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-1/2 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
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 ϕ -->
{\displaystyle \phi }
, the Pauli equation reads:
Pauli equation (general)
[
1
2
m
(
σ σ -->
⋅ ⋅ -->
(
p
^ ^ -->
− − -->
q
A
)
)
2
+
q
ϕ ϕ -->
]
|
ψ ψ -->
⟩ ⟩ -->
=
i
ℏ ℏ -->
∂ ∂ -->
∂ ∂ -->
t
|
ψ 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 -->
=
(
σ σ -->
x
,
σ σ -->
y
,
σ 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 ℏ -->
∇ ∇ -->
{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }
is the momentum operator in position representation. The state of the system,
|
ψ 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 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 σ -->
⋅ ⋅ -->
(
p
^ ^ -->
− − -->
q
A
)
]
2
+
q
ϕ 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 Π -->
=
p
− − -->
q
A
{\displaystyle \mathbf {\Pi } =\mathbf {p} -q\mathbf {A} }
, where now
Π 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 σ -->
⋅ ⋅ -->
a
)
(
σ σ -->
⋅ ⋅ -->
b
)
=
a
⋅ ⋅ -->
b
+
i
σ 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 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 ψ -->
{\displaystyle \psi }
:
[
(
p
^ ^ -->
− − -->
q
A
)
× × -->
(
p
^ ^ -->
− − -->
q
A
)
]
ψ ψ -->
=
− − -->
q
[
p
^ ^ -->
× × -->
(
A
ψ ψ -->
)
+
A
× × -->
(
p
^ ^ -->
ψ ψ -->
)
]
=
i
q
ℏ ℏ -->
[
∇ ∇ -->
× × -->
(
A
ψ ψ -->
)
+
A
× × -->
(
∇ ∇ -->
ψ ψ -->
)
]
=
i
q
ℏ ℏ -->
[
ψ ψ -->
(
∇ ∇ -->
× × -->
A
)
− − -->
A
× × -->
(
∇ ∇ -->
ψ ψ -->
)
+
A
× × -->
(
∇ ∇ -->
ψ ψ -->
)
]
=
i
q
ℏ ℏ -->
B
ψ 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 ψ -->
⟩ ⟩ -->
=
[
1
2
m
[
(
p
^ ^ -->
− − -->
q
A
)
2
− − -->
q
ℏ ℏ -->
σ σ -->
⋅ ⋅ -->
B
]
+
q
ϕ ϕ -->
]
|
ψ ψ -->
⟩ ⟩ -->
=
i
ℏ ℏ -->
∂ ∂ -->
∂ ∂ -->
t
|
ψ 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
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 ϕ -->
]
|
ψ ψ -->
⟩ ⟩ -->
=
i
ℏ ℏ -->
∂ ∂ -->
∂ ∂ -->
t
|
ψ 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 ℏ -->
σ 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 μ -->
⋅ ⋅ -->
B
{\textstyle -{\boldsymbol {\mu }}\cdot \mathbf {B} }
which is the usual interaction between a magnetic moment
μ 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 μ -->
B
g
J
m
j
|
B
|
− − -->
e
ϕ ϕ -->
]
|
ψ ψ -->
⟩ ⟩ -->
=
i
ℏ ℏ -->
∂ ∂ -->
∂ ∂ -->
t
|
ψ 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 μ -->
B
=
e
ℏ 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 -->
(
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 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
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-1/2 particles.[ 4]
Derivation
Dirac equation can be written as:
i
ℏ ℏ -->
∂ ∂ -->
t
(
ψ ψ -->
1
ψ ψ -->
2
)
=
c
(
σ σ -->
⋅ ⋅ -->
Π Π -->
ψ ψ -->
2
σ σ -->
⋅ ⋅ -->
Π Π -->
ψ ψ -->
1
)
+
q
ϕ ϕ -->
(
ψ ψ -->
1
ψ ψ -->
2
)
+
m
c
2
(
ψ ψ -->
1
− − -->
ψ 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 ψ -->
1
,
ψ ψ -->
2
{\displaystyle \psi _{1},\psi _{2}}
are two-component spinor , forming a bispinor .
Using the following ansatz:
(
ψ ψ -->
1
ψ ψ -->
2
)
=
e
− − -->
i
m
c
2
t
ℏ ℏ -->
(
ψ ψ -->
χ 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 ψ -->
,
χ χ -->
{\displaystyle \psi ,\chi }
, the equation becomes
i
ℏ ℏ -->
∂ ∂ -->
t
(
ψ ψ -->
χ χ -->
)
=
c
(
σ σ -->
⋅ ⋅ -->
Π Π -->
χ χ -->
σ σ -->
⋅ ⋅ -->
Π Π -->
ψ ψ -->
)
+
q
ϕ ϕ -->
(
ψ ψ -->
χ χ -->
)
+
(
0
− − -->
2
m
c
2
χ 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 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 χ -->
≈ ≈ -->
σ σ -->
⋅ ⋅ -->
Π Π -->
ψ 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 ℏ -->
∂ ∂ -->
t
ψ ψ -->
=
[
(
σ σ -->
⋅ ⋅ -->
Π Π -->
)
2
2
m
+
q
ϕ ϕ -->
]
ψ ψ -->
.
{\displaystyle i\hbar \,\partial _{t}\,\psi =\left[{\frac {({\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }})^{2}}{2\,m}}+q\,\phi \right]\psi .}
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
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 γ -->
μ μ -->
p
μ μ -->
→ → -->
γ γ -->
μ μ -->
p
μ μ -->
− − -->
q
γ γ -->
μ μ -->
A
μ μ -->
+
a
σ σ -->
μ μ -->
ν ν -->
F
μ μ -->
ν 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 μ -->
{\displaystyle p_{\mu }}
is the four-momentum operator,
A
μ μ -->
{\displaystyle A_{\mu }}
is the electromagnetic four-potential ,
a
{\displaystyle a}
is proportional to the anomalous magnetic dipole moment ,
F
μ μ -->
ν ν -->
=
∂ ∂ -->
μ μ -->
A
ν ν -->
− − -->
∂ ∂ -->
ν ν -->
A
μ μ -->
{\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }}
is the electromagnetic tensor , and
σ σ -->
μ μ -->
ν ν -->
=
i
2
[
γ γ -->
μ μ -->
,
γ γ -->
ν 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 γ -->
μ 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
^ The formula used here is for a particle with spin-1/2, 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 -->
(
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
^ 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. p. 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 .
Books
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 .