6.18

Electron Spin and Magnetic Dipole Moment

Intrinsic Spin Angular Momentum

• Electrons possess an intrinsic property called spin angular momentum, analogous to mass and charge.
• The concept of spin is supported by experimental evidence.

Magnetic Dipole Moment

• Electrons have a magnetic dipole moment proportional to their spin angular momentum.
• Recall from previous discussions that \mu (magnetic dipole moment) is related to l (orbital angular momentum).
• This relationship was derived from a model of an electron as a rotating ring, leading to a current I and area A to calculate \mu .
• \mu = I \times A
• q represents the magnitude of the electron's charge.
• The vector direction of \mu is opposite to the angular momentum vector.

Spin Magnetic Dipole Moment

• \mu_l represents the magnetic dipole moment associated with the electron's motion.
• Contrarily, \mu_s represents the spin magnetic dipole moment, an intrinsic property unrelated to electron motion around the atom.
• The relationship between \mus and spin angular momentum S is: \mus = - \frac{q}{m} S
• Notably, there's a missing factor of 2 in this relationship, which is not fully understood due to the intrinsic nature of electron properties.

Hamiltonian with Magnetic Field

• The Hamiltonian is modified when a magnetic field is introduced by including two additional terms:
  • -\mu_l \cdot B
  • -\mu_s \cdot B
• In an experiment with only the z-component of the magnetic field present (Bz), the Hamiltonian becomes: H = H0 - \mu{lz} Bz - \mu{sz} Bz
• The spin operators behave similarly to previous angular momentum operators.
• The new energy levels are the sum of the old energy levels and the additional terms.
• Old energy: -\frac{E_1}{n^2}

Energy Levels

• The term involving \mul contributes \pm \frac{q}{2m} Lz B to the energy, where Lz = \hbar m{lz} .
• Which can be rewritten as:
  \hbar \omegal m{lz}
  Where the Larmor frequency is given by:
  \omega_l = \frac{qB}{2m}
• The spin term contributes 2\hbar \omegal m{sz} to the energy, with an extra factor of 2.
• Each energy level is split into two due to spin.

Allowed Values of Spin Angular Momentum

• The allowed values for the magnitude of spin angular momentum are given by the quantum number s = \frac{1}{2} .
• Unlike the orbital angular momentum quantum number l , which ranges from 0 to integers, the spin angular momentum magnitude has only one value.
• The magnitude of the spin angular momentum squared is:
  S^2 = s(s+1) \hbar^2
  = \frac{1}{2} (\frac{1}{2} + 1) \hbar^2 = \frac{3}{4} \hbar^2
• Thus, the magnitude of spin angular momentum S = \sqrt{\frac{3}{4}} \hbar = \frac{\sqrt{3}}{2} \hbar .
• Every electron possesses this intrinsic spin angular momentum magnitude.
• Even a free electron with zero momentum will have quantum numbers for its spin angular momentum.

Z-Component of Spin Angular Momentum

• The z-component quantum number ms follows the rule of ranging from -s to +s, yielding two allowed states: ms = +\frac{1}{2}, -\frac{1}{2} , referred to as spin up and spin down.
• The angle between the spin angular momentum vector and its z-component is constant.

Quantum Numbers for Hydrogen Atom

• Five quantum numbers are needed to define the state of an electron in a hydrogen atom: n, l, ml, s = \frac{1}{2}, ms .
• Examples of allowed states: (3, 2, -1, \frac{1}{2}, -\frac{1}{2}) and (2, 0, 0, \frac{1}{2}, +\frac{1}{2}) .

Representing Spin States

• Additional symbols are needed to represent spin up and spin down states.
• Spin-up state: \begin{bmatrix} 1 \ 0 \end{bmatrix} .
• Spin-down state: \begin{bmatrix} 0 \ 1 \end{bmatrix} .
• Before measurement, particles can be in a mixed state (linear combinations of eigenstates), like \frac{1}{\sqrt{3}} \begin{bmatrix} 1 \ 0 \end{bmatrix} + \sqrt{\frac{2}{3}} \begin{bmatrix} 0 \ 1 \end{bmatrix} .

Size of Splitting

• The additional splitting due to spin needs to be considered alongside previous energy level diagrams.
• The m_s quantum numbers (+\frac{1}{2} and -\frac{1}{2}) are multiplied by 2 due to the relationship between angular momentum and magnetic dipole moment.
• The size of the spin-related energy jump is the same as the orbital angular momentum splitting.
• Each state splits into two: spin up and spin down.
• Spin up and spin down correspond to the direction of the magnetic field.
• Energy level diagrams now need to incorporate five quantum numbers.
• The spin part of the Hamiltonian causes a split equal in size to the splitting from orbital angular momentum.
• The splitting also occurs for l = 0 states because spin is an intrinsic property.
• Selection rules for m_s are the same: +\frac{1}{2} to -\frac{1}{2}.

Electron Spin Resonance (ESR)

• To study electron spin in isolation, use an electron in an s-state (l = 0) within a long-chain molecule.
• Adjust the frequency on the ESR controller such that \hbar \omega matches the \Delta E (energy difference) for resonance.
• Resonance in quantum mechanics means that an incoming photon is absorbed, and the electron transitions from spin down to spin up.
• In classical mechanics, resonance occurs when the frequency of an external force matches the natural frequency of an object; in quantum mechanics, it's the energy of the photon matching the energy difference between states.
• The ESR experiment involves tuning the frequency of a coil, changing the energy of photons interacting with a DPPH chemical.
• Helmholtz coils provide a constant magnetic field that causes energy splitting.
• The splits only occur when a magnetic field is present.
• The frequency of photons must match the energy difference for an interaction to occur.
• Analogy to index of refraction: UV light cannot pass through glass because its frequency matches the natural frequencies of electrons in the glass.

Allowed Interactions of Nature

• Two possible processes:
  • A photon with energy \hbar \omega is absorbed, and an electron transitions from a low-energy state to a higher-energy state.
  • A photon interacts with an electron in the upstate, and two photons are emitted as the electron goes to the down state.
• Both processes are allowed because of energy conservation.