1st major update to lecture note 4

src/4_vector_spaces_QM.md

@@ -40,29 +40,39 @@ can be applied to describe physical states in quantum mechanics.
 The state of a physical system in quantum mechanics is represented by a vector belonging to a *complex vector space*.
 This vector space is known as the *state space* of the system.
 
+### Ket
+
 !!! info "Ket"
      A physical state of a quantum system is represented by a symbol $$|~~\rangle$$ known as a **ket**. 
-     This notation is known as the *Dirac notation*, and it is very prominent in the description of quantum mechanics. Note that a *ket* is also refered to as a state vector, *ket* vector, or just a state. 
+     This notation is known as the *Dirac notation*, and it is very prominent in the description of quantum mechanics. 
+     Note that a *ket* is also refered to as a state vector, *ket* vector, or just a state. 
+
+### Hilbert space
 
 The set of all possible state vectors describing a given physical system forms a complex vector space $\mathcal{H}$, which is known as the *Hilbert space* of the system. You can think of the Hilbert space as the space populated by all possible states that a quantum system can be found on. Hilbert spaces inherit a number of the important properties of general vector spaces:
     
-- A linear combination (or superposition) of two or more state vectors $|{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle$, is also a state of the quantum system. Therefore, a linear combination $|{\Psi}\rangle$ of the form
-     $$|{\Psi}\rangle=c_1|{\psi_1}\rangle+c_2|{\psi_1}\rangle+c_3|{\psi_3}\rangle+...+c_n|{\psi_n}\rangle = \sum_{i=1}^n c_i|{\psi_i}\rangle  $$
-where $c_1, c_2, c_3, ...$ are general complex numbers will also be a physically allowed state vector of the quantum system.
+!!! info "" 
+     A linear combination (or superposition) of two or more state vectors $|{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle$, is also a state of the quantum system. Therefore, a linear combination $|{\Psi}\rangle$ of the form $$|{\Psi}\rangle=c_1|{\psi_1}\rangle+c_2|{\psi_1}\rangle+c_3|{\psi_3}\rangle+...+c_n|{\psi_n}\rangle = \sum_{i=1}^n c_i|{\psi_i}\rangle$$ 
+     where $c_1, c_2, c_3, ...$ are general complex numbers will also be a physically allowed state vector of the quantum system.
      
-- If a physical state of the system is given by a vector $|{\Psi}\rangle$, then the same physical state can also be represented by the vector $c|{\Psi}\rangle$ where $c$ is a non-zero complex number. The reason for this is that the overall normalisation of the state vector *does not change the physics* of the system (or in other words, does not modify the *information content* of the state vector). As we will discuss below, in quantum mechanics it is advantageous to work with  *normalised vectors*, that is, whose *length* is one.
-We will define in a while what do we mean by length.
-
-- A set of vectors $|{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle$ is said to be  *complete* if every state of the quantum system can be represented as a linear combination of them.
-In such a case, it becomes possible to express  *any* state vector $|{\Psi}\rangle$ of the system's Hilbert space as a superposition of these $n$ vectors,
-$$ |{\Psi}\rangle=\sum_{i=1}^n c_i|{\psi_i}\rangle$$
-for some specific choice of coefficients $c_i$. The set of vector \{$|{\psi_i}\rangle$\} are then said to *span* the Hilbert space of the quantum system.
+!!! info ""
+     If a physical state of the system is given by a vector $|{\Psi}\rangle$, then the same physical state can also be represented by the vector $c|{\Psi}\rangle$ where $c$ is a non-zero complex number. The reason for this is that the overall normalisation of the state vector *does not change the physics* of the system (or in other words, does not modify the *information content* of the state vector). As we will discuss below, in quantum mechanics it is advantageous to work with  *normalised vectors*, that is, whose *length* is one. 
+     We will define in a while what do we mean by length.
+
+!!! info ""
+     A set of vectors $|{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle$ is said to be  *complete* if every state 
+     of the quantum system can be represented as a linear combination of them.
+     In such a case, it becomes possible to express  *any* state vector $|{\Psi}\rangle$ of the system's Hilbert space as a superposition of these $n$ vectors,
+     $$ |{\Psi}\rangle=\sum_{i=1}^n c_i|{\psi_i}\rangle$$
+     for some specific choice of coefficients $c_i$. The set of vector \{$|{\psi_i}\rangle$\} are then said to *span* the Hilbert space of the quantum system.
     
-- A set of vectors \{$|{\psi_i}\rangle$\} is said to form a basis for the state space if the set of vectors is *complete* and if in addition they are  *linearly independent*. The latter condition means essentially that one cannot express a given basis vector as a linear combination of the rest of basis vectors.
-Linear independence can also be expressed as the requirement that if one has that
-$$\sum_{i=1}^n c_i |{\psi_i}\rangle=0\;\text{then}\; c_i=0\;\text{for all}\; i$$
+!!! info ""
+     A set of vectors \{$|{\psi_i}\rangle$\} is said to form a basis for the state space if the set of vectors is *complete* and if in addition they are  *linearly independent*. The latter condition means essentially that one cannot express a given basis vector as a linear combination of the rest of basis vectors.
+     Linear independence can also be expressed as the requirement that if one has that
+     $$\sum_{i=1}^n c_i |{\psi_i}\rangle=0\;\text{then}\; c_i=0\;\text{for all}\; i$$
 
-- The minimum number of vectors needed to form a complete set of basis states is known as the *dimensionality* of the state space. In quantum mechanis you will encounter systems whose Hilbert spaces have very different dimensionality, from the spin-1/2 particle (a $n=2$ vector space) to the free particle (whose state vectors live in an infinite vector space).
+!!! info ""
+     The minimum number of vectors needed to form a complete set of basis states is known as the *dimensionality* of the state space. In quantum mechanis you will encounter systems whose Hilbert spaces have very different dimensionality, from the spin-1/2 particle (a $n=2$ vector space) to the free particle (whose state vectors live in an infinite vector space).
 
 ### Bra vectors