1st major update to lecture note 4

src/4_vector_spaces_QM.md

@@ -10,9 +10,9 @@ The lecture on vector spaces in quantum mechanics consists of the following part
 
 - [4.2. Inner product of state vectors](#42-inner-product-of-state-vectors)
 
-- [4.3. Matrix representation of ket and bra vectors](#43-matrix-representation-ket-bra-vectors)
+- [4.3. Matrix representation of ket and bra vectors](#43-matrix-representation-ket-and-bra-vectors)
 
-- [4.4. A two-dimensional Hilbert space](#44-two-dimensional-hilbert-space)
+- [4.4. A two-dimensional Hilbert space](#44-a-two-dimensional-hilbert-space)
 
 and at the end of the lecture there is a set of exercises 
 
@@ -95,7 +95,7 @@ between them, $\langle{\psi}|{\phi}\rangle$, as follows.
 
 The inner product in quantum mechanics is the analog of the usual scalar product that one encounters in vector spaces, and which we reviewed in the previous lecture. As in usual vector spaces, the inner product of two state vectors is a  *scalar* and in this case a complex number in general. 
 
-!!! tip "Meaning of the inner product in quantum mechanics"
+!!! tip "Interpretation of the inner product in quantum mechanics"
      1.   The value of the inner product $\langle{\psi}|{\phi}\rangle$ indicates the **probability amplitude** (not the probability) of measuring a system, which characterised by the state $|{\phi}\rangle$, to be in the state $|{\psi}\rangle$. 
      2.   This inner product can also be understood as measuring the **overlap** between the state vectors $|{\psi}\rangle$ and $|{\phi}\rangle$. 
      3.   Then the  **probability of observing the system to be in the state $|\psi\rangle$** given that it is in the state $|\phi\rangle$ will be given by $$|\langle \psi | \phi \rangle|^2 \, .$$ Since the latter quantity is a probability, we know that it should satisfy the condition that 
@@ -208,7 +208,7 @@ $$
 (c_+,c_-) = (i,-4) \, ,\qquad
 (c_+,c_-) = (2,5) \, .\qquad 
 $$
-!!! danger ""
+!!! warning ""
      Note however that many other bases are possible, and that the physics of a quantum system do not depend on the basis that we choose.
 
 The bra vectors associated to these ket vectors will be given by