Pauli equation

The Pauli equation goes back to the Austrian physicist Wolfgang Pauli (1900–1958). It describes the development over time of a charged spin 1/2 particle, for example an electron that moves so slowly in the electromagnetic field that the field energy and the kinetic energy are small compared to the rest energy, i.e. no relativistic effects occur. In addition to the terms in the Schrödinger equation for spinless particles, the Pauli equation contains a term that couples the spin with the magnetic field and which has no equivalent in classical physics. With this term one can understand the behavior of the silver atoms in the Stern-Gerlach experiment . If they fly through an inhomogeneous magnetic field, they are split into two partial beams depending on the spin direction.

The Pauli equation is:

${\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\varphi =\underbrace {\left({\frac {({\vec {p}}-q{\vec {A}})^{2}}{2\,m}}+q\,\phi \right)} _{\text{Hamiltonoperator ohne Spin}}\,\varphi -g\,\underbrace {{\frac {q\,\hbar }{2\,m}}\,{\frac {\vec {\sigma }}{2}}\cdot {\vec {B}}} _{\text{Spin-Magnetfeld}}\,\varphi \,}$

Marked here

• ${\displaystyle \varphi ={\begin{pmatrix}\varphi _{\uparrow }(t,{\vec {x}})\\\varphi _{\downarrow }(t,{\vec {x}})\end{pmatrix}}}$ the two-component spatial wave function,
• ${\displaystyle p^{i}=-\mathrm {i} \hbar \partial _{x^{i}}\,,i\in \{1,2,3\}\,,}$the -th component of the momentum, ${\displaystyle i}$
• ${\displaystyle \,q}$the electrical charge and the mass of the particle, ${\displaystyle \,m}$
• ${\displaystyle \,\phi }$the scalar electrical potential and the vector potential , ${\displaystyle {\vec {A}}}$
• ${\displaystyle \,g}$the gyromagnetic factor ,
• ${\displaystyle {\vec {\sigma }}}$the Pauli matrices (with the spin operator ), ${\displaystyle {\vec {S}}=\hbar \,{\frac {\vec {\sigma }}{2}}}$
• ${\displaystyle {\vec {B}}={\text{rot}}\,{\vec {A}}}$the magnetic field .

In a weak, homogeneous magnetic field , according to the Pauli equation, the spin couples more strongly to the magnetic field by the gyromagnetic factor than an orbital angular momentum of the same size ${\displaystyle {\vec {B}}}$ ${\displaystyle g}$ ${\displaystyle {\vec {L}}\,,}$

${\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\varphi ={\frac {{\vec {p}}^{2}}{2\,m}}\varphi -{\frac {q}{2\,m}}\,{\bigl (}{\vec {L}}+g\,{\vec {S}}{\bigr )}\cdot {\vec {B}}\,\varphi \,.}$

The Pauli equation is also obtained as a non-relativistic limiting case from the Dirac equation , which describes the behavior of elementary spin 1/2 particles with or without charge. The Dirac equation predicts the value for the gyromagnetic factor of electrons . This value can also be calculated from the linearization of the Schrödinger equation without including relativistic assumptions . Quantum electrodynamics corrects this value ${\displaystyle g=2}$

${\displaystyle g=2,002\,319\,304\,8(8)\,.}$

The theoretical value for the electron corresponds to the measured value in the first 10 decimals.

Derived from the Dirac equation

Based on the Dirac equation for a particle in the electromagnetic field, split into two two spinors,

${\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\left({\begin{array}{c}\varphi _{1}\\\varphi _{2}\end{array}}\right)=c\,\left({\begin{array}{c}{\vec {\sigma }}\cdot {\vec {\pi }}\,\varphi _{2}\\{\vec {\sigma }}\cdot {\vec {\pi }}\,\varphi _{1}\end{array}}\right)+q\,\phi \,\left({\begin{array}{c}\varphi _{1}\\\varphi _{2}\end{array}}\right)+m\,c^{2}\,\left({\begin{array}{c}\varphi _{1}\\-\varphi _{2}\end{array}}\right)}$   With   ${\displaystyle {\vec {\pi }}={\vec {p}}-q\,{\vec {A}}}$

one assumes that after splitting off the rapid development of time, which originates from the resting energy,

${\displaystyle \left({\begin{array}{c}\varphi _{1}\\\varphi _{2}\end{array}}\right)=\mathrm {e} ^{-\mathrm {i} {\frac {\displaystyle m\,c^{2}\,t}{\displaystyle \hbar }}}\left({\begin{array}{c}\varphi \\\chi \end{array}}\right)}$

the time derivative of the two spinors and is small. ${\displaystyle \varphi }$ ${\displaystyle \chi }$

${\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\left({\begin{array}{c}\varphi \\\chi \end{array}}\right)=c\,\left({\begin{array}{c}{\vec {\sigma }}\cdot {\vec {\pi }}\,\chi \\{\vec {\sigma }}\cdot {\vec {\pi }}\,\varphi \end{array}}\right)+q\,\phi \,\left({\begin{array}{c}\varphi \\\chi \end{array}}\right)+\left({\begin{array}{c}0\\-2\,m\,c^{2}\,\chi \end{array}}\right)}$

According to the assumption, the time derivative in the row is small and the kinetic energies and the electrostatic energy small compared to the rest energy. Therefore small is compared to and approximately equal ${\displaystyle \partial _{t}\chi }$ ${\displaystyle m\,c^{2}\,.}$ ${\displaystyle \chi }$ ${\displaystyle \varphi }$

${\displaystyle \chi \approx {\frac {{\vec {\sigma }}\cdot {\vec {\pi }}\,\varphi }{2\,m\,c}}\,.}$

Inserted in the first line results

${\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\varphi ={\frac {({\vec {\sigma }}\cdot {\vec {\pi }})^{2}}{2\,m}}\,\varphi +q\,\phi \,\varphi \,.}$

For the product of the Pauli matrices one obtains

${\displaystyle ({\vec {\sigma }}\cdot {\vec {\pi }})^{\,2}=\sigma ^{i}\,\sigma ^{j}\pi ^{i}\pi ^{j}=(\delta ^{ij}+\mathrm {i} \varepsilon ^{ijk}\sigma ^{k})\pi ^{i}\pi ^{j}={\vec {\pi }}^{2}-q\,\hbar \,{\vec {\sigma }}\cdot {\vec {B}}\,.}$

The spinor therefore satisfy the Pauli equation , ${\displaystyle \varphi }$ ${\displaystyle g=2}$

${\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\varphi ={\frac {{\vec {\pi }}^{\,2}}{2\,m}}\,\varphi +q\,\phi \,\varphi -{\frac {q\,\hbar }{2\,m}}\,{\vec {\sigma }}\cdot {\vec {B}}\,\varphi \,.}$

In a homogeneous magnetic field, and with the aid of the interchangeability rules of the late product, it follows ${\displaystyle \phi =0\,,\,{\vec {A}}={\frac {1}{2}}\,{\vec {B}}\times {\vec {x}}\,,}$

${\displaystyle ({\vec {p}}-q{\vec {A}})^{2}={\vec {p}}^{\,2}-q\,{\vec {x}}\times {\vec {p}}\cdot {\vec {B}}={\vec {p}}^{\,2}-q\,{\vec {L}}\cdot {\vec {B}}\,,}$

if one neglects terms that are quadratic in . Then the Pauli equation says ${\displaystyle {\vec {B}}}$

${\displaystyle \mathrm {i} \,\hbar \,\partial _{t}\,\varphi ={\frac {{\vec {p}}^{\,2}}{2\,m}}\,\varphi -{\frac {q}{2\,m}}\,({\vec {L}}+g\,{\vec {S}})\cdot {\vec {B}}\,\varphi \,.}$

The magnetic field therefore not only couples to the orbital angular momentum and not only contributes to the energy. The factor is called the magneton of the particle. In the special case of the electron one also speaks of the Bohr magneton . ${\displaystyle {\vec {L}}}$ ${\displaystyle -{\frac {q\,\hbar }{2\,m}}\,{\frac {\vec {L}}{\hbar }}\cdot {\vec {B}}}$ ${\displaystyle {\frac {q\,\hbar }{2\,m}}}$

In angular momentum eigenstates there is an integer multiple of the magnetic field strength. In contrast, a half- integer multiple results , which only becomes an integer after multiplication by g . In the case of isolated atoms or ions , the total orbital angular momentum and the total spin angular momentum of the atom or ion have to be added to a total angular momentum J (= L + S ) and the so-called Landé factor g ( L , S , J ) is obtained . This is 1 for pure total orbital angular momentum and 2 for pure total spindle angular momentum, and otherwise has values ​​different from 1 and 2. Furthermore, if the atoms concerned are built into a solid, additional contributions are obtained which can significantly change g . ${\displaystyle {\frac {\vec {L}}{\hbar }}\cdot {\vec {B}}}$ ${\displaystyle |{\vec {B}}|\,.}$ ${\displaystyle {\frac {\vec {S}}{\hbar }}\cdot {\vec {B}}}$

literature

• Franz Schwabl : Quantum Mechanics (QM I). 5th expanded edition. Springer, Berlin et al. 1998, ISBN 3-540-63779-6 ( Springer textbook ).
• Franz Schwabl: Quantum Mechanics for Advanced Students (QM II). Springer, Berlin et al. 1997, ISBN 3-540-63382-0 ( Springer textbook ).
• Claude Cohen-Tannoudji , Bernard Diu, Franck Laloe: Quantum Mechanics. Volume 2. Wiley et al., New York NY et al. 1977, ISBN 0-471-16435-6 ( A Wiley-Interscience Publication ).

Individual evidence

1. ↑ Wolfgang Pauli: To the quantum mechanics of the magnetic electron . In: Journal of Physics . tape 43 , 1927, pp. 601-623 , doi : 10.1007 / BF01397326 .
2. ^ Walter Greiner : Quantum Mechanics. Introduction. Volume 4, ISBN 3-8171-1765-5 .