1st major update to lecture note 4

src/4_vector_spaces_QM.md

@@ -105,25 +105,21 @@ The inner product in quantum mechanics is the analog of the usual scalar product
 
 The inner product (probability amplitude) $\langle \psi | \phi \rangle$ exhibits the following properties:
       
-!!! info "Complex conjugate:" 
-     $$\langle \psi | \phi \rangle=\langle \phi | \psi \rangle^*$$
-        
-!!! info "Distributivity and associativity" 
-     $$\langle \psi |\{c_1 |\phi_1\rangle+c_2 |\phi_2 \rangle\}=c_1\langle \psi | \phi_1\rangle+c_2\langle \psi | \phi_2\rangle$$
-
-!!! info "Positivity" 
-     $$\langle \psi | \psi \rangle\geq0$$. If $\langle \psi | \psi \rangle = 0$$ 
-     then this implies that the state vector $|\psi\rangle=0$ is the null element of the Hilbert space.
-
-!!! info "Orthogonality" 
-     Two states $|\psi \rangle$ and $|\phi \rangle$ are said to be *orthogonal* if 
-     $$\langle \psi | \phi\rangle=0 \, .$$ 
-     By analogy with regular vector spaces, we can think of these two state vectors $|\psi \rangle$ and $|\phi \rangle$ as being *perpendicular* to each other. Note that for a quantum system occupying a certain state, there is a vanishing probability of it being observed in a state orthogonal to it.
-        
-!!! info "Norm:"
-     The quantity $\sqrt{\langle \psi | \psi \rangle}$ is known as the  *length* or the *norm* of the state vector $|\psi\rangle$. You can see from the properties of complex algebra that this length must be a real number.
-
-A physically valid state $|\psi \rangle$ must be normalized to unity, that is $\langle \psi | \psi \rangle=1$. Note that a state that cannot be normalized to unity does not represent a physically acceptable state.
+!!! info 
+     1. Complex conjugate:
+          $$\langle \psi | \phi \rangle=\langle \phi | \psi \rangle^*$$
+     2. Distributivity and associativity:
+          $$\langle \psi |\{c_1 |\phi_1\rangle+c_2 |\phi_2 \rangle\}=c_1\langle \psi | \phi_1\rangle+c_2\langle \psi | \phi_2\rangle$$
+     3. Positivity:
+          $$\langle \psi | \psi \rangle\geq0 \, .$$ 
+          If $\langle \psi | \psi \rangle = 0$ then, this implies that the state vector $|\psi\rangle=0$ is the null element of the Hilbert space.
+     4. Orthogonality:
+          Two states $|\psi \rangle$ and $|\phi \rangle$ are said to be *orthogonal* if 
+     $$\langle \psi | \phi\rangle=0 \, .$$
+          By analogy with regular vector spaces, we can think of these two state vectors $|\psi \rangle$ and $|\phi \rangle$ as being *perpendicular* to each other. Note that for a quantum system occupying a certain state, there is a vanishing probability of it being observed in a state orthogonal to it.
+     5. Norm:
+          The quantity $\sqrt{\langle \psi | \psi \rangle}$ is known as the  *length* or the *norm* of the state vector $|\psi\rangle$. 
+          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.
        
 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 $i\neq j$.