;i=1,2,...,n\}$, as follows
-$$
-|\psi\rangle=\sum_{i=1}^n c_i |\phi_i\rangle \, ,
-$$
+As we have discussed, in quantum mechanics a general state vector $|\psi\rangle$ can be represented in terms of the basis vectors, $\{|\phi_i\rangle;i=1,2,...,n\}$, as 
+$$ |\psi\rangle=\sum_{i=1}^n c_i |\phi_i\rangle $$
 for some values of the complex coefficients $\{ c_i\}$. To determine the values of these coefficients, we can take the inner product between the bra basis vector $\langle \phi_j|$ and the ket state vector $|\psi\rangle$ and use the orthogonality properties of the basis vectors:
-$$
-\langle \phi_j|\psi\rangle = \langle \phi_j|\sum_{i=1}^n c_i |\phi_i\rangle = \sum_{i=1}^n c_i\langle \phi_j|\phi_i\rangle = \sum_{i=1}^n c_i\delta_{ij} = c_j \, .
-$$
+$$ \langle \phi_j|\psi\rangle = \langle \phi_j|\sum_{i=1}^n c_i |\phi_i\rangle = \sum_{i=1}^n c_i\langle \phi_j|\phi_i\rangle = \sum_{i=1}^n c_i\delta_{ij} = c_j \, .$$
 Therefore, if we now denote the coefficients $\{ c_i\}$ of the state vector $|\psi\rangle$ by $\{ \psi_i\}$, we have the expansion
-$$
-|\psi\rangle=\sum_{i=1}^n \psi_i |\phi_i\rangle= \sum_{i=1}^n \left( \langle \phi_i|\psi\rangle \right) |\phi_i\rangle \, .
-$$
-By analogy with the Euclidean case, we can understand the coefficients $\psi_i$
-as the *components* of the state vector $ |\psi\rangle$ along the $n$ directions
-spanned by the basis vectors. Here note that in this notation $\psi_i$ is an *scalar* (just a number) and not a vector. Further note that, as opposed to the Euclidean space, the coefficients $\psi_i$ will be in general complex numbers.
+$$ |\psi\rangle=\sum_{i=1}^n \psi_i |\phi_i\rangle= \sum_{i=1}^n \left( \langle \phi_i|\psi\rangle \right) |\phi_i\rangle \, .$$
+By analogy with the Euclidean case, we can understand the coefficients $\psi_i$ as the *components* of the state vector $ |\psi\rangle$ along the $n$ directions spanned by the basis vectors. Here, note also that in this notation $\psi_i$ is an *scalar* (just a number) and not a vector. Furthermore, note that, as opposed to the Euclidean space, the coefficients $\psi_i$ will generally be complex numbers.
 
 This analogy with the case of ordinary vectors allows us to write the state $|\psi\rangle$ as a *column vector* with respect to the set of basis vectors $\{|\phi_i\rangle;i=1,2,...,n\}$, which are kept implicit: 
-$$
-|\psi\rangle= \begin{pmatrix} \psi_1\\\psi_2\\\psi_3\\\vdots\\\psi_n\end{pmatrix}.
-$$
-We can also express the basis vectors in this manner. Given that the basis vectors are *orthonormal* among them,
+$$ |\psi\rangle= \begin{pmatrix} \psi_1\\\psi_2\\\psi_3\\\vdots\\\psi_n\end{pmatrix} \, . $$
+
+We can also express the basis vectors in this manner. Given that the basis vectors are *orthonormal* among themselves,
 the basis state $|\phi_i\rangle$ will have as component in the $j$ direction
-$$
-(\phi_i)_j=\langle \phi_j|\phi_i\rangle=\delta_{ji} \, ,
-$$
+$$ (\phi_i)_j=\langle \phi_j|\phi_i\rangle=\delta_{ji} \, ,$$ 
 and thus the vector column expression of the basis vectors will be very simple
-$$
+$$ 
 |\phi_1\rangle= \begin{pmatrix} 1\\0\\0 \\\vdots\end{pmatrix} \;, \quad
 |\phi_2\rangle= \begin{pmatrix} 0\\1\\0 \\\vdots\end{pmatrix} \;, \ldots
 $$
 
-Let us show how we can use the matrix representation to evaluate the inner
-product (bracket) between two state vectors when expanded in terms of their components in the same basis:
+Let us show how we can use the matrix representation to evaluate the inner product (bracket) between two state vectors when expanded in terms of their components in the same basis:
 $$
 |\psi\rangle=\sum_{i=1}^n \psi_i |\phi_i\rangle \,, \qquad
 |\chi\rangle=\sum_{i=1}^n \chi_i |\phi_i\rangle\, .