Introduction | Notion

Scale and units

⚛️ Atomic Physics

Size: $\sim 10^{-10} \to 10^{-9} \,\rm m$

Energies: $\sim 1\to 100 \, \rm eV$

☢️ Nuclear Physics

Size: $\sim 10^{-15}\, \rm m$ or $1 \,\rm fm$ (Fermi or Femtometer)

Energy: $\sim 10^6 \, \rm eV$ or $1 \rm \, MeV$

mass: $\sim \rm \, GeV/c^2$

🗒️ Note: we will work in natural units $c=\hbar = 1$

Useful quantities:

$$ \begin{aligned} \hbar c &\approx 200 \rm\, MeV \, fm \\ \text{fine structure constant: } \quad\alpha &=\frac{1}{4\pi \epsilon_0}\frac{e^2}{\hbar c} = \frac{1}{137} \; \text{ (dimensionless) }\;\;

\end{aligned} $$

⚠️ Warning: don’t use SI units

Wave particle duality and its consequences

Nuclei exhibit wave-like Phenomena, thus they follow the following relations:

De Broglie relation: $\lambda = \frac{h}{p}$
Heisenberg uncertainty principle: $\Delta x \Delta p_x \ge \frac{\hbar}{2}$

🗒️ Note: equality only holds for ideal situations
For a non-relativistic particle:
- Kinetic energy: $E_\text{kin}=\frac{p^2}{2M}$
- Schrodinger equation:
  
  $$ -\frac{\hbar ^2}{2M} \frac{ \text d^2 \psi(x)}{\text dx^2} + V(x) \psi (x)=E \psi(x) $$
  
  which can be rewritten using the Hamiltonian as
  
  $$ \hat H \psi(x)=E\psi(x) $$
- Wave function with a time dependence: $\Psi(x,t)= \psi(x) e^{iEt/\hbar}$
- Must respect unitary:
  
  $$ \int^\infin_{-\infin} P(x) \, \text dx = \int ^\infin _{-\infin} \Psi^* (x,t) \Psi(x,t) \, \text dx =1 $$

Antiparticles

Antiparticles: have the same properties but reversed charges. Postulated by Dirac they come from the fact that the total energy of a particle $E^2 =p^2 c^2+ m_0^2 c^4$ has a negative energy solution: $E =\pm \sqrt{p^2 c^2+ m_0^2 c^4}$

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Properties:

When they come into contact with particles they annihilate typically releasing $\gamma$ radiation
They can bond to form antimatter

Angular momentum

$$ \vec L = \vec r \times \vec p \Rightarrow \left \{ \begin{aligned} \hat L_x&=-i \hbar \left ( y\frac{\partial }{\partial z}-z\frac{\partial}{\partial y} \right ) \\ \hat L_y&=i \hbar \left ( x\frac{\partial }{\partial z}-z\frac{\partial}{\partial x} \right ) \\ \hat L_z&=-i \hbar \left ( x\frac{\partial }{\partial y}-y\frac{\partial}{\partial x} \right ) \end{aligned} \right . $$