Pauli equation

The Pauli equation goes back to the Austrian physicist Wolfgang Pauli. It describes the time evolution of a charged Spin-1/2-Teilchens, such as an electron, the so- slowly moving in the electromagnetic field that the field energy and the kinetic energy is small compared to the rest energy, so no relativistic effects. In addition to the terms in the Schrödinger equation for spinless particles, the Pauli equation contains a term which couples the spin with the magnetic field and in classical physics has no equivalent. With this term, one can understand the behavior of the silver atoms in the Stern-Gerlach experiment. If you are flying through an inhomogeneous magnetic field, so they are split depending on the spin direction into two partial beams.

The Pauli equation is:

Referred to here

• The two-component spatial wave function,
• The th component of the pulse,
• The electrical charge and the mass of the particle,
• The scalar electric potential and the vector potential,
• The gyromagnetic factor,
• The Pauli matrices ( with the spin operator),
• The magnetic field.

In a weak, uniform magnetic field coupled by the Pauli equation of the spin to the gyromagnetic factor more strongly to the magnetic field than an equally large orbital angular momentum

The Pauli equation is obtained as the nonrelativistic limit of the Dirac equation, which describes the behavior of elementary Spin-1/2-Teilchen with or without charge. The Dirac equation predicts the value of the gyromagnetic factor of electron. This value can also be calculated without the inclusion of relativistic assumptions from the linearization of the Schrödinger equation. Quantum electrodynamics corrects this value to

The theoretical value is consistent with the electron with the measured value in the first 10 decimal places.

Derivation of the Dirac equation

Starting from the Dirac equation for a particle in the electromagnetic field is split into two Zweierspinoren,

Subordinated to that after removal of the fast development time, resulting from the rest energy

The time derivative of Zweierspinoren and small.

In the line, the time derivative is small by assumption and the kinetic energies and the electrostatic energy small compared to the rest energy therefore is against small and approximately equal

Inserted in the first line is the

For the product of the Pauli matrices obtained

Therefore, the spinor satisfies the Pauli equation,

Applies in the homogeneous magnetic field and follows with the aid of the commutation relations of the Spatproduktes

If one neglects terms which are quadratic in. Then says the Pauli equation

Consequently, the magnetic field coupled not only to the angular momentum and not only contributes to energy at. The factor is referred to as the particle magneton. In the special case of the electron is commonly known as the Bohr magneton.

In angular momentum eigenstates an integer multiple of the magnetic field strength is the other hand, results in a half-integral multiple, which is an integer after multiplying by g. In isolated atoms or ions have to add the total orbital angular momentum and the total spin angular momentum of the atom or ion to a total angular momentum J ( = L S) and is given the so-called Landé factor g ( L, S, J). It is 1 for pure total orbital angular momentum and 2 with pure total spin angular momentum, and is otherwise 1 and 2 different values. Further, if the affected atoms are incorporated into a solid, one obtains additional contributions that can g change significantly.