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Pauli equation

From Wikipedia, the free encyclopedia
Quantum mechanical equation of motion of charged particles in magnetic field
Part of a series of articles about
Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }

Inquantum mechanics, thePauli equation orSchrödinger–Pauli equation is the formulation of theSchrödinger equation forspin-1/2 particles, which takes into account the interaction of the particle'sspin with an externalelectromagnetic field. It is the non-relativistic limit of theDirac equation and can be used where particles are moving at speeds much less than thespeed of light, so that relativistic effects can be neglected. It was formulated byWolfgang Pauli in 1927.[1] In its linearized form it is known asLévy-Leblond equation.

Equation

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For a particle of massm{\displaystyle m} and electric chargeq{\displaystyle q}, in anelectromagnetic field described by themagnetic vector potentialA{\displaystyle \mathbf {A} } and theelectric scalar potentialϕ{\displaystyle \phi }, the Pauli equation reads:

Pauli equation (general)

[12m(σ⋅(p^−qA))2+qϕ]|ψ⟩=iℏ∂∂t|ψ⟩{\displaystyle \left[{\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot (\mathbf {\hat {p}} -q\mathbf {A} ))^{2}+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }

Hereσ=(σx,σy,σz){\displaystyle {\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})} are thePauli operators collected into a vector for convenience, andp^=−iℏ∇{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla } is themomentum operator in position representation. The state of the system,|ψ⟩{\displaystyle |\psi \rangle } (written inDirac notation), can be considered as a two-componentspinorwavefunction, or acolumn vector (after choice of basis):

|ψ⟩=ψ+|↑⟩+ψ−|↓⟩=⋅[ψ+ψ−]{\displaystyle |\psi \rangle =\psi _{+}|{\mathord {\uparrow }}\rangle +\psi _{-}|{\mathord {\downarrow }}\rangle \,{\stackrel {\cdot }{=}}\,{\begin{bmatrix}\psi _{+}\\\psi _{-}\end{bmatrix}}}.

TheHamiltonian operator is a 2 × 2 matrix because of thePauli operators.

H^=12m[σ⋅(p^−qA)]2+qϕ{\displaystyle {\hat {H}}={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot (\mathbf {\hat {p}} -q\mathbf {A} )\right]^{2}+q\phi }

Substitution into theSchrödinger equation gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. SeeLorentz force for details of this classical case. Thekinetic energy term for a free particle in the absence of an electromagnetic field is justp22m{\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}} wherep{\displaystyle \mathbf {p} } is thekinetic momentum, while in the presence of an electromagnetic field it involves theminimal couplingΠ=p−qA{\displaystyle \mathbf {\Pi } =\mathbf {p} -q\mathbf {A} }, where nowΠ{\displaystyle \mathbf {\Pi } } is thekinetic momentum andp{\displaystyle \mathbf {p} } is thecanonical momentum.

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

(σ⋅a)(σ⋅b)=a⋅b+iσ⋅(a×b){\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 operatorp^−qA=−iℏ∇−qA{\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 functionψ{\displaystyle \psi }:

[(p^−qA)×(p^−qA)]ψ=−q[p^×(Aψ)+A×(p^ψ)]=iqℏ[∇×(Aψ)+A×(∇ψ)]=iqℏ[ψ(∇×A)−A×(∇ψ)+A×(∇ψ)]=iqℏBψ{\displaystyle {\begin{aligned}\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 \end{aligned}}}

whereB=∇×A{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} } is the magnetic field.

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

Pauli equation (standard form)

H^|ψ⟩=[12m[(p^−qA)2−qℏσ⋅B]+qϕ]|ψ⟩=iℏ∂∂t|ψ⟩{\displaystyle {\hat {H}}|\psi \rangle =\left[{\frac {1}{2m}}\left[\left(\mathbf {\hat {p}} -q\mathbf {A} \right)^{2}-q\hbar {\boldsymbol {\sigma }}\cdot \mathbf {B} \right]+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }

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

Weak magnetic fields

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For the case of where the magnetic field is constant and homogenous, one may expand(p^−qA)2{\textstyle (\mathbf {\hat {p}} -q\mathbf {A} )^{2}} using thesymmetric gaugeA^=12B×r^{\textstyle \mathbf {\hat {A}} ={\frac {1}{2}}\mathbf {B} \times \mathbf {\hat {r}} }, wherer{\textstyle \mathbf {r} } is theposition operator and A is now an operator. We obtain

(p^−qA^)2=|p^|2−q(r^×p^)⋅B+14q2(|B|2|r^|2−|B⋅r^|2)≈p^2−qL^⋅B,{\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} \,,}

whereL^{\textstyle \mathbf {\hat {L}} } is the particleangular momentum operator and we neglected terms in the magnetic field squaredB2{\textstyle B^{2}}. Therefore, we obtain

Pauli equation (weak magnetic fields)

[12m[|p^|2−q(L^+2S^)⋅B]+qϕ]|ψ⟩=iℏ∂∂t|ψ⟩{\displaystyle \left[{\frac {1}{2m}}\left[|\mathbf {\hat {p}} |^{2}-q(\mathbf {\hat {L}} +2\mathbf {\hat {S}} )\cdot \mathbf {B} \right]+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }


whereS=ℏσ/2{\textstyle \mathbf {S} =\hbar {\boldsymbol {\sigma }}/2} is thespin of the particle. The factor 2 in front of the spin is known as the Diracg-factor. The term inB{\textstyle \mathbf {B} }, is of the form−μ⋅B{\textstyle -{\boldsymbol {\mu }}\cdot \mathbf {B} } which is the usual interaction between a magnetic momentμ{\textstyle {\boldsymbol {\mu }}} and a magnetic field, like in theZeeman effect.

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

[|p|22m+μBgJmj|B|−eϕ]|ψ⟩=iℏ∂∂t|ψ⟩{\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 }

whereμB=eℏ2m{\textstyle \mu _{\rm {B}}={\frac {e\hbar }{2m}}} is theBohr magneton andmj{\textstyle m_{j}} is themagnetic quantum number related toJ{\textstyle \mathbf {J} }. The termgJ{\textstyle g_{J}} is known as theLandé g-factor, and is given here by

gJ=32+34−ℓ(ℓ+1)2j(j+1),{\displaystyle g_{J}={\frac {3}{2}}+{\frac {{\frac {3}{4}}-\ell (\ell +1)}{2j(j+1)}},}[a]

whereℓ{\displaystyle \ell } is theorbital quantum number related toL2{\displaystyle L^{2}} andj{\displaystyle j} is the total orbital quantum number related toJ2{\displaystyle J^{2}}.

From Dirac equation

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The Pauli equation can be inferred from the non-relativistic limit of theDirac equation, which is the relativistic quantum equation of motion for spin-1/2 particles.[4]

Derivation

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The Dirac equation can be written as:iℏ∂t(ψ1ψ2)=c(σ⋅Πψ2σ⋅Πψ1)+qϕ(ψ1ψ2)+mc2(ψ1−ψ2),{\displaystyle i\hbar \,\partial _{t}{\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}=c\,{\begin{pmatrix}{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi _{2}\\{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi _{1}\end{pmatrix}}+q\,\phi \,{\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}+mc^{2}\,{\begin{pmatrix}\psi _{1}\\-\psi _{2}\end{pmatrix}},}

where∂t=∂∂t{\textstyle \partial _{t}={\frac {\partial }{\partial t}}} andψ1,ψ2{\displaystyle \psi _{1},\psi _{2}} are two-componentspinor, forming abispinor.

Using the following ansatz:(ψ1ψ2)=e−imc2tℏ(ψχ),{\displaystyle {\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}=e^{-i{\tfrac {mc^{2}t}{\hbar }}}{\begin{pmatrix}\psi \\\chi \end{pmatrix}},}with two new spinorsψ,χ{\displaystyle \psi ,\chi }, the equation becomesiℏ∂t(ψχ)=c(σ⋅Πχσ⋅Πψ)+qϕ(ψχ)+(0−2mc2χ).{\displaystyle i\hbar \partial _{t}{\begin{pmatrix}\psi \\\chi \end{pmatrix}}=c\,{\begin{pmatrix}{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\chi \\{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi \end{pmatrix}}+q\,\phi \,{\begin{pmatrix}\psi \\\chi \end{pmatrix}}+{\begin{pmatrix}0\\-2\,mc^{2}\,\chi \end{pmatrix}}.}

In the non-relativistic limit,∂tχ{\displaystyle \partial _{t}\chi } and the kinetic and electrostatic energies are small with respect to the rest energymc2{\displaystyle mc^{2}}, leading to theLévy-Leblond equation.[5] Thusχ≈σ⋅Πψ2mc.{\displaystyle \chi \approx {\frac {{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi }{2\,mc}}\,.}

Inserted in the upper component of Dirac equation, we find Pauli equation (general form):iℏ∂tψ=[(σ⋅Π)22m+qϕ]ψ.{\displaystyle i\hbar \,\partial _{t}\,\psi =\left[{\frac {({\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }})^{2}}{2\,m}}+q\,\phi \right]\psi .}

From a Foldy–Wouthuysen transformation

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

Pauli coupling

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

γμpμ→γμpμ−qγμAμ+aσμνFμν{\displaystyle \gamma ^{\mu }p_{\mu }\to \gamma ^{\mu }p_{\mu }-q\gamma ^{\mu }A_{\mu }+a\sigma _{\mu \nu }F^{\mu \nu }}

wherepμ{\displaystyle p_{\mu }} is thefour-momentum operator,Aμ{\displaystyle A_{\mu }} is theelectromagnetic four-potential,a{\displaystyle a} is proportional to theanomalous magnetic dipole moment,Fμν=∂μAν−∂νAμ{\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }} is theelectromagnetic tensor, andσμν=i2[γμ,γν]{\textstyle \sigma _{\mu \nu }={\frac {i}{2}}[\gamma _{\mu },\gamma _{\nu }]} are the Lorentzian spin matrices and the commutator of thegamma matricesγμ{\displaystyle \gamma ^{\mu }}.[7][8] In the context of non-relativistic quantum mechanics, instead of working with the Schrödinger equation, Pauli coupling is equivalent to using the Pauli equation (or postulatingZeeman energy) for an arbitraryg-factor.

See also

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Footnotes

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  1. ^The formula used here is for a particle with spin-1/2, with ag-factorgS=2{\textstyle g_{S}=2} and orbitalg-factorgL=1{\textstyle g_{L}=1}. More generally it is given by:gJ=32+ms(ms+1)−ℓ(ℓ+1)2j(j+1).{\displaystyle g_{J}={\frac {3}{2}}+{\frac {m_{s}(m_{s}+1)-\ell (\ell +1)}{2j(j+1)}}.} wherems{\displaystyle m_{s}} is thespin quantum number related toS^{\displaystyle {\hat {S}}}.

References

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  1. ^Pauli, Wolfgang (1927)."Zur Quantenmechanik des magnetischen Elektrons".Zeitschrift für Physik (in German).43 (9–10):601–623.Bibcode:1927ZPhy...43..601P.doi:10.1007/BF01397326.ISSN 0044-3328.S2CID 128228729.
  2. ^Bransden, BH; Joachain, CJ (1983).Physics of Atoms and Molecules (1st ed.). Prentice Hall. p. 638.ISBN 0-582-44401-2.
  3. ^Sidler, Dominik; Rokaj, Vasil; Ruggenthaler, Michael; Rubio, Angel (2022-10-26)."Class of distorted Landau levels and Hall phases in a two-dimensional electron gas subject to an inhomogeneous magnetic field".Physical Review Research.4 (4) 043059.Bibcode:2022PhRvR...4d3059S.doi:10.1103/PhysRevResearch.4.043059.hdl:10810/58724.ISSN 2643-1564.S2CID 253175195.
  4. ^abGreiner, Walter (2012-12-06).Relativistic Quantum Mechanics: Wave Equations. Springer.ISBN 978-3-642-88082-7.
  5. ^Greiner, Walter (2000-10-04).Quantum Mechanics: An Introduction. Springer Science & Business Media.ISBN 978-3-540-67458-0.
  6. ^Fröhlich, Jürg; Studer, Urban M. (1993-07-01)."Gauge invariance and current algebra in nonrelativistic many-body theory".Reviews of Modern Physics.65 (3):733–802.Bibcode:1993RvMP...65..733F.doi:10.1103/RevModPhys.65.733.ISSN 0034-6861.
  7. ^Das, Ashok (2008).Lectures on Quantum Field Theory. World Scientific.ISBN 978-981-283-287-0.
  8. ^Barut, A. O.; McEwan, J. (January 1986)."The four states of the Massless neutrino with pauli coupling by Spin-Gauge invariance".Letters in Mathematical Physics.11 (1):67–72.Bibcode:1986LMaPh..11...67B.doi:10.1007/BF00417466.ISSN 0377-9017.S2CID 120901078.

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