Intrinsic Spin

incomplete

• Spin in QM has no analogy in classical dynamics

• Consider case where there is no angular momentum, $L=0$

• Contribution to Hamiltonian from spin:

$$H=-\vec{\mu }\cdot \vec{B}=-g\frac{q{B}_z}{2m_0}\cdot s\hslash$$

• Where $s$ is the spin quantum number. For an electron, the spin values can be $+/-\frac{1}{2}$

• “g value” for intrinsic spin for electrons is 2. For protons is 5.6

• G value is a “fudge factor”

Hydrogen Hamiltonian is comprised of the potential term, kinetic term, spin term and angular momentum term.

Silver screen experiment

MRI

Hamiltonian for Spin

$$H=-{\vec{\mu }}_s\cdot \vec{B}={\omega }_0{S}_z$$ $${\omega }_0=-\frac{g|e|B_z}{2m_0};g=+5.6$$

for protons

Gives use to 2 level system in ${\omega }_0=$ both $+/-\frac{\hslash }{2}$

• Stern-Gerlach Experiment
• g-factor (fudge factor)

$$\vec{\mu }=g\frac{\mu B}{\hslash }\vec{s}$$ $$|\vec{s}|=\sqrt{\frac{1}{2}\left(\frac{1}{2}+l\right)}\hslash$$ $$s=\frac{1}{2}$$ $$\Psi (x,t)\to \Psi (x,t)\chi (t)$$ $${\hat{H}}^{\prime \prime }=-{\vec{\mu }}_s\cdot \vec{B}$$

${\hat{H}}^{\prime \prime }$ is the spin derivative of the hamiltonian

$$\hat{H}=H_0+H^{\prime }+H^{\prime \prime }$$