Update 4_vector_spaces_QM.md

src/4_vector_spaces_QM.md

@@ -151,7 +151,7 @@ Therefore, we can present  bra vector $\bra{\chi}$ as row vectors and ket vector
 
 As a practical example to illustrate the concepts presented above, we will consider a quantum system which is fundamental for quantum mechanics and its applications. This system corresponds to the possible states that the intrinsic angular momentum of an electron, known as {\it spin}, can occupy. As you will see in following courses, the Hilbert space for the electron spin
 has dimension $n=2$, meaning that we can found an electron {\it pointing} either
-in the up direction, denoted by $\ket{+}$, or the down direction, denoted by $\ket{-}$.
+in the up direction, denoted by $|+\rangle$, or the down direction, denoted by $|-\rangle$.
 
 The general state vector of this system will be expressed as a linear superposition of the {\it up} and {\it down} states,
 $$
@@ -234,12 +234,12 @@ $$
     \frac{1}{\sqrt{2}} \left( 1 , i  \right) \left( \begin{array}{c}0 \\1 \end{array} \right) =
 \frac{i}{\sqrt{2}} 
 $$
-  and the associated probability will be given by
- $$
-    |\braket{\psi}{\phi}|^2 = \frac{1}{2}
- $$
-  meaning that if I measure the state $\ket{\phi}$, I will have a 50\% probability
-  of finding it in the state $\ket{\psi}$.
+and the associated probability will be given by
+$$
+|\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 physical
   sense. Note also that I am using normalised vectors, you can check yourselves
@@ -292,18 +292,18 @@ Demonstrate this result.
 
 ***
 
-**4)** *Basis Vectors*
+**4)** *Basis vectors*
 
 *(a)* Show that the following vectors are linearly dependent:
 
 $$
-\rangle {a}=\begin{pmatrix} 1\\ 2\\1\end{pmatrix},~~\rangle{b}=\begin{pmatrix} 0\\ 1\\0\end{pmatrix},~~\rangle{c}=\begin{pmatrix} -1\\ 0\\-1\end{pmatrix}
+|a\rangle=\begin{pmatrix} 1\\ 2\\1\end{pmatrix},~~|b\rangle=\begin{pmatrix} 0\\ 1\\0\end{pmatrix},~~|c\rangle=\begin{pmatrix} -1\\ 0\\-1\end{pmatrix}
 $$
 
 *(b)* Is the following set of vectors linearly independent?
 
 $$
-\rangle{a}=\begin{pmatrix} 2\\ 0\\0\end{pmatrix},~~\rangle{b}=\begin{pmatrix} 0\\ -1\\0\end{pmatrix},~~\rangle{c}=\begin{pmatrix} 0\\ 0\\-1\end{pmatrix}
+|a\rangle=\begin{pmatrix} 2\\ 0\\0\end{pmatrix},~~|b\rangle=\begin{pmatrix} 0\\ -1\\0\end{pmatrix},~~|c\rangle=\begin{pmatrix} 0\\ 0\\-1\end{pmatrix}
 $$
 
 ***
@@ -318,13 +318,13 @@ $$
 |\Phi\rangle=3|u_1\rangle-2|u_2\rangle+4|u_3\rangle
 $$
 
-*(a)* Find $\bra{\Psi}$ and $\bra{\Phi}$.
+*(a)* Find $\langle \Psi|$ and $\langle \Phi|$.
 
-*(b)* Compute the inner product $\braket{\Phi}{\Psi}$.
+*(b)* Compute the inner product $\langle \Phi | \Psi \rangle$.
 
-*(c)* Show that $\braket{\Phi}{\Psi}=\braket{\Psi}{\Phi}^*$$.
+*(c)* Show that $\langle \Phi | \Psi\rangle=\langle \Psi|\Phi\rangle^*$$.
 
-*(d)* Write the column vector representing the vector $\ket{\Psi}$ in the given basis. Then write down the row vector that represents $\bra{\Psi}$ in the given basis as well.
+*(d)* Write the column vector representing the vector $|\Psi\rangle$ in the given basis. Then write down the row vector that represents $\langle \Psi|$ in the given basis as well.
 
 ***
 
@@ -333,10 +333,10 @@ $$
 The state vector for a spin half particle that passes through a magnetic field oriented in the direction $\^n$ and exists with its spin component in the direction of the magnetic field, i.e. $S=\vec{S}\cdot\^n=\frac{1}{2}\hbar$ is given by
 
 $$
-\ket{S}=cos(\frac{1}{2} \theta)\ket{+}+sin(\frac{1}{2} \theta) \, e^{i\phi}\ket{-}
+|S\rangle =cos(\frac{1}{2} \theta) |+\rangle +sin(\frac{1}{2} \theta) \, e^{i\phi} |-\rangle
 $$
 
-where $\^n=sin\theta \,cos\phi \, \^i +sin\theta \, sin\phi \,\^j+cos\theta \, \^k$. 
+where $\hat{n}=sin\theta \,cos\phi \, \hat{i} +sin\theta \, sin\phi \,\hat{j}+cos\theta \, \^k$. 
 
 *(a)* What is the corresponding bra vector?