AcademyAtomic Quantum Structure
Academy
Electron Spin
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
Electron spin is intrinsic angular momentum. It is not literal rotation of a tiny sphere, but it behaves like angular momentum and gives the electron a magnetic moment.
Spin adds new quantum numbers and explains two-state outcomes in measurements such as Stern-Gerlach experiments.
Notation
\(s\)
spin quantum number for electron
1
\(m_s\)
spin projection quantum number
1
\(S\)
spin angular momentum magnitude
\(\mathrm{J\,s}\)
\(S_z\)
spin angular momentum projection
\(\mathrm{J\,s}\)
\(\mu_s\)
spin magnetic moment
\(\mathrm{J\,T^{-1}}\)
\(g_s\)
electron spin g-factor
1
Method
Derivation 1: Quantize spin magnitude
Spin angular momentum follows the same angular-momentum algebra as orbital angular momentum, but the electron has fixed \(s=1/2\).
Electron spin quantum number
\[s=\frac12\]
Spin magnitude
\[S=\sqrt{s(s+1)}\hbar\]
Derivation 2: Quantize spin projection
For an electron, measuring spin along a chosen axis gives two possible projections.
Allowed projections
\[m_s=+\frac12,\,-\frac12\]
Projection
\[S_z=m_s\hbar\]
Derivation 3: Connect spin to magnetism
Electron spin has a magnetic moment, so external magnetic fields can split spin states.
Spin magnetic moment scale
\[\mu_s\approx g_s\mu_Bm_s\]
Electron spin g-factor
\[g_s\approx2\]
Rules
Electron spin quantum number
\[s=\frac12\]
Spin magnitude
\[S=\sqrt{s(s+1)}\hbar\]
Spin projection
\[S_z=m_s\hbar\]
Spin projections
\[m_s=\pm\frac12\]
Examples
Question
What are the possible measured values of \(S_z\) for an electron?
Answer
\[S_z=+\frac{\hbar}{2}\]
or \[S_z=-\frac{\hbar}{2}\]
Checks
- Electron spin has only two projections along any chosen axis.
- Spin is intrinsic, not classical spinning motion.
- Spin affects magnetic interactions.
- Spin is required to explain electron filling in atoms.