Quantum mechanical equation of motion of charged particles in magnetic field
In quantum mechanics , the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger 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 Lévy-Leblond equation .
Equation [ edit ] For a particle of mass m {\displaystyle m} and electric charge q {\displaystyle q} , in an electromagnetic field described by the magnetic vector potential A {\displaystyle \mathbf {A} } and the electric scalar potential ϕ {\displaystyle \phi } , the Pauli equation reads:
Pauli equation (general) [ 1 2 m ( σ ⋅ ( p ^ − q A ) ) 2 + q ϕ ] | ψ ⟩ = i ℏ ∂ ∂ 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 }
Here σ = ( σ x , σ y , σ z ) {\displaystyle {\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})} are the Pauli operators collected into a vector for convenience, and p ^ = − i ℏ ∇ {\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla } is the momentum operator in position representation. The state of the system, | ψ ⟩ {\displaystyle |\psi \rangle } (written in Dirac notation ), can be considered as a two-component spinor wavefunction , or a column 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}}} . The Hamiltonian operator is a 2 × 2 matrix because of the Pauli operators .
H ^ = 1 2 m [ σ ⋅ ( p ^ − q A ) ] 2 + q ϕ {\displaystyle {\hat {H}}={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot (\mathbf {\hat {p}} -q\mathbf {A} )\right]^{2}+q\phi } Substitution into the Schrödinger 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 p 2 2 m {\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}} where p {\displaystyle \mathbf {p} } is the kinetic momentum , while in the presence of an electromagnetic field it involves the minimal coupling Π = p − q A {\displaystyle \mathbf {\Pi } =\mathbf {p} -q\mathbf {A} } , where now Π {\displaystyle \mathbf {\Pi } } is the kinetic momentum and p {\displaystyle \mathbf {p} } is the canonical momentum .
The Pauli operators can be removed from the kinetic energy term using the Pauli vector identity :
( σ ⋅ a ) ( σ ⋅ b ) = a ⋅ b + i σ ⋅ ( 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 operator p ^ − q A = − i ℏ ∇ − q A {\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 ψ {\displaystyle \psi } :
[ ( p ^ − q A ) × ( p ^ − q A ) ] ψ = − q [ p ^ × ( A ψ ) + A × ( p ^ ψ ) ] = i q ℏ [ ∇ × ( A ψ ) + A × ( ∇ ψ ) ] = i q ℏ [ ψ ( ∇ × A ) − A × ( ∇ ψ ) + A × ( ∇ ψ ) ] = i q ℏ B ψ {\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 B = ∇ × 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 ⟩ = [ 1 2 m [ ( p ^ − q A ) 2 − q ℏ σ ⋅ B ] + q ϕ ] | ψ ⟩ = i ℏ ∂ ∂ 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 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 ( p ^ − q A ) 2 {\textstyle (\mathbf {\hat {p}} -q\mathbf {A} )^{2}} using the symmetric gauge A ^ = 1 2 B × r ^ {\textstyle \mathbf {\hat {A}} ={\frac {1}{2}}\mathbf {B} \times \mathbf {\hat {r}} } , where r {\textstyle \mathbf {r} } is the position operator and A is now an operator. We obtain
( p ^ − q A ^ ) 2 = | p ^ | 2 − q ( r ^ × p ^ ) ⋅ B + 1 4 q 2 ( | B | 2 | r ^ | 2 − | B ⋅ r ^ | 2 ) ≈ p ^ 2 − q L ^ ⋅ 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} \,,} where L ^ {\textstyle \mathbf {\hat {L}} } is the particle angular momentum operator and we neglected terms in the magnetic field squared B 2 {\textstyle B^{2}} . Therefore, we obtain
Pauli equation (weak magnetic fields) [ 1 2 m [ | p ^ | 2 − q ( L ^ + 2 S ^ ) ⋅ B ] + q ϕ ] | ψ ⟩ = i ℏ ∂ ∂ 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 }
where S = ℏ σ / 2 {\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 B {\textstyle \mathbf {B} } , is of the form − μ ⋅ B {\textstyle -{\boldsymbol {\mu }}\cdot \mathbf {B} } which is the usual interaction between a magnetic moment μ {\textstyle {\boldsymbol {\mu }}} and a magnetic field, like in the Zeeman 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 momentum J = L + S {\textstyle \mathbf {J} =\mathbf {L} +\mathbf {S} } and Wigner-Eckart theorem . Thus we find
[ | p | 2 2 m + μ B g J m j | B | − e ϕ ] | ψ ⟩ = i ℏ ∂ ∂ 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 } where μ B = e ℏ 2 m {\textstyle \mu _{\rm {B}}={\frac {e\hbar }{2m}}} is the Bohr magneton and m j {\textstyle m_{j}} is the magnetic quantum number related to J {\textstyle \mathbf {J} } . The term g J {\textstyle g_{J}} is known as the Landé g-factor , and is given here by
g J = 3 2 + 3 4 − ℓ ( ℓ + 1 ) 2 j ( j + 1 ) , {\displaystyle g_{J}={\frac {3}{2}}+{\frac {{\frac {3}{4}}-\ell (\ell +1)}{2j(j+1)}},} [a] where ℓ {\displaystyle \ell } is the orbital quantum number related to L 2 {\displaystyle L^{2}} and j {\displaystyle j} is the total orbital quantum number related to J 2 {\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:
i ℏ ∂ t ( ψ 1 ψ 2 ) = c ( σ ⋅ Π ψ 2 σ ⋅ Π ψ 1 ) + q ϕ ( ψ 1 ψ 2 ) + m c 2 ( ψ 1 − ψ 2 ) , {\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}}} and ψ 1 , ψ 2 {\displaystyle \psi _{1},\psi _{2}} are two-component spinor , forming a bispinor .
Using the following ansatz:
( ψ 1 ψ 2 ) = e − i m c 2 t ℏ ( ψ χ ) , {\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 spinors
ψ , χ {\displaystyle \psi ,\chi } , the equation becomes
i ℏ ∂ t ( ψ χ ) = c ( σ ⋅ Π χ σ ⋅ Π ψ ) + q ϕ ( ψ χ ) + ( 0 − 2 m c 2 χ ) . {\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, ∂ t χ {\displaystyle \partial _{t}\chi } and the kinetic and electrostatic energies are small with respect to the rest energy m c 2 {\displaystyle mc^{2}} , leading to the Lévy-Leblond equation .[5] Thus
χ ≈ σ ⋅ Π ψ 2 m c . {\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):
i ℏ ∂ t ψ = [ ( σ ⋅ Π ) 2 2 m + q ϕ ] 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 [ 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 O ( 1 / m c ) {\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 O ( 1 / ( m c ) 2 ) {\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
γ μ p μ → γ μ p μ − q γ μ A μ + a σ μ ν F μ ν {\displaystyle \gamma ^{\mu }p_{\mu }\to \gamma ^{\mu }p_{\mu }-q\gamma ^{\mu }A_{\mu }+a\sigma _{\mu \nu }F^{\mu \nu }} where p μ {\displaystyle p_{\mu }} is the four-momentum operator, A μ {\displaystyle A_{\mu }} is the electromagnetic four-potential , a {\displaystyle a} is proportional to the anomalous magnetic dipole moment , F μ ν = ∂ μ A ν − ∂ ν A μ {\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }} is the electromagnetic tensor , and σ μ ν = i 2 [ γ μ , γ ν ] {\textstyle \sigma _{\mu \nu }={\frac {i}{2}}[\gamma _{\mu },\gamma _{\nu }]} are the Lorentzian spin matrices and the commutator of the gamma matrices γ μ {\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 postulating Zeeman energy ) for an arbitrary g -factor.
See also [ edit ] ^ The formula used here is for a particle with spin ½, with a g -factor g S = 2 {\textstyle g_{S}=2} and orbital g -factor g L = 1 {\textstyle g_{L}=1} . More generally it is given by: g J = 3 2 + m s ( m s + 1 ) − ℓ ( ℓ + 1 ) 2 j ( j + 1 ) . {\displaystyle g_{J}={\frac {3}{2}}+{\frac {m_{s}(m_{s}+1)-\ell (\ell +1)}{2j(j+1)}}.} where m s {\displaystyle m_{s}} is the spin quantum number related to S ^ {\displaystyle {\hat {S}}} . References [ edit ] ^ 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 . ^ Bransden, BH; Joachain, CJ (1983). Physics of Atoms and Molecules (1st ed.). Prentice Hall. pp. 638–638. ISBN 0-582-44401-2 . ^ 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 . ^ a b Greiner, Walter (2012-12-06). Relativistic Quantum Mechanics: Wave Equations . Springer. ISBN 978-3-642-88082-7 . ^ Greiner, Walter (2000-10-04). Quantum Mechanics: An Introduction . Springer Science & Business Media. ISBN 978-3-540-67458-0 . ^ 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 . ^ Das, Ashok (2008). Lectures on Quantum Field Theory . World Scientific. ISBN 978-981-283-287-0 . ^ 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 . Schwabl, Franz (2004). Quantenmechanik I . Springer. ISBN 978-3540431060 . Schwabl, Franz (2005). Quantenmechanik für Fortgeschrittene . Springer. ISBN 978-3540259046 . Claude Cohen-Tannoudji; Bernard Diu; Frank Laloe (2006). Quantum Mechanics 2 . Wiley, J. ISBN 978-0471569527 .