Quantum mechanics/Magnetic fields and spin

Some classical physics

If a 'classical' electron rotates around a point with angular momentum ${\vec {L}}$ , the system will have a magnetic dipole moment ${\vec {\mu }}$ (a vector perpendicular to the plane of rotation) proportional to the angular momentum:

${\vec {\mu }}=-{\frac {e}{2M}}{\vec {L}}=-\gamma {\vec {L}}$ (Eq. 1)

γ is called the gyromagnetic constant and M is the mass of the electron. In classical physics, a magnetic needle can be approximated as magnetic dipole moment and it has the tendency to align itself to a magnetic field ${\vec {B}}$ . More quantitatively, the energy of a magnetic dipole moment ${\vec {\mu }}$ in a field ${\vec {B}}$ is $E=-{\vec {\mu }}\cdot {\vec {B}}$ (scalar product) and if we consider the field aligned along the z axis

$E_{\mathrm {magnetic} }=-\mu _{z}B_{z}=\gamma L_{z}B_{z}$ (Eq. 2)

The hydrogen atom in a magnetic field

The angular momentum along the z direction for an electron in the hydrogen atom is Lz = mħ with m the magnetic quantum number. If a hydrogen atom is put where there is a magnetic field, there is an extra term of the energy, dependent on the quantum number m:

$E_{\mathrm {magnetic} }=-\mu _{z}B_{z}=m\hbar \gamma B_{z}$ (Eq. 3)

The previously degenerate levels of orbital 2p-1, 2p0, 2p+1 are split by the external magnetic field. This is called the Zeeman effect.

The Zeeman effect does not explain all the observations. You need another ingredient to explain the finer detail of the hydrogen spectrum.

The Stern-Gerlach experiment

A beam of particles with same mass, same velocity and different magnetic dipole moments passing through a region with inhomogeneous magnetic field will follow a trajectory that depends on the magnetic dipole moment.

The hydrogen atom in the ground state does not have magnetic dipole moment due to the orbital n=1, l=0, m=0. Nevertheless it is found that two different trajectories are followed by hydrogen atoms: there is an additional, previously unknown, magnetic dipole moment.

It was postulated (Goudsmit and Ulhembeck) that the electron possesses an intrinsic angular moment and (being charged) an intrinsic magnetic dipole moment.

Spin

The intrinsic angular momentum of the electron is called spin. The operators ${\hat {S}}^{2},{\hat {S}}_{z}$ can be defined in analogy to the angular momentum operators ${\hat {L}}^{2},{\hat {L}}_{z}$ with the eigenfunction σ satisfying

${\hat {S}}^{2}\omega =s(s+1)\hbar ^{2}\omega$ (Eq. 4)

• s is the angular quantum number for the spin

${\hat {S}}_{z}\omega =m_{s}\hbar \omega$ (Eq. 5)

• ms is the magnetic quantum number for the spin

However, as the experiment shows that there are only two possible values of magnetic dipole moments, we have to acknowledge that the only possible quantum numbers are:

s = ½ ; ms = ±½

So there are only two possible spin wavefunctions for the electron, conventionally called spin up (or α) and spin down (or β).

Consequences of the existence of spin

The extra magnetic dipole affects the energy of the atom under magnetic field (that is how it was discovered) but the details are not essential if you are not specifically interested in magnetic properties.

The angular momentum of the electron and the angular momentum of the spin combine together in a non-trivial way (the total energy is not exactly the sum of the energies of the two rotors). This effect is called spin-orbit coupling and it is essential to understand the details of the spectra of multi-electron atoms. The spin-orbit coupling is small and important only for heavy atoms.

The existence of spin changes completely the behavior of multi-electron systems. It gives rise to the periodic table and all the phenomenology of chemical bonding and reactions. This is important for all chemists and will be investigated in the next lesson. Exercise Calculate ħγBz for Bz = 1 Tesla and for Bz equal to the typical magnetic field on the earth surface. What is the electron spin? Describe the Zeeman effect.

Next: Lesson 11 - Many electron systems