Update 4_vector_spaces_QM.md

src/4_vector_spaces_QM.md

@@ -194,6 +194,8 @@ $$ |{\Psi}\rangle = \left( \begin{array}{c}i \\ -4 \end{array} \right) \, , \qqu
 \langle{\Psi}|=\left(  -i , -4  \right)\, , $$
 $$|{\Psi}\rangle=  \left( \begin{array}{c}2 \\ 5 \end{array} \right) \, , \qquad
 \langle{\Psi}|=  \left( 2 , 5  \right) \, .$$
+Note however that the above vectors are not normalised (the inner product with themselves is different from unity), and thus
+cannot represent physical states. We show below an explicit example of a normalised state vector belonging to this Hilbert space.
 
 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
 $$
@@ -301,7 +303,7 @@ $$
 
 The state vector for a spin half particle that passes through a magnetic field oriented in the direction $\hat{n}$ and exists with its spin component in the direction of the magnetic field, i.e. $S=\vec{S}\cdot\hat{n}=\frac{1}{2}\hbar$ is given by
 $$
-|S\rangle =\cos(\frac{1}{2} \theta) |+\rangle +\sin(\frac{1}{2} \theta) \, e^{i\phi} |-\rangle
+|S\rangle =\cos(\theta/2) |+\rangle +\sin(\theta/2) \, e^{i\phi} |-\rangle
 $$
 where $\hat{n}=\sin\theta \,\cos\phi \, \hat{i} +\sin\theta \, \sin\phi \,\hat{j}+\cos\theta \, \hat{k}$.