🤖 Bosons: particles with $\Z$ spin
- Spin $0$: $^1 \rm H$ and $\rm ^4 He$ in ground state, pion, Higgs boson
- Spin 1: $^1 \rm H$ and $^4 \rm He$ in first excited state, $\rho$ meson, photon $W$ and $Z$ bosons, gluons
- Spin 2: $^{16} \rm O$ in ground state, graviton
🍹 Fermions: particles with $\frac 12 \Z$ spin
- Spin $\frac 12$: $^3 \rm He$ in the ground state, proton, neutron, quark, electron, neutrino
- Spin $\frac 32$: $^5 \rm He$ in ground state, $\Delta$ baryons (excitations of protons and neutrons)
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/3bb920d9-03db-42e9-b536-5e51a1844f77/Spin_of_a_composite_particle.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/3bb920d9-03db-42e9-b536-5e51a1844f77/Spin_of_a_composite_particle.png" width="40px" /> Spin of a composite particle: total angular momentum usually $J$ not $S$, $J$ is the quantised vector sum of all the spins and all the orbital angular momentum of all the constituents
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<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/1301850c-bca0-43ed-a4af-4f0f32ea2a5b/Pauli_exclusion_principle.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/1301850c-bca0-43ed-a4af-4f0f32ea2a5b/Pauli_exclusion_principle.png" width="40px" /> Pauli exclusion principle: no more than one fermion can occupy a single quantum state ($m_s$ is part of the description of the state)
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<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/3fe8e92d-d31a-4668-88d7-ed8b419e09f5/Pauli_Exclusion_principle_tougher.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/3fe8e92d-d31a-4668-88d7-ed8b419e09f5/Pauli_Exclusion_principle_tougher.png" width="40px" /> Pauli exclusion principle tougher: the overall wave function of a system of identical fermions must be antisymmetric under exchange of any pair
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💼 Case: consider 2 spin-$\frac 12$ particles in the same spatial state
The overall wave function needs to be ‘
$$ \Psi(\vec r_1, \vec r_2,m_1,m_2)=\frac 1 {\sqrt{2}} \phi_{1s}(\vec r_1)\phi_{1s} (\vec r_2 )(\uparrow \downarrow - \downarrow \uparrow ) $$
Spatial part symmetric, spin antisymmetric
💼 Case: 2-fermion spin-state such as $\uparrow \uparrow$
The particle have to be in different spatial states
$$ \Psi(\vec r_1, \vec r_2 ,m_1,m_2)=\frac 1 {\sqrt{2}} [ \phi_{1s}(\vec r_1 ) \phi_{2s} (\vec r_2)-\phi_{2s}(\vec r_1)\phi_{1s} (\vec r_2)]\uparrow \uparrow $$
🗒️ Note: identical bosons don’t follow Pauli exclusion, they’re symmetric under pair exchange
Spinless bosons
$$ \Psi(\vec r_1,\vec r_2,\vec r_3\ldots)=\frac{1}{\sqrt{2}}\phi(\vec r_1)\phi(\vec r_2)\phi(\vec r_3) $$
🗒️ Note: nothing observable can change if we swap particles for bosons
$$ \begin{aligned} |\Psi(\vec r_1,\vec r_2,m_1,m_1)|^2=|\Psi(\vec r_2,\vec r_2,m_2,m_1)|^2 \\ \Psi(\vec r_1,\vec r_2,m_1,m_1)=e^{i\alpha}\Psi(\vec r_2,\vec r_2,m_2,m_1) \end{aligned} $$
Since the swaps need to return the same state without any phase change $\alpha=\{\pi,2\pi\}$
💳 Take-away: All particles in nature are either bosons or fermions. Their statistical properties are very different: no two fermions can be in the same state, but there is no such restriction on bosons.
🗒️ Note: our past assumption $n_Q\gg n$ is invalid for fermions thus we use the Gibbs distribution
If we consider the partition function of a single energy level $\epsilon_r$, if it has $n$ particles
$$ \mathcal{Z} r=1+e^{(\mu-\epsilon_r)\beta}+e^{2(\mu-\epsilon_r)\beta}\ldots\equiv \sum{n=0} e^{n(\mu-\epsilon_r)\beta} $$
Thus the grand partition function is
$$ \mathcal Z=\mathcal Z_1 \mathcal Z_2 \mathcal Z_3 \ldots = \prod_r \mathcal Z_r $$
🗒️ Note: For fermions the sum in $\mathcal Z_r$ is restricted to the first two terms
