Winter 2013 Chem 356: Introductory Quantum Mechanics Chapter 7b – Electron Spin and Spin-Orbit Coupling..................96 H-atom in a Magnetic Field: Electron Spin..........................96 Total Angular Momentum............................................103 Chapter 7b – Electron Spin and Spin-Orbit Coupling H-atom in a Magnetic Field: Electron Spin If electron in orbital has angular momentum , one has a magnetic moment This magnetic moment can interact with a magnetic field and the interaction energy is given by is measured in Tesla 1T = 1 Newton/(ampere meter) If we take to be in direction then And the total Hamiltonian would be The wavefunctions we obtained are also eigenfunctions of including the magnetic field each level splits in sublevels 96
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We can indicate the -quantum number as s, p, d and get functions
, , We could then write, including spin
, ,
, , In what follows we will focus on one particular quantum number, which we can suppress. Moreover I
can indicate the -spin function by an overbar.
Then we get the 6 p-functions
, , , , , Or the d-functions
, , , , , , , , ,
This indicates the quantum numbers
In this way we can label the exact eigenstates of the (non-relativistic) Hamiltonian.
The splitting of the lines in a magnetic field would then be determined by the Hamiltonian
where
This factor determining the ratio between spin and orbital interactions with the magnetic field, can be calculated using relativistic quantum field theory. (Schwinger, Tomanaga, Feynman). Far beyond our aim.
It agrees to about 10 digits with the experimental value! (that is like measuring the distance from here to New York up to a millimeter!)
The , functions are eigenfunctions of this magnetic Hamiltonian, and we can easily calculate the energy splitting.
Unfortunately, this does not give correct results!!
The splitting due to the magnetic field is very small. There are other corrections to the energy levels in Hydrogen atom due to relativity. They are of at least comparable importance, and cannot be neglected when discussing magnetic effects.
Due to all -levels go up by all -levels go down by . Since transitions cannot change spin,
, I get same transitions as without spin!!
Our conclusion thus far.
If one does not consider spin levels split in a magnetic field using
3 equal spaced levels
5 equal spaced levels
If we include spin, for single electron states then all -states shift up by 1 unit of , all -levels shift
down by 1 unit of , and the transition energies are not affected by spin. Moreover all
multiples are split by the same amount . We would not see the effects of spin.
This is what was originally observed. It is called the ‘normal Zeeman effect’ and it was explained by Lorentz (two Dutch physicists).
However we do observe the effects of spin in emission spectra! The story is more complicated.
The complications occur already for the H-atom without a magnetic field. There is a substantial correction due to what is called spin-orbit interaction.
A good way to think about this is as follows:We usually think of the electron as wizzing around the nucleus. From the point of view of the
electron we can just as easily think that the nucleus is wizzing about the electron. This moving nucleus, with its angular momentum generates a magnetic field. This magnetic field interacts with the spin of the electron.
Compare the electron with your position on the spinning earth. The sun rises and sets from our standing still point of view, and moves with incredible velocities, in this frame. For a charged particle the magnetic force would be large.
Spin-orbit coupling is usually said to be a relativistic effect. This is because it arises in a natural way from the fully relativistic Dirac equation. So does spin; it arises naturally. And so do particles and antiparticles, which also arise from the Dirac equation.
Let me say something more about spin. The spin operators are best represented by matrices.
, ,
These matrices satisfy the commutation relations of angular momentum
Moreover
Hence
matrices have only 2 eigenvectors. This is why we have only ,
The Dirac equation is a matrix equation and we get (2 spin 2 mass) solutions.
The splitting due to the magnetic field is very small. There are other corrections to the energy levels in Hydrogen atom due to relativity. They are of at least comparable importance, and cannot be neglected when discussing magnetic effects.
The relativistic Hydrogen atom is described by the Dirac equation. This is far more complicated than we wish to discuss.
One can approximate the effects by including spin-orbit interaction in the Hamiltonian.
Hence we can derive without too much trouble that the operators and all commute.
Moreover these operators commute with , and also with .
It then follows that the angular momentum operators , commute with the relativistic
Hamiltonian . And we can classify the states with quantum numbers
As for the non-relativistic case, the angular momentum problem, defining is independent from the radial equation, and can be solved once and for all.
These equations hold for the Hydrogen atom, but later on we will see that they are very similar for many-electron atoms, which also have spherical symmetry.
Can we deduce what the eigenstates of and might be?