Update 4_vector_spaces_QM.md

src/4_vector_spaces_QM.md

@@ -189,11 +189,11 @@ $$
 Note however that many other bases are possible, and that the physics of a quantum system do not depend on the basis that we choose.
 The bra vectors associated to these ket vectors will be given by
 $$ |{\Psi}\rangle=   \left( \begin{array}{c}3 \\ -2i \end{array} \right) \, , \qquad
-\langle{\Psi}|=\left( 3, 2i \right) $$
+\langle{\Psi}|=\left( 3, 2i \right) \, ,$$
 $$ |{\Psi}\rangle = \left( \begin{array}{c}i \\ -4 \end{array} \right) \, , \qquad
-\langle{\Psi}|=\left(  -i , -4  \right) $$
+\langle{\Psi}|=\left(  -i , -4  \right)\, , $$
 $$|{\Psi}\rangle=  \left( \begin{array}{c}2 \\ 5 \end{array} \right) \, , \qquad
-$$ \langle{\Psi}|=  \left( 2 , 5  \right)$$
+\langle{\Psi}|=  \left( 2 , 5  \right) \, .$$
 
 We also know how we can evaluate the inner product between any two state vectors belonging to this Hilbert space. If we have two state vectors given by
 $$
@@ -210,10 +210,10 @@ and the associated probability will be given by
 $$
 |\langle \psi|\phi\rangle|^2 = \frac{1}{2}
 $$
-meaning that if I measure the state $\ket{\phi}$, I will have a 50\% probability
-of finding it in the state $\ket{\psi}$. Recall that probabilities must always be smaller than 1 to make physicalsense. Note also that I am using normalised vectors, you can check yourselves that
+meaning that if I measure the state $| \phi \rangle $, I will have a 50\% probability
+of finding it in the state $| \psi \rangle$. Recall that probabilities must always be smaller than 1 to make physicalsense. Note also that I am using normalised vectors, you can check yourselves that
 $$
-\braket{\psi}{\psi} =  \braket{\phi}{\phi} = 1 \, .
+\langle \psi |\psi \rangle =  \langle \phi | \phi\rangle = 1 \, .
 $$