Quantum Theory
Spin
Spin is actually a confusing term. An electron has a fundamental property that is an intrinsic angular momentum. Its direction can change but its magnitude can’t. Mass, charge, and spin are intrinsic properies of electrons and other fermions. Electrons are like tiny bar magnets and spin is an angular momentum pointing in the direction of the North pole.
There are several issues associted with this from the classical perspective.
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There are upper limits on the size of an electron. It would have to be spinning faster than light to achieve its measured angularar mimentum.
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Likewise the electron’s charge would have to be spinning faster than light to achive its measured magnetic moment.
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Clasical calculation of an electron’s gyromagnetic ratio gives about half of the measured value and requires a g-factor.
The Einstein-de Haas effect can be explained by spin. If an iron cylinder is suspended by a thin cord and a magentic field is applied to the cylinder its starts to rotate. Unpaired electrons align with the magnetic field. This changes the angular momentum. The cylinder rotates to conserve angular momentum.
An electron has a magnetic field. This a result of its spin \(m_s\). It can be in one of two directions, \(m_s = \frac{1}{2}\hbar\) up or \(m_s = -\frac{1}{2}\hbar\) down. If two electrons have the same spin then their magnetic fields combine. If they have opposite spin, their magnetic fields cancel out.
Gyromagnetic Ratio
The gyromagnetic ration is the ratio of a particle’s magnetic moment to its angular momentum. Its units are \(s^{-1}T^{-1}\) or equivalently \(Ckg^{-1}\).
g-factor
Spin is a form of angualar momentum. A charged particle with angular momentum generates a magnetic moment. The magnetic moment due to electron, or other fermion, spin has been accurately measured to be more than twice the expected value. The g-factor is a dimensionless multiplier that corrects for this. It varies between fermions.
Where:
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\vec{\mu} is the magnetic moment
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g = 2.00231930436092(36) is the electron g-factor
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e is the electron charge
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m is the electron mass
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\vec{S} is the angular momentum where \(|\vec{S}| = \hbar/2\)
Spin Statistics Theorem
The spin statistics theorem is fundamental to quantum theory. It was originally formulated by Wolfgang Pauli. It is very badly named because arguably there is no spin and no statistics involved!
If a particle’s wave function is asymmetric - it is a fermion with the Fermi-Dirac statistics. The particle has spin \(n + \frac{1}{2}, n \in \mathbb{Z}\) and satisfies the Dirac equation. The wave functions anti-commute \(\{\psi_i(x), \psi_j(x')\} = 0\).
If a particle’s wave function is symmetric - it is a boson with the Bose-Eisntein statistics. The particle has spin \(n, n \in \mathbb{Z}\) and satisfies the relativistic Klein-Gordon equation. The wave functions commute \([\psi_i(x), \psi_j(x')] = 0\).
These particles are mutually exclusive as if \(\{\psi_i(x), \psi_j(x')\} = 0\) then \([\psi_i(x), \psi_j(x')] \ne 0\) unless \(\psi_i(x)\psi_j(x') = 0\).
It would be better to call it the spin-commutation theorem!
Pauli Exclusion Principle
Consider two identical particles in states \(a\) and \(b\). The probability amplitude of the first particle being in state \(a\) is \(\psi_1(a)\).
For a boson, the probability amplitude that the particles are in the two states is.
If \(a = b\) then this is allowed.
For a fermion, the probability amplitude that the particles are in the two states is.
If \(a = b\) then the probabilty is zero. This means that they can’t be in the same state!