1st major update to lecture note 4

src/4_vector_spaces_QM.md

@@ -2,38 +2,47 @@
 title: Vector spaces in quantum mechanics
 ---
 
-# Vector spaces in quantum mechanics
+# 4. Vector spaces in quantum mechanics
 
 The lecture on vector spaces in quantum mechanics consists of the following parts:
 
-- [Dirac notation and Hilbert spaces](#dirac-notation-and-hilbert-spaces)
+- [4.1. Dirac notation and Hilbert spaces](#41-dirac-notation-and-hilbert-spaces)
 
-- [Inner product of state vectors](#inner-product-of-state-vectors)
+- [4.2. Inner product of state vectors](#42-inner-product-of-state-vectors)
 
-- [Matrix representation of ket and bra vectors](#matrix-representation-ket-bra-vectors)
+- [4.3. Matrix representation of ket and bra vectors](#43-matrix-representation-ket-bra-vectors)
 
-- [A two-dimensional Hilbert space](#two-dimensional-hilbert-space)
+- [4.4. A two-dimensional Hilbert space](#44-two-dimensional-hilbert-space)
 
-and at the end of the lecture one can find the corresponding [Problems](#problems)
+and at the end of the lecture one can find the exercises 
+
+- [4.5. Problems](#45-problems)
+
+---
 
 The contents of this lecture are summarised in the following **videos**:
 
-- [4_vector_spaces_QM_video1](https://www.dropbox.com/s/mnccmpff33pre9r/linear_algebra_04.mov?dl=0)
+- [1. Dirac notation and properties of Hilbert spaces](https://www.dropbox.com/s/mnccmpff33pre9r/linear_algebra_04.mov?dl=0)
 
-- [4_vector_spaces_QM_video2](https://www.dropbox.com/s/709bh9j083y7d0s/linear_algebra-06.mov?dl=0)
+- [2. Algebra with Dirac notation - bras and kets](https://www.dropbox.com/s/709bh9j083y7d0s/linear_algebra-06.mov?dl=0)
 
-- [4_vector_spaces_QM_video3](https://www.dropbox.com/s/k9plspkonnk3nc0/linear_algebra-07.mov?dl=0)
+- [3. Finding expansion coefficients for Dirac notation](https://www.dropbox.com/s/k9plspkonnk3nc0/linear_algebra-07.mov?dl=0)
 
+**Total lenght of the videos: ~14 minutes**
 
-## Dirac notation and Hilbert spaces
+---
+
+## 4.1. Dirac notation and Hilbert spaces
 
-In the previous lecture we reviewed the basic properties of linear vector spaces. We now discuss how the same formalism
+In the previous lecture, we reviewed the basic properties of linear vector spaces. Next, we will discuss how the same formalism
 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 is known as the *state space* of the system.
-Such 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 state. 
+This vector space is known as the *state space* of the system.
+
+!!! 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. 
 
 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: