'magnetic moment'

ch05/note05.md

@@ -3,7 +3,10 @@
 :::{admonition} What you need to know
 :class: note
 
-- Experiment of Stern and Herlach provdies estblaishes the existance of spin for electrons. Contrary to the suggestive name, *spin* is an intrinsic magnetic momentum which is permanently attached to a subatomic particle and has nothing to do with "spinning" or motion. Spins is a fundamental property of particles just like mass or charge. 
+- **Magnetism** results from the circular motion of charged particles.
+- It is expected that atoms that have electrons with non-zero value of orbital angular momentum will be affected by the external magnetic field. 
+- Experiment of Stern and Herlach estblaished the existance of intrinsic magnetic moment of lectrons termed spin that are affected by magnetic field regardless of orbital angular momentum. 
+- Contrary to the suggestive name, *spin* is an intrinsic magnetic momentum which is permanently attached to a subatomic particle and has nothing to do with "spinning" or motion. Spins is a fundamental property of particles just like mass or charge. 
 - Particles with half-integer spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer spins, such as 0, 1, 2, are known as bosons . The two families of particles obey different rules. Fermions obey the Pauli exclusion principle:  there cannot be two identical fermions simultaneously having the same quantum numbers, e.g, having the same position, velocity and spin direction. In contrast, bosons  have no such restriction, so they may "bunch together" even if in identical states.
 - *Normal Zeeman effect*: splitting of singlet states with spin zero in the magnetic field is due to electron's angular momentum can be understood classically. 
 - *Anomalous Zeeman effect*: Is a more general case of electron spin and angular mometnum both contributing to splitting of energy levels.
@@ -27,18 +30,36 @@ $${\vec{F} = q\left( \vec{E} + \vec{v}\times \vec{B}\right)}$$
 
 ### Rotating charge generates mangetic moment
 
--  A moving charge interacts with an external magnetic field. When an electron is in a state with $l > 0$, it can be thought to be in quantum mechanical circular motion around the nucleus and generate its own magnetic field. Note that this motion is not classical but here we are just trying to obtain a wire frame model based on classical interpretation. The electron has now a magnetic moment given by (see your physics lecture notes):
+:::{figure-md} markdown-fig
+<img src="./images/electro1.gif" alt="electro" class="bg-primary mb-1" width="300px">
 
-![](./images/magnetic.jpg)
+A GIF demonstrating the existence of a magnetic field around a current carrying wire.
+:::
+
+
+-  A moving charge interacts with an external magnetic field. 
+- When an electron is in a state with $l > 0$, one can think of a circular motion of charge (of the wavefunction describing electron) around the nucleus and generate its own magnetic field. 
+- Note that this motion is not classical but here we are just trying to obtain a wire frame model based on classical interpretation.
+
+:::{figure-md} markdown-fig
+<img src="./images/magnetic.jpg" alt="electro" class="bg-primary mb-1" width="300px">
+
+Shown are magnetic moment and angular momentum generated by a charge moving on an obrit. 
+:::
+
+### Magnetic moment of  electron 
+
+According to classical mechanics a charged particle like an electron that is rotating around an axis has a magnetic moment given by:
 
 $${\vec{\mu} = \gamma_e\vec{L}}$$
 
-- where $\gamma_e$ is the magnetogyric ratio of the electron ($-\frac{e}{2m_e}$). We choose the external magnetic field to lie along the $z$-axis and therefore it is important to consider the $z$ component of $\vec{\mu}$:
+- where $\gamma_e$ is the magnetogyric ratio of the electron expressed via fundamental constants ($-\frac{e}{2m_e}$) We choose the external magnetic field to lie along the $z$-axis and therefore it is important to consider the $z$ component of $\vec{\mu}$:
 
 $${\mu_z = -\left(\frac{e}{2m_e}\right)L_z = -\left(\frac{e\hbar}{2m_e}\right)m \equiv -\mu_B m}$$
 
 - where $\mu_B$ is the Bohr magneton as defined above. The interaction between a magnetic moment and an external magnetic field is given by (classical expression):
 
+
 $${U = -\vec{\mu}\cdot\vec{B} = -|\vec{\mu}||\vec{B}|\cos(\alpha)}$$
 
 - Where $\alpha$ is the angle between the two magnetic field vectors. This gives the energy for a bar magnet in presence of an external magnetic field:
@@ -52,7 +73,7 @@ $${U = -\mu_z B = \frac{eB}{2m_e}L_z}$$
 
 ### Effect of magnetic field on atoms
 
-- In quantum mechanics, a magnetic moment (here corresponding to a non $s$ orbital electron) may only take specific orientations.
+- In quantum mechanics, a magnetic moment (here corresponding to a non $s$ orbital electron) may only take specific orientations!
 
 - The $z$-axis is often called the quantization axis. The eigenvalues of $\hat{L}_z$ essentially give the possible orientations of the magnetic moment with respect to the external field. For example, consider an electron on $2p$ orbital in a hydrogenlike atom. The electron may reside on any of $2p_{+1}$, $2p_0$ or $2p_{-1}$ orbitals (degenerate without the field). For these orbitals $L_z$ may take the following values ($+\hbar, 0, -\hbar$):
 
@@ -215,3 +236,5 @@ $${E_{n,m_l,m_s} = -\frac{m_ee^4Z^2}{2(2\pi\epsilon_0)^2\hbar n^2} + \frac{eB\hb
 | $L=\hbar\sqrt{l(l+1)}$<br>$L_z=\hbar m$                      | $S=\hbar\sqrt{s(s+1)}=\hbar\sqrt{3/4}$<br>$S_z=\hbar m_s= \pm \hbar/2$ |
 | $\mu_L=- g_l \frac{e}{2m_e}L$<br>$g_l=1$                     | $\mu_S = g_s \frac{e}{2m_e}S$ <br> $g_s \approx 2$           |
 |                                                              |                                                              |
+### Problems
+