Physics 361Principles of Modern Physics

Lecture 22

The quantum mechanical model of the hydrogen atom

This lecture• Spin-Orbit Coupling – Fine Structure

We can perform a semi-classical approximation to estimate spin-orbit coupling in a hydrogen atom. Let’s start with electron going around the nucleus.

We could take the frame ofreference of the electron,in which case is sees a positive nucleus circulatingaround it.

This will lead to a magnetic field at the location of the electron, regardless of whether an external field is applied.

Without going through the E&M derivation, the field at the center where the electron is located is .

Spin-Orbit Coupling (Fine Structure). The Electron Frame.

This field at the electron will be quantized in Bohr orbits.We must fit an integer number of waves in the circumference,

so

This can be inserted in the B-field relation .

, which is a quantization condition for a magnetic field at the location of the electron.

Fine Structure Semi-classical Field

From energy for a magnetic moment and moment due to spin,

we have,

Thus,

We can also solve for the radius in terms of the principal quantum number,

, where we inserted the semi-classical Bohr atom result.

Fine structure energy differences due to spin-orbit coupling

Inserting the quantized orbital radius gives

which is typically rewritten,

with the dimensionless “fine structure constant”

.

This fine structure constant pops up frequently. For example, the electron g-factor, .Precise measurement and quantum electrodynamics show that it is in fact slightly greater than 2, with the first order correction given by the fine structure constant, .

Fine structure constant