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 equations:pauli_equation [2018/03/13 11:12]jakobadmin equations:pauli_equation [2018/04/16 09:09] (current)jakobadmin [Intuitive] Both sides previous revision Previous revision 2018/04/16 09:09 jakobadmin [Intuitive] 2018/03/28 10:22 jakobadmin 2018/03/28 10:22 jakobadmin 2018/03/26 16:40 jakobadmin 2018/03/26 16:38 jakobadmin 2018/03/13 11:14 jakobadmin [Why is it interesting?] 2018/03/13 11:14 jakobadmin 2018/03/13 11:13 jakobadmin 2018/03/13 11:12 jakobadmin 2018/03/13 11:12 jakobadmin 2017/12/04 08:01 external edit2017/10/21 15:29 jakobadmin created Next revision Previous revision 2018/04/16 09:09 jakobadmin [Intuitive] 2018/03/28 10:22 jakobadmin 2018/03/28 10:22 jakobadmin 2018/03/26 16:40 jakobadmin 2018/03/26 16:38 jakobadmin 2018/03/13 11:14 jakobadmin [Why is it interesting?] 2018/03/13 11:14 jakobadmin 2018/03/13 11:13 jakobadmin 2018/03/13 11:12 jakobadmin 2018/03/13 11:12 jakobadmin 2017/12/04 08:01 external edit2017/10/21 15:29 jakobadmin created Line 1: Line 1: - ====== Pauli Equation ====== + ​$\left( \frac{1}{2m}(\vec \sigma ( \vec p - q\vec A))^2 + q\phi \right) \Psi = i \hbar \partial_t \Psi$​ - $$\left( \frac{1}{2m}(\vec \sigma ( \vec p - q\vec A))^2 + q\phi \right) \Psi = i \hbar \partial_t \Psi$$ + ====== Pauli Equation ​ ====== - -->​Details # + - $\Psi$ is the wave function, $m$ the mass of the particle, $q$ the charge of the particle, $\vec{\sigma}$ the Pauli matrices, $\vec{p}$ the momentum, $\vec A$ the vector potential and $\phi$ the electric scalar potential. ​ + - Take note that $\vec \sigma$ is only a convenient short-hand notation for the sums that appear in the equation. For example, $\vec \sigma \vec p = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3.$ - <-- - ​ -  ​ - The Pauli equation is the correct non-relativistic equation to describe spin $1/2$ particles. ​ +  ​ -  ​ + The Pauli equation describes how the state of a quantum system with [[basic_notions:​spin|half-integer spin]] changes in time. - + In contrast, the [[equations:​schroedinger_equation|Schrödinger equation]] describes the time evolution of systems without spin. - Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. + <​tabbox ​Concrete> - ​ + - ​ + - <​tabbox ​Student> + The Pauli equation is the non-relativistic limit of the [[equations:​dirac_equation|Dirac equation]]. ​ The Pauli equation is the non-relativistic limit of the [[equations:​dirac_equation|Dirac equation]]. ​ - <​tabbox ​Researcher> + <​tabbox ​Abstract> * Nonrelativistic particles and wave equations by Jean-Marc Lévy-Leblond * Nonrelativistic particles and wave equations by Jean-Marc Lévy-Leblond - --> Common Question 1# - +  ​ - <-- + - --> Common Question ​2# + The Pauli equation is the correct __non-relativistic__ equation to describe spin $1/2$ particles. ​ - + <​tabbox ​Definitions> - <-- + - + - <​tabbox ​Examples> + - --> Example1# + * $\Psi$ is the wave function, - + * $m$ the mass of the particle, - + * $q$ the charge of the particle, - <-- + * $\vec{\sigma}$ the Pauli matrices, - + * $\vec{p}$ the momentum, - --> Example2:#​ + * $\vec A$ the vector potential, - + ​* $\phi$ the electric scalar potential and - + * $\hbar$ the reduced Planck constant. - <-- + - + -  ​ + + Take note that $\vec \sigma$, a "​vector of matrices"​ is only used as a convenient short-hand notation for the sums that appear in the equation. For example, $\vec \sigma \vec p = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3.$