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In quantum mechanics , the Pauli equation or Schrodinger?Pauli equation is the formulation of the Schrodinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field . It is the non- relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light , so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927. ^[1] In its linearized form it is known as Levy-Leblond equation .

Equation [ edit ]

For a particle of mass ${\displaystyle m}$ and electric charge ${\displaystyle q}$, in an electromagnetic field described by the magnetic vector potential ${\displaystyle \mathbf {A} }$ and the electric scalar potential ${\displaystyle \phi }$, the Pauli equation reads:

Here ${\displaystyle {\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})}$ are the Pauli operators collected into a vector for convenience, and ${\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }$ is the momentum operator in position representation. The state of the system, ${\displaystyle |\psi \rangle }$ (written in Dirac notation ), can be considered as a two-component spinor wavefunction , or a column vector (after choice of basis):

${\displaystyle |\psi \rangle =\psi _{+}|{\mathord {\uparrow }}\rangle +\psi _{-}|{\mathord {\downarrow }}\rangle \,{\stackrel {\cdot }{=}}\,{\begin{bmatrix}\psi _{+}\\\psi _{-}\end{bmatrix}}}$.

The Hamiltonian operator is a 2 × 2 matrix because of the Pauli operators .

${\displaystyle {\hat {H}}={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot (\mathbf {\hat {p}} -q\mathbf {A} )\right]^{2}+q\phi }$

Substitution into the Schrodinger equation gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. See Lorentz force for details of this classical case. The kinetic energy term for a free particle in the absence of an electromagnetic field is just ${\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}}$ where ${\displaystyle \mathbf {p} }$ is the kinetic momentum , while in the presence of an electromagnetic field it involves the minimal coupling ${\displaystyle \mathbf {\Pi } =\mathbf {p} -q\mathbf {A} }$, where now ${\displaystyle \mathbf {\Pi } }$ is the kinetic momentum and ${\displaystyle \mathbf {p} }$ is the canonical momentum .

The Pauli operators can be removed from the kinetic energy term using the Pauli vector identity :

${\displaystyle ({\boldsymbol {\sigma }}\cdot \mathbf {a} )({\boldsymbol {\sigma }}\cdot \mathbf {b} )=\mathbf {a} \cdot \mathbf {b} +i{\boldsymbol {\sigma }}\cdot \left(\mathbf {a} \times \mathbf {b} \right)}$

Note that unlike a vector, the differential operator ${\displaystyle \mathbf {\hat {p}} -q\mathbf {A} =-i\hbar \nabla -q\mathbf {A} }$ has non-zero cross product with itself. This can be seen by considering the cross product applied to a scalar function ${\displaystyle \psi }$:

${\displaystyle \left[\left(\mathbf {\hat {p}} -q\mathbf {A} \right)\times \left(\mathbf {\hat {p}} -q\mathbf {A} \right)\right]\psi =-q\left[\mathbf {\hat {p}} \times \left(\mathbf {A} \psi \right)+\mathbf {A} \times \left(\mathbf {\hat {p}} \psi \right)\right]=iq\hbar \left[\nabla \times \left(\mathbf {A} \psi \right)+\mathbf {A} \times \left(\nabla \psi \right)\right]=iq\hbar \left[\psi \left(\nabla \times \mathbf {A} \right)-\mathbf {A} \times \left(\nabla \psi \right)+\mathbf {A} \times \left(\nabla \psi \right)\right]=iq\hbar \mathbf {B} \psi }$

where ${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$ is the magnetic field.

For the full Pauli equation, one then obtains ^[2]

for which only a few analytic results are known, e.g., in the context of Landau quantization with homogenous magnetic fields or for an idealized, Coulomb-like, inhomogeneous magnetic field. ^[3]

Weak magnetic fields [ edit ]

For the case of where the magnetic field is constant and homogenous, one may expand ${\textstyle (\mathbf {\hat {p}} -q\mathbf {A} )^{2}}$ using the symmetric gauge ${\textstyle \mathbf {\hat {A}} ={\frac {1}{2}}\mathbf {B} \times \mathbf {\hat {r}} }$, where ${\textstyle \mathbf {r} }$ is the position operator and A is now an operator. We obtain

${\displaystyle (\mathbf {\hat {p}} -q\mathbf {\hat {A}} )^{2}=|\mathbf {\hat {p}} |^{2}-q(\mathbf {\hat {r}} \times \mathbf {\hat {p}} )\cdot \mathbf {B} +{\frac {1}{4}}q^{2}\left(|\mathbf {B} |^{2}|\mathbf {\hat {r}} |^{2}-|\mathbf {B} \cdot \mathbf {\hat {r}} |^{2}\right)\approx \mathbf {\hat {p}} ^{2}-q\mathbf {\hat {L}} \cdot \mathbf {B} \,,}$

where ${\textstyle \mathbf {\hat {L}} }$ is the particle angular momentum operator and we neglected terms in the magnetic field squared ${\textstyle B^{2}}$. Therefore, we obtain

where ${\textstyle \mathbf {S} =\hbar {\boldsymbol {\sigma }}/2}$ is the spin of the particle. The factor 2 in front of the spin is known as the Dirac g -factor . The term in ${\textstyle \mathbf {B} }$, is of the form ${\textstyle -{\boldsymbol {\mu }}\cdot \mathbf {B} }$ which is the usual interaction between a magnetic moment ${\textstyle {\boldsymbol {\mu }}}$ and a magnetic field, like in the Zeeman effect .

For an electron of charge ${\textstyle -e}$ in an isotropic constant magnetic field, one can further reduce the equation using the total angular momentum ${\textstyle \mathbf {J} =\mathbf {L} +\mathbf {S} }$ and Wigner-Eckart theorem . Thus we find

${\displaystyle \left[{\frac {|\mathbf {p} |^{2}}{2m}}+\mu _{\rm {B}}g_{J}m_{j}|\mathbf {B} |-e\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }$

where ${\textstyle \mu _{\rm {B}}={\frac {e\hbar }{2m}}}$ is the Bohr magneton and ${\textstyle m_{j}}$ is the magnetic quantum number related to ${\textstyle \mathbf {J} }$. The term ${\textstyle g_{J}}$ is known as the Lande g-factor , and is given here by

${\displaystyle g_{J}={\frac {3}{2}}+{\frac {{\frac {3}{4}}-\ell (\ell +1)}{2j(j+1)}},}$^[a]

where ${\displaystyle \ell }$ is the orbital quantum number related to ${\displaystyle L^{2}}$ and ${\displaystyle j}$ is the total orbital quantum number related to ${\displaystyle J^{2}}$.

From Dirac equation [ edit ]

The Pauli equation can be inferred from the non-relativistic limit of the Dirac equation , which is the relativistic quantum equation of motion for spin-½ particles. ^[4]

Derivation [ edit ]

Dirac equation can be written as:

where ${\textstyle \partial _{t}={\frac {\partial }{\partial t}}}$ and ${\displaystyle \psi _{1},\psi _{2}}$ are two-component spinor , forming a bispinor .

Using the following ansatz:

with two new spinors ${\displaystyle \psi ,\chi }$, the equation becomes

In the non-relativistic limit, ${\displaystyle \partial _{t}\chi }$ and the kinetic and electrostatic energies are small with respect to the rest energy ${\displaystyle mc^{2}}$, leading to the Levy-Leblond equation . ^[5] Thus

Inserted in the upper component of Dirac equation, we find Pauli equation (general form):

From a Foldy?Wouthuysen transformation [ edit ]

The rigorous derivation of the Pauli equation follows from Dirac equation in an external field and performing a Foldy?Wouthuysen transformation ^[4] considering terms up to order ${\displaystyle {\mathcal {O}}(1/mc)}$. Similarly, higher order corrections to the Pauli equation can be determined giving rise to spin-orbit and Darwin interaction terms, when expanding up to order ${\displaystyle {\mathcal {O}}(1/(mc)^{2})}$ instead. ^[6]

Pauli coupling [ edit ]

Pauli's equation is derived by requiring minimal coupling , which provides a g -factor g =2. Most elementary particles have anomalous g -factors, different from 2. In the domain of relativistic quantum field theory , one defines a non-minimal coupling, sometimes called Pauli coupling, in order to add an anomalous factor

${\displaystyle \gamma ^{\mu }p_{\mu }\to \gamma ^{\mu }p_{\mu }-q\gamma ^{\mu }A_{\mu }+a\sigma _{\mu \nu }F^{\mu \nu }}$

where ${\displaystyle p_{\mu }}$ is the four-momentum operator, ${\displaystyle A_{\mu }}$ is the electromagnetic four-potential , ${\displaystyle a}$ is proportional to the anomalous magnetic dipole moment , ${\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }}$ is the electromagnetic tensor , and ${\textstyle \sigma _{\mu \nu }={\frac {i}{2}}[\gamma _{\mu },\gamma _{\nu }]}$ are the Lorentzian spin matrices and the commutator of the gamma matrices ${\displaystyle \gamma ^{\mu }}$. ^{[7] [8]} In the context of non-relativistic quantum mechanics, instead of working with the Schrodinger equation, Pauli coupling is equivalent to using the Pauli equation (or postulating Zeeman energy ) for an arbitrary g -factor.

See also [ edit ]

• Semiclassical physics
• Atomic, molecular, and optical physics
• Group contraction
• Gordon decomposition

Footnotes [ edit ]

References [ edit ]

Books [ edit ]

• Schwabl, Franz (2004). Quantenmechanik I . Springer. ISBN   978-3540431060 .
• Schwabl, Franz (2005). Quantenmechanik fur Fortgeschrittene . Springer. ISBN   978-3540259046 .
• Claude Cohen-Tannoudji; Bernard Diu; Frank Laloe (2006). Quantum Mechanics 2 . Wiley, J. ISBN   978-0471569527 .